==== TABLE OF CONTENTS ====
A-pointwise-2-converse-for-elliptic-curves-with-rational-two-torsion//coefficients.tex
A-pointwise-2-converse-for-elliptic-curves-with-rational-two-torsion//cyclotomic.tex
A-pointwise-2-converse-for-elliptic-curves-with-rational-two-torsion//graphs.tex
A-pointwise-2-converse-for-elliptic-curves-with-rational-two-torsion//introduction.tex
A-pointwise-2-converse-for-elliptic-curves-with-rational-two-torsion//nonvanishing.tex
A-pointwise-2-converse-for-elliptic-curves-with-rational-two-torsion//paper.tex
A-pointwise-2-converse-for-elliptic-curves-with-rational-two-torsion//pointwise.tex
A-pointwise-2-converse-for-elliptic-curves-with-rational-two-torsion//preliminaries.tex
A-pointwise-2-converse-for-elliptic-curves-with-rational-two-torsion//proof-roadmap-figure.tex
A-pointwise-2-converse-for-elliptic-curves-with-rational-two-torsion//ring-determinants.tex
A-pointwise-2-converse-for-elliptic-curves-with-rational-two-torsion//ring-limits.tex
A-pointwise-2-converse-for-elliptic-curves-with-rational-two-torsion//references.bib
===========================
==== BEGIN A-pointwise-2-converse-for-elliptic-curves-with-rational-two-torsion//coefficients.tex ====
\section{Coefficient systems and their contraction identities}
\label{co:section}
We construct Fourier coefficients detecting central values and Heegner
points, then use arithmetic lower bounds to normalize their least nonzero
valuation. The resulting binary tests satisfy the contraction and isolation
identities used in Section~\ref{gr:section}. The even lower bound is already
available from Section~\ref{cy:section}; the odd detector is constructed
without one, and its normalization remains conditional until
Proposition~\ref{pw:odd-lower}.
Throughout, $S$ contains $2$ and the primes dividing the
conductor $N$. A filter prescribes local unit squareclasses at $S$, together
with the infinite sign, and excludes finitely many further primes. Its
indices are $h=\sigma D$, where $D$ is positive and squarefree. All primes
adjoined subsequently are odd, good, and outside the current support.
Put $F=\mathbb Q(E[2])$. In the present residual range, $F$ is trivial or
quadratic. A prime split in $F$ has weight $1$, and a prime inert in $F$,
called simple, has weight $1/2$. We write $w(D)$ for the sum of these weights.
All valuations satisfy $v(2)=1$.
We use the theta multiplier cubed for weight $3/2$. Local unit characters
and their associated Dirichlet characters are related by the usual inverse
idele convention. Thus, if a finite Schwartz function satisfies
$f(ux)=\alpha(u)f(x)$, its scalar classical character on the lower-right
entry is the Dirichlet character associated to $\alpha^{-1}$.
This convention removes a possible inverse ambiguity in the formulas below.
\subsection{Even coefficients and exact comparison of ratios}
\begin{proposition}[Even coefficient system]\label{co:even}
Fix an even-sign filter and a nonvanishing index $D_0$ in it.
There is a holomorphic cuspidal form $g=\sum_{n\geq1}a(n)q^n$ of weight
$3/2$, orthogonal to unary theta series, of level supported on a fixed
enlargement of $S$, and characters $\eta,\psi$ of $2$-power order such that
$\psi/\eta=\chi_{-\sigma}$ and
\begin{equation}\label{co:recurrence}
a(p^2n)=\bigl(\eta(p)a_p(E)-\psi(p)(-n/p)\bigr)a(n)
-p\eta(p)^2a(n/p^2)\qquad(p\notin S).
\end{equation}
Its coefficients are zero off the unit-squareclass filter. After
normalization at $D_0$, every squarefree coefficient is rational up to a
$2$-power root of unity. For two nonzero squarefree coefficients,
\begin{equation}\label{co:exact-ratio}
2v\!\left(\frac{a(D)}{a(D_0)}\right)
=v\!\left(\frac{L_*(\sigma D)}{L_*(\sigma D_0)}\right).
\end{equation}
Moreover $a(D)\neq0$ exactly when $L(E^{(\sigma D)},1)\neq0$.
Consequently any previously established bound
$v(L_*(\sigma D))\geq2w(D)+s(\sigma D)-C$ gives
\begin{equation}\label{co:even-bound}
v(a(D))\geq w(D)+\tfrac12s(\sigma D)-C_g
\end{equation}
with $C_g$ fixed independently of $D$.
\end{proposition}
\begin{proof}
Use the Shimura packet of $E^{(-\sigma)}$. Its positive index $D$ tests
the twist by $\chi_{-D}$, which is $E^{(\sigma D)}$.
The nonvanishing at $D_0$ selects a globally occurring member admitting
that Fourier functional. Take its spherical vector outside $S$ and its
holomorphic vector of weight $3/2$ at infinity.
Here is the support-local vector selection. Start with a vector on which
the $D_0$ Fourier functional is nonzero. Projection under a sufficiently
large compact group of upper unipotents to additive frequency $D_0$
restricts the potentially nonzero unit translates to arbitrarily small
neighborhoods of the two solutions of $u^2=1$. Choose a unit character
agreeing with the forced action at $-1$. It can have $2$-power order:
the image of $-1$ is detected in the $2$-primary quotient of the unit
group. The neighborhoods of $1$ and $-1$ then contribute with the same
sign, so this projection preserves the nonzero functional. Upper
integral translations act trivially, and smoothness gives sufficiently
deep lower-unipotent invariance. The vector therefore defines a
classical form at a level supported on $S$, allowing arbitrary depth at
$2$ and at the bad primes. The unit-squareclass and coprimality sieves
commute with good $T(p^2)$, since their conditions are unchanged by
multiplication of the index by $p^2$. The classical Hecke formula
\cite{Shimura1973HalfIntegral} gives
\eqref{co:recurrence}, with $\eta=\psi\chi_{-\sigma}$.
At the level just chosen, let $V$ be the full space of weight-$3/2$
cusp forms orthogonal to unary theta series, with scalar character
$\psi$ and good $T(p^2)$ eigenvalues $\eta(p)a_p(E)$. It contains
the constructed form, whose coefficient at $D_0$ is nonzero. For
$D/D_0$ a unit square at every support prime, Waldspurger's comparison
\cite[Corollary~2, pp.~483--484]{Waldspurger1981} gives, for every
$f\in V$,
\[
a_f(D)^2=
\psi(D/D_0)\frac{L_*(\sigma D)}{L_*(\sigma D_0)}a_f(D_0)^2.
\]
Its hypothesis (H1) follows from the nonzero eigenspace
\cite[\S I.2, Proposition~2]{Waldspurger1981}; the final paragraph of
the comparison proof removes (H2) for these local squareclasses.
Factoring the displayed quadratic identity in the two coefficient
functionals shows that they are proportional on all of $V$, with one
scalar independent of $f$. This includes oldvectors and their linear
combinations; if the central value at $D$ vanishes, the first functional
is simply zero.
Choose an algebraic form $g$ in the filtered subspace of $V$ with
$a(D_0)\ne0$. To justify this choice, multiplication by ordinary theta
embeds the space into an integral-weight space, and divisibility by
theta is specified by algebraic cusp conditions
\cite[\S\S3--4]{Ramsey2006HalfIntegral}. The character, good-Hecke
and filter conditions are algebraic linear conditions as well.
Since evaluation at $D_0$ is nonzero after extension to $\mathbb C$,
it is nonzero on some algebraic form.
Let $u$ be an automorphism of $\overline{\mathbb Q}$ acting on the
character values by an odd power $e$, and put $\nu=\eta^{(e-1)/2}$.
The forms $g^u$ and $g\otimes\nu$ have the same scalar character
$\psi^e=\psi\nu^2$ and the same good eigenvalues
$\eta(p)^e a_p(E)=\nu(p)^2\eta(p)a_p(E)$; here $\psi/\eta$ is
quadratic. They belong to one good-Hecke eigenspace at a common level
still supported on $S$. Applying the preceding proportionality in
that nonzero space and normalizing at $D_0$ gives
\[
u(\xi)=\nu(D/D_0)\xi,\qquad \xi=a(D)/a(D_0).
\]
Local square roots of $D/D_0$ at the conductor of $\eta$ give a value
$z$ of $\eta$ with $\eta(D/D_0)=z^2$. Then $u(z)=z^e$ and
$\nu(D/D_0)=z^{e-1}$, so $\xi/z$ is fixed by every algebraic
automorphism and is rational. Moreover the same local-square condition
gives $\psi(D/D_0)=\eta(D/D_0)=z^2$. The comparison therefore reads
\[
(\xi/z)^2=\frac{L_*(\sigma D)}{L_*(\sigma D_0)},
\]
which proves \eqref{co:exact-ratio} since $v(z)=0$.
There is no product over the new prime factors: the unramified
squarefree formula includes both local valuations zero and one.
No exceptional local representation at $2$ is excluded by the
same-local-class comparison. The vanishing assertion and
\eqref{co:even-bound} follow.
\end{proof}
\subsection{A detecting Kummer functional}
Fix now the positive odd-sign filter and its original support, including
the auxiliary class-group tests needed for arithmetic realization.
Choose a fixed negative odd
fundamental discriminant $h_*$ with $L(E^{(h_*)},1)\neq0$, whose local
squareclasses agree with those of the filter at $2N$. Adjoin its prime
factors to $S$, exclude them from the variable indices, and refine the
fixed unit tests there as required, all before varying $D$.
Choose it so that $|h_*|>4$ and one of its ramified primes
excludes the CM field of $E$, if there is one. Thus
\[
K_D=\mathbb Q(\sqrt{Dh_*})
\]
is imaginary, split at $2N$, and has only the units $\{1,-1\}$.
Fix a modular parametrization $\varphi$
with $\varphi(\infty)=0$ throughout. With this normalization the
cuspidal constants in its Hecke and Atkin--Lehner relations are torsion;
see \cite{Manin1972,Drinfeld1973}. An arbitrary translation by a
rational point is not permitted.
Let $H_D$ be the Hilbert class field of $K_D$. Choose an
$\mathcal O_{K_D}$-ideal $\mathfrak n_D$ with
$\mathcal O_{K_D}/\mathfrak n_D\simeq\mathbb Z/N\mathbb Z$,
fixing one orientation at each level prime, and let
$x_D\in X_0(N)(H_D)$ be its conductor-one CM point. The unramified
genus character $\chi_D$ attached to
$K_D(\sqrt{h_*})/K_D$ satisfies
$\chi_D([\mathfrak a])=\chi_{h_*}(\operatorname{Norm}\mathfrak a)$
on ideals prime to $Dh_*$. Define the unaveraged character sum
\[
P_D=\sum_{[\mathfrak a]\in\operatorname{Pic}(\mathcal O_{K_D})}
\chi_D([\mathfrak a])\varphi(x_D^{[\mathfrak a]}).
\]
These conductor-one genus sums over the varying fields $K_D$ are
distinct from the conductor-$|h|$ ring-class sums over a fixed field
$K$ defined in Section~\ref{rl:section}.
The point $P_D$ lies in $E(\mathbb Q(\sqrt D,\sqrt{h_*}))$, with the quadratic
interpretation when $D=1$. Its genus-character component over $K_D$
is the sum of the $D$- and $h_*$-twist components over $\mathbb Q$.
The latter has rank zero because $L(E^{(h_*)},1)\ne0$.
Thus the free part of $P_D$ lies in the $D$-twist component, a line
when $E^{(D)}$ has a simple zero. We retain the unaveraged sum in the
theta comparison; a fixed integer $L$ kills bounded-degree torsion
and makes its comparison with a rational twist point integral.
At a simple zero write $j(P_D)$ for its index valuation in the free
rational twist line, allowing the fixed restriction and projection
indices. Put $j(P_D)=+\infty$ if the product derivative vanishes.
\begin{lemma}[Kummer detection, including CM]\label{co:kummer}
Let $C$ act on $T=T_2E$ as complex conjugation, and choose a nonzero
integral row $b:T\longrightarrow\mathbb Z_2$ with $bC=b$.
For a fixed positive simple-zero witness $D_w$, there are arbitrarily
large precisions $M$ and good primes $\ell$ such that
\begin{equation}\label{co:ell-conditions}
\rho_T(\operatorname{Fr}_\ell)\equiv C\pmod {2^M},\qquad
(h_*/\ell)=-1,\qquad(D_w/\ell)=1.
\end{equation}
Evaluation at Frobenius squared defines an integral homomorphism
\[
\lambda_M:E(\mathbb F_{\ell^2})\longrightarrow\mathbb Z/2^M\mathbb Z,
\qquad \lambda_M=b\kappa_M,
\]
of Frobenius sign $+1$. The valuation of
$\lambda_M(\operatorname{red}_\ell(LP_{D_w}))$ is bounded independently
of $M$, for a fixed nonzero integer $L$ making the twist comparison
integral. Extra fixed finite-field restrictions can be retained when they agree
with the chosen complex-conjugation coset, or when a compatible lift
has been specified.
\end{lemma}
\begin{proof}
For every elliptic curve $A/\mathbb Q$, the rational Tate module $V_2A$
is irreducible. Indeed Faltings' semisimplicity and endomorphism theorem
\cite[\S1, assertions (a),(b)]{Faltings1983} would turn a proper
summand into a nontrivial idempotent in
$\operatorname{End}_{\mathbb Q}(A)\otimes\mathbb Q_2=\mathbb Q_2$.
The equality of this endomorphism ring with $\mathbb Z$ includes CM
curves: a nonreal CM endomorphism is not defined over $\mathbb Q$.
The Tate image contains a nonidentity central scalar, by the homothety
theorem \cite[\S2, Theorem~3 and paragraph~7]{Bogomolov1981Points}.
Let $Q\in A(\mathbb Q)$ be nontorsion and let $c_Q$ be its division
cocycle. On the kernel of the Tate action its image is a closed
$\mathbb Z_2$-submodule of $T_2A$, stable under the full Galois image.
If its rational span were zero, the rational Kummer class would factor
through the Tate image. A central scalar $z\neq1$ annihilates its
first cohomology: for a cocycle $c$ the equality of $c(zg)$ and $c(gz)$
gives $(z-1)c(g)=(g-1)c(z)$, so $c$ is a coboundary over $\mathbb Q_2$.
This contradicts Kummer injectivity and the finite generation of
$A(\mathbb Q)$. Irreducibility consequently makes the translation
image contain $2^aT_2A$ for some $a$. Intersecting with the kernel of
additional finite data retains a full lattice.
Apply this to the rational twist point supplied by $LP_{D_w}$ on
$A=E^{(D_w)}$. Since $D_w>0$, conjugation acts on its Tate module as
$C$. Modify a conjugation lift by an element of the simultaneous
kernel whose translation is $v$. The square of the modified element
has translation $(1+C)v$, and $b(1+C)v=2b(v)$. The full-lattice
property allows a choice with a fixed finite nonzero valuation.
Conjugation acts trivially on $\mathbb Q(\sqrt{D_w})$ and nontrivially
on $\mathbb Q(\sqrt{h_*})$. Chebotarev in each finite torsion,
division, and character extension gives \eqref{co:ell-conditions}
and the chosen finite nonzero detection.
For good $\ell$, Lang's theorem and finite-field Kummer theory give
\[
E(\mathbb F_{\ell^2})/2^M E(\mathbb F_{\ell^2})
\simeq T/\bigl(2^MT+(\operatorname{Fr}_\ell^2-1)T\bigr)=T/2^MT.
\]
This is evaluation of a cocycle at arithmetic Frobenius squared.
Its equivariance and $bC=b$ prove the sign assertion. Reduction of
the division cocycle gives precisely the detection already arranged.
Finally $\det C=-1$ implies $\ell\equiv-1\pmod {2^M}$.
\end{proof}
\subsection{The finite Schwartz function and its transformation}
\begin{lemma}[Explicit weighted theta construction]\label{co:theta-bridge}
For $\ell$ as in Lemma~\ref{co:kummer}, the reduced modular
parametrization followed by $\lambda_M$ admits an integral weighted
ternary theta lift. After extraction of the coefficients at multiples
of $|h_*|$ it gives a classical holomorphic weight-$3/2$ form
$g_M=\sum a_M(n)q^n$, of level supported on $S\cup\{\ell\}$, with
\eqref{co:recurrence} modulo $2^M$. The associated characters have
$2$-power order and quadratic ratio. For squarefree admissible $D$,
\begin{equation}\label{co:weighted-comparison}
a_M(D)=c_0\,2^{\omega(N)}\zeta_M(D)
\lambda_M\bigl(\operatorname{red}_\ell(LP_D)\bigr)
\pmod {2^M},
\end{equation}
where $\zeta_M(D)$ is a $2$-power root of unity and $c_0,L$ are fixed
integers independent of $D,M,\ell$. The extra index restriction at
$\ell$ is $(D/\ell)=1$.
\end{lemma}
\begin{proof}
Let $B$ be ramified at $\ell$ and infinity, $R$ an Eichler order of
level $N$, and $V=B^0$ with $Q(x)=-x^2=\operatorname{Nrd}(x)$.
The root characters at the fixed level primes below are chosen once.
Let $\mathcal O$ be the valuation ring at a place above $2$ of the
field generated by their values. The other local weights are
quadratic, so this field is independent of $M$ and $\ell$.
In this construction integrality means membership in $\mathcal O$,
and congruences mean congruences modulo $2^M\mathcal O$.
\par\smallskip\noindent\emph{The supersingular coefficient function.}
The supersingular pairs on $X_0(N)$ are the finite set
\[
X=B^\times\backslash\widehat B^\times/\widehat R^\times.
\]
This description permits every $N$ prime to $\ell$, including arbitrary
depth at $2$; see \cite[\S\S1.1--1.2,2.4--2.6]{JetchevKane2010}.
For $z\in X$ put $R_z=B\cap z\widehat Rz^{-1}$.
Choose $L$ killing the finite subgroup generated by the images of the
cuspidal constants in the Atkin--Lehner relations, and also all the
bounded-degree torsion used below. Then
\[
P_M(z)=\lambda_M(L\varphi_\ell(z)),
\]
where $\varphi_\ell$ is the specialization of $\varphi$ to the
good-reduction special fiber. This function has Brandt eigenvalues
$a_p(E)$ away from $N\ell$, Atkin--Lehner
eigenvalue $w_{p^e}$ at $p^e\Vert N$, and eigenvalue $+1$ under the
right $\ell$-uniformizer. These identities are modulo $2^M$.
Choose any integer lifts of this finite list of values.
\par\smallskip\noindent\emph{The local Schwartz weights.}
We now define the local functions, rather than appealing to an
unspecified weighted theta theorem. Off $2Nh_*\ell$ take the
characteristic function of $R_p^0$. At $p^e\Vert N$, use a model
\[
R_p=\left\{\begin{pmatrix}a&b\\c&d\end{pmatrix}
:a,b,d\in\mathbb Z_p, c\in p^e\mathbb Z_p\right\}.
\]
Require the split fundamental unit squareclass and an integral
embedding. Its action on the cyclic level quotient selects one root
$r\in\mathbb Z_p^\times$ of $x^2$. Give $x$ weight $\alpha_p(r)$,
where $\alpha_p$ is a finite unit character of $2$-power order with
\begin{equation}\label{co:root-parity}
\alpha_p(-1)=w_{p^e}\chi_{h_*}(p)^e.
\end{equation}
Such a character exists for either prescribed sign. The root is a
locally constant function to any fixed character precision: the two
idempotents of the split integral quadratic algebra distinguish the
two summands of the cyclic quotient. Unit scaling $x\mapsto ux$
multiplies this weight by $\alpha_p(u)$, and order-unit conjugation
preserves it.
At $2$ require $(1+x)/2\in R_2$ and $x^2\in\mathbb Z_2^{\times2}$.
If $2\mid N$, the preceding selected-root rule is imposed using
$e_x=(1+x)/2$. Its two eigenvalues $(1+r)/2,(1-r)/2$ differ by the
odd unit $r$; hence the two idempotents remain distinct modulo $2$.
This verifies the construction at every exponent of $2$ in $N$.
Scaling by an odd $u$ preserves integrality, since
$(1+ux)/2=u(1+x)/2+(1-u)/2$. This is an invertible affine change of
the operator $e_x$, so it preserves its two summands individually.
The selected summand is selected by the cyclic level quotient, not
by declaring its eigenvalue to be $0$ or $1$ modulo $2$. On that same
summand $x$ acts as $r$ and $ux$ acts as $ur$. Even when
$u\equiv-1\pmod4$ interchanges the two eigenvalue parities, the root
weight therefore changes by exactly $\alpha_2(u)$.
If $2\nmid N$ there is no root weight at $2$.
For $p\mid h_*$ require $x\in M_2(\mathbb Z_p)^0$ and
$v_p(Q(x))=1$. Its reduction is a nonzero nilpotent. Fix
$X_0=\left(\begin{smallmatrix}0&1\\0&0\end{smallmatrix}\right)$ and
define
\[
\epsilon_p(gX_0g^{-1})=\left(\frac{\det g}{p}\right).
\]
The centralizer consists of
$\left(\begin{smallmatrix}a&b\\0&a\end{smallmatrix}\right)$, with
determinant $a^2$, so this is well-defined on the entire nonzero
nilpotent orbit. It obeys
\[
\epsilon_p(uyu^{-1})=(\det u/p)\epsilon_p(y),\qquad
\epsilon_p(ay)=(a/p)\epsilon_p(y).
\]
The second equality follows by conjugating $X_0$ by
$\operatorname{diag}(a,1)$. Set $f_p(x)=\epsilon_p(\bar x)$ on the
specified support. Thus the right conjugation character is precisely
the local component of $\chi_{h_*}\circ\operatorname{Nrd}$.
At $\ell$, let $\mathfrak P$ be the maximal ideal of the maximal
order. Its residue field is $\mathbb F_{\ell^2}$, and the reduction of
an integral trace-zero element lies on its one-dimensional trace-zero
line. Fix a nonzero basis vector $\beta$ on that line and put
\[
f_\ell(x)=\left(\frac{t}{\ell}\right)
\quad\text{if }\bar x=t\beta\neq0,
\qquad f_\ell(x)=0\quad\text{otherwise}.
\]
Order units act trivially by conjugation on the residue field.
A uniformizer acts by its nontrivial automorphism and sends
$\beta$ to $-\beta$. Therefore its multiplier is $-1$, since
$\ell\equiv-1\pmod4$; scaling has character $(u/\ell)$.
Moreover $\beta^2$ is a nonsquare and $-1$ is a nonsquare modulo
$\ell$, so $Q(x)=-t^2\beta^2$ is a nonzero square. Because
$(|h_*|/\ell)=1$, the extracted indices automatically satisfy
$(D/\ell)=1$. No average over $\mathbb F_\ell^\times$ is used.
All these functions have compact support and are locally constant.
Let $f=\prod f_p$, incorporating any additional fixed unit-squareclass
conditions by their characteristic functions. Those conditions are
stable under unit squares and change none of the scaling characters.
For $u\in\widehat R^\times$,
\begin{equation}\label{co:order-equivariance}
f(u^{-1}xu)=\chi_{h_*}(\operatorname{Nrd}u)^{-1}f(x).
\end{equation}
The product scaling character is even. Indeed its value at $-1$ is
\[
\left(\prod_{p^e\Vert N}w_{p^e}\chi_{h_*}(p)^e\right)
\chi_{h_*}(-1)(-1)=1;
\]
the parenthesized factor is the even functional sign of $E^{(h_*)}$.
Denote its associated classical character by $\eta_B$. In the
inverse-idele convention fixed above, if $\alpha$ is this finite
scaling character, then $\eta_B(d)=\alpha(d)$ for positive integers
$d$ prime to the support: the inverse idele character becomes the
product of the finite unit values at the lower-right entry.
\par\smallskip\noindent\emph{The scalar transformation law.}
Here are all the Weil-representation checks needed for scalar
modularity. For an additive character of conductor $\mathbb Z_p$,
write $n(b)=\left(\begin{smallmatrix}1&b\\0&1\end{smallmatrix}\right)$,
$m(u)=\operatorname{diag}(u,u^{-1})$, and
$n^-(c)=\left(\begin{smallmatrix}1&0\\c&1\end{smallmatrix}\right)$.
Use a Weyl lift $w$, its compatible self-dual Fourier transform, and
the Weil factors $\gamma_p(u),\gamma_p$ for this additive character.
The defining formulas are
\[
\omega(n(b))f_p(x)=\mathbf e_p(bQ(x))f_p(x),\quad
\omega(m(u))f_p(x)=\gamma_p(u)|u|_p^{3/2}f_p(ux),\quad
\omega(w)f_p=\gamma_p\widehat f_p.
\]
They follow from the Weil construction and Poisson summation
\cite[I, no.~13]{Weil1964}. Integral $b$ fixes $f_p$, since $Q$ is integral on
its support. If $f_p$ is invariant under translations by a lattice
$L_p'$, its Fourier transform is supported on the dual lattice
$(L_p')^\vee$. Choose $A_p$ so that
$p^{A_p}Q((L_p')^\vee)\subset\mathbb Z_p$. Then $n(p^{A_p}b)$ fixes
$\widehat f_p$, and conjugating by $w$ proves that every
$n^-(p^{A_p}b)$ fixes $f_p$. This argument works without change at
$2$ and for all finite character depths.
Relative to the ordinary theta multiplier cubed, the diagonal Weil
factor is $(\det Q,u)_p$. Indeed diagonalizing $Q$ and multiplying
the one-dimensional Weil-index identities
\cite[II, nos.~25, 28]{Weil1964} gives the Hilbert symbol
of the product of its three coefficients. In a presentation
$B=(-a,-b)$ that product is the determinant of
$aX^2+bY^2+abZ^2$, namely $(ab)^2$. Thus this additional factor is
$1$. The remaining diagonal action is exactly the scaling character
already computed. Increasing $A_2$ to the depth needed for the
standard metaplectic splitting, the factorization
\[
\begin{pmatrix}a&b\\c&d\end{pmatrix}
=n^-(c/a)m(a)n(b/a),\qquad c\in p^{A_p}\mathbb Z_p, a\in\mathbb Z_p^\times,
\]
proves scalar transformation on the required local $\Gamma_0$.
Conjugation by $z$ commutes with these Weil actions and preserves $Q$.
Consequently the same level and scalar character work for every
summand. The Gaussian at infinity gives holomorphic weight $3/2$;
positive definiteness gives nonnegative exponents at every cusp.
Choose an integer $c_0$ divisible by every possible $|R_z^\times|$.
Such a uniform choice exists because these unit groups embed in
$\operatorname{GL}_4(\mathbb Z)$ and their orders are bounded.
The theta series is
\begin{equation}\label{co:theta-sum}
H_M(n)=c_0\sum_{z\in X}
\frac{P_M(z)\chi_{h_*}(\operatorname{Nrd}z)}{|R_z^\times|}
\sum_{\substack{x\in B^0\\Q(x)=n}} f(z^{-1}xz).
\end{equation}
Equation~\eqref{co:order-equivariance} proves independence of right
representatives; rational reduced norms are positive, so the norm
character is trivial on the left rational action. The preceding Weil
calculation proves modularity \cite[III, Theorem~6]{Weil1964}, and
the choice of $c_0$ proves integral Fourier coefficients. More
precisely, each $c_0/|R_z^\times|$ is an integer, so changing a lift
of $P_M(z)$ by $2^M$ changes every coefficient by an element of
$2^M\mathcal O$. No precision is lost to a stabilizer denominator.
The chosen integer lifts need satisfy no characteristic-zero
eigenrelation.
\par\smallskip\noindent\emph{The good-prime Hecke relation.}
For completeness, the good-prime relation can be checked in a matrix
order. Neighbors correspond to the lines in $\mathbb F_p^2$.
A nonzero primitive nilpotent reduction has exactly one invariant
line. For an integral element of nonzero reduction and norm $n$,
the number of invariant lines is $1+(-n/p)$; zero reduction contributes
the additional $p$ lines. If a primitive $y$ has $p^2\mid Q(y)$,
choose a basis in which its reduction is $X_0$. Its lower-left entry
is divisible by $p^2$, and exactly the neighbor for that invariant
line makes $y/p$ integral. These counts partition the three terms
of the theta Hecke operator. Scaling by $p$ at the support contributes
$\eta_B(p)$, and the reduced norm of a Brandt neighbor contributes
$\chi_{h_*}(p)$. Thus, with $A(n)=H_M(n)$,
\[
A(p^2n)+\eta_B(p)(-n/p)A(n)+p\eta_B(p)^2A(n/p^2)
=\eta_B(p)\chi_{h_*}(p)a_p(E)A(n)\pmod {2^M}.
\]
The orbit-stabilizer weights in \eqref{co:theta-sum} are exactly the
weights in this neighbor count. Extracting
$a_M(n)=H_M(|h_*|n)$ gives
$\eta=\eta_B\chi_{h_*}$ and
$\psi=\eta_B(|h_*|/\cdot)$, proving \eqref{co:recurrence}.
Fixed-support coefficient sieves are exact zero-or-retain operations;
they preserve coefficient integrality and commute with this relation.
\par\smallskip\noindent\emph{The weighted coefficient comparison.}
It remains to prove the weighted comparison; modularity alone does
not supply it. At $n=|h_*|D$ the counted elements are precisely the
optimal embeddings $\mathcal O_{K_D}\hookrightarrow R_z$, sending
$\sqrt{Dh_*}$ to $x$. The condition at $2$ is exactly the condition
on the integral generator $(1+\sqrt{Dh_*})/2$.
At a level prime there are two orientations. Indeed a stable lattice
splits into its two integral idempotent summands, and a cyclic quotient
of order $p^e$ lies wholly in one summand. At $\ell$ there are two
orientations, the two residue-field embeddings. At all remaining
primes there is a single local optimal-embedding orbit.
Fix a rational embedding $x_0$ and an initial orientation tuple $z_0$.
The orbits with these orientations are indexed by
$\operatorname{Pic}(\mathcal O_{K_D})$, with representatives $az_0$
for $a\in\widehat K_D^\times$. The equality
\[
(az_0)^{-1}x_0(az_0)=z_0^{-1}x_0z_0
\]
shows that the Schwartz weight is constant along that orbit. Its
remaining relative weight is
$\chi_{h_*}(\operatorname{Norm}_{K_D/\mathbb Q}a)$, exactly the
unramified genus character. The reduction correspondence commutes
with this ideal-class action and remembers the cyclic level subgroup
\cite[\S2]{JetchevKane2010}; a Galois-equivariant description of the
lift, class action and reduction is given in
\cite[\S3.2, \S\S4.6, 4.9--4.10 and Corollary~4.1]{CornutJetchev2013}.
We use these correspondences only for their geometry; the weight
identity has just been computed, and the multiplicity is computed next.
An embedding stabilizer is $\mathcal O_{K_D}^\times=\{\pm1\}$.
An orbit in $R_z$ therefore contains $|R_z^\times|/2$ elements, giving
the factor $1/2$ in \eqref{co:theta-sum}. At $p^e\Vert N$, reversing
orientation multiplies its Schwartz factor by
$\alpha_p(-1)=w_{p^e}\chi_{h_*}(p)^e$, and its norm-weighted
parametrization value by the same sign. At $\ell$ both multipliers
are $-1$: one is the residue-line calculation and the other is
$\chi_{h_*}(\ell)$ times the Frobenius eigenvalue $+1$.
Every one of the $2^{\omega(N)+1}$ orientation tuples consequently
has the same contribution. The initial tuple has a $2$-power unit
phase. Combining this count with its factor $1/2$ gives exactly
\eqref{co:weighted-comparison}. There is no division by a class
number, and none of its rational multipliers grows with $\ell$ or
the number of factors of $D$.
\end{proof}
\begin{proposition}[Odd coefficient detection]\label{co:odd}
The forms in Lemma~\ref{co:theta-bridge}, after a fixed adjustment of
precision, satisfy
\[
v(a_M(D))\geq\min\{M,j(P_D)-C\},\qquad
v(a_M(D))<M\ \Longrightarrow\ j(P_D)\leq v(a_M(D))+C.
\]
A visible coefficient detects $a(E^{(D)})=1$, and a fixed positive
witness has bounded coefficient valuation as $M\to\infty$.
All constants are independent of $M,\ell,D$.
\end{proposition}
\begin{proof}
Gross--Zagier identifies the height of the genus sum with the product
derivative \cite{GrossZagier1986}. Since $L(E^{(h_*)},1)\neq0$,
vanishing of the derivative makes $P_D$ torsion, and a nontorsion
$P_D$ is equivalent to a simple zero of $E^{(D)}$.
The genus sum lies in a biquadratic field, so a fixed integer kills
all possible torsion: reduce prime-to-residue-characteristic torsion
at two fixed good primes of different residue characteristic; the
residue degrees are at most four. Include that integer in $L$.
On the free twist line, divisibility by $2^{j(P_D)}$ remains divisibility
after the integral maps of reduction and $\lambda_M$.
Restriction and the two quadratic projections have composite
multiplication by $2$, so their integral indices have a fixed bound.
Equation~\eqref{co:weighted-comparison} therefore proves both
valuation assertions, with fixed losses only. Lemma~\ref{co:kummer}
gives the witness.
\end{proof}
\subsection{Minimum depth and integral coefficient operators}
We now pass from detection to normalization. For the even source,
the cyclotomic lower bound gives \eqref{co:even-bound} at every
nonzero squarefree coefficient. For the odd source, assume for now
that every simple-zero parameter satisfies
\[
j(P_D)\geq w(D)+s(D)/2-C_*.
\]
Proposition~\ref{pw:odd-lower} will prove this using only the even
construction. Proposition~\ref{co:odd} then gives
\[
v(a_M(D))\geq\min\{M,w(D)+s(D)/2-C'\}
\]
at simple zeros, where $s(D)$ is defined. At every other odd-sign
parameter the product derivative vanishes and $a_M(D)=0$ modulo
$2^M$. In particular the coarser bound
$v(a_M(D))\geq\min\{M,w(D)-C'\}$ holds at every admissible $D$.
The remaining coefficient calculus is conditional on these lower
bounds; it does not establish them.
For the operator and graph arguments, at each actual odd stage we
write $S$ for the original fixed support together with the primes of
$h_*$ and the current reduction prime $\ell$. The latter is stationary
during that stage and is never an element of a newly adjoined set
$U$. Thus every statement below about ramification outside $S\cup U$
includes $\ell$. This notation does not change the original fixed
data controlling the constants, or choose a new minimum when a vertex
is omitted.
\begin{lemma}[Stabilized minimum depth]\label{co:minimum}
Assume the uniform arithmetic lower bounds just stated. Each coefficient source
defines a binary symbol
\begin{equation}\label{co:symbol}
A(D)=\begin{cases}
1,&v(a(D))=\delta+w(D),\\
0,&v(a(D))>\delta+w(D)\text{ or }D\text{ is off the filter},
\end{cases}
\end{equation}
with a fixed minimum normalized depth $\delta$ and at least one
bounded-weight witness. For the odd source $a=a_M$ at an actual stage:
the numerical minimum and its divisibility statements stabilize at
every fixed finite weight bound. The finite symbol tables on arithmetic
labels will be stabilized separately in Section~\ref{pw:odd-stages}.
At a unit symbol,
\[
v(L_*(\sigma D))\leq2w(D)+O(1)
\quad\text{or}\quad j(P_D)\leq w(D)+O(1),
\]
respectively, and the analytic order is respectively zero or one.
\end{lemma}
\begin{proof}
For the even source normalize $a(D_0)=1$; the phase-removed squarefree
coefficients are rational by Proposition~\ref{co:even}. For the odd
source, \eqref{co:weighted-comparison} makes every visible squarefree
coefficient an integer times a root of unity modulo $2^M$.
Their finite valuations are therefore integers in both cases, not
arbitrary elements of a ramified coefficient field's value group.
The fractional part of $w(D)$ is fixed on the filter,
because the parity of the number of simple primes is its fixed
quadratic discriminant character. The lower bounds and a nonzero
witness therefore give a minimum for the even source.
For the odd source choose a nonprincipal ultrafilter on the increasing
precisions. Minimize $v(a_M(D_M))-w(D_M)$ over sequences with bounded
weight and visible coefficient on an ultrafilter-large set. The
uniform lower bound, discrete values, and fixed witness give a finite
attained minimum. At each fixed weight cutoff $W$, failure of its
divisibility on an ultrafilter-large set would allow a choice of a
violating $D_M$ with $w(D_M)\leq W$, contrary to the minimum.
Hence every prescribed finite list of these divisibilities holds at
a sufficiently advanced actual stage. If larger weights arise in a
coefficient-integrality check, the coarse bound
$\min\{M,w(D)-C\}$ already pays any fixed normalization once both
$M$ and $w(D)$ exceed its fixed threshold. This also explains why
no infinite-precision modular lift is asserted or needed.
The last conclusions follow from the two coefficient comparisons.
\end{proof}
Write $U_p\sum c(n)q^n=\sum c(pn)q^n$ and
$V_p\sum c(n)q^n=\sum c(n)q^{pn}$. For a squarefree set $U$ put
\begin{equation}\label{co:BU}
B_U=2^{-\lceil\delta+w(U)\rceil}
\prod_{p\in U}(U_p-\eta(p)V_p)g,\qquad
e_U=\lceil\delta+w(U)\rceil-\delta-w(U).
\end{equation}
We call $U$ matching if $e_U=0$ and opposite if $e_U=1/2$.
These operators introduce each new prime to level exponent one:
$V_p$ changes the character by $(p/\cdot)$ and raises the level by
$p$, while $U_p=U_{p^2}V_p$ on that raised level.
Thus the new character is $\psi_U=\psi\prod_{p\in U}(p/\cdot)$.
\begin{lemma}[Coefficient bounds and unary removal]\label{co:operators}
For the even source let $U$ be arbitrary. For the odd source, fix
an integer $R\ge1$ and a weight bound $W$; all assertions below are
made at a sufficiently advanced actual stage, simultaneously for every
$U$ with $w(U)\le W$ and every Fourier index.
In these respective settings the coefficients of $B_U$ are integral.
Write uniquely
$n=Jrm^2$, with $J\subseteq U$, $r$ squarefree, and $(r,U)=1$.
The coefficient vanishes unless $D=(U/J)r$ belongs to the filter,
and for the even source otherwise
\begin{equation}\label{co:BU-bound}
v(B_U[n])\geq w(r)+\tfrac12\#\{p\in J:p\text{ simple}\}-e_U.
\end{equation}
At that stage the odd source satisfies this inequality for every
$U$ with $w(U)\le W$ and every $n$, with its right side replaced
by its minimum with $R$.
In particular the untruncated bound holds on any prescribed bounded
range of squarefree weights, after choosing $R$ and the stage.
For a split $p\in J$ the difference factor, after its unit phase and
one factor of $2$ are removed, has residue
\begin{equation}\label{co:cp}
c_p(D)=\frac{a_p(E)-(\psi/\eta)(p)(-D/p)-1}{2}\pmod2
\end{equation}
at every square shell. For a simple $p$ this formula holds on the
first shell $p\nmid m$.
Choose $i\in\{0,1\}$ such that $\eta\chi_{-4}^i$ is odd, and set
\[
\Theta_J=\sum_{\substack{m>0\\(m,S)=1}}
\eta(m)\chi_{-4}(m)^i m q^{Jm^2}.
\]
For matching $U$ the modular form
\begin{equation}\label{co:HU}
H_U=\frac12\left(B_U-
\sum_{\substack{J\subseteq U\\J\text{ split only}}}
B_U[J]\Theta_J\right)
\end{equation}
has integral coefficients, with the same stage and weight restriction
in the odd case. For opposite $U$ put $H_U=B_U$.
The new primes in the level of $\Theta_J$ have exponent one, and its
character is $\eta\chi_{-4}^i(-J/\cdot)$.
\end{lemma}
\begin{proof}
After removing the powers of $\eta(p)$, the square-index multipliers
in \eqref{co:recurrence} are
\[
b_0=1,\quad b_1=a_p-\kappa,\quad
b_{s+1}=a_pb_s-pb_{s-1},\qquad
\kappa=(\psi/\eta)(p)(-D/p).
\]
If $p\mid D$, then $\kappa=0$ and $b_s$ alternates between $1$ and
$0$ modulo $2$, because $a_p$ is even. Thus the difference from
$p\in U/J$ is odd. If $p\nmid D$, all $b_s$ are odd and the
difference from $p\in J$ is even. Subtracting the normalization in
\eqref{co:BU} gives exactly
\[
\delta+w((U/J)r)+\#J-\lceil\delta+w(U)\rceil
=w(r)+\tfrac12\#\{p\in J:p\text{ simple}\}-e_U.
\]
This is an integer on the allowed filter. Its only potential
negative value before imposing parity is $-1/2$, so it is nonnegative.
For the odd source this calculation first applies on a fixed bounded
range of squarefree weights, with enough precision to pay the fixed
normalizing power of $2$. To obtain the global truncated assertion,
fix $R$ and a bound $W$ for $w(U)$. The normalizing exponents
$\lceil\delta+w(U)\rceil$ are uniformly bounded. Above a sufficiently large weight
threshold, the coarse bound $\min\{M,w(D)-C\}$ from
Lemma~\ref{co:minimum} pays that normalizing power and at least $R$
further factors of $2$, once $M$ is large enough. Below the threshold
use the stabilized minimum and increase the precision again if needed.
The recurrence has integral multipliers, so errors modulo $2^M$
remain divisible by $2^M$ at every square shell before normalization.
Thus at the chosen stage the truncated bound holds at every
coefficient for every $U$ with $w(U)\le W$, giving integrality
without asserting an unbounded valuation estimate for a fixed
finite-precision lift.
For split $p$, rationality of $E[2]$ over $\mathbb F_p$ gives
$a_p\equiv p+1\pmod4$. With $\Delta_s=b_{s+1}-b_s$, every
$\Delta_s$ is even and
$\Delta_{s+1}=a_p\Delta_s-p\Delta_{s-1}$, so
$\Delta_{s+1}\equiv\Delta_{s-1}\pmod4$.
Direct calculation gives
\[
\Delta_1-\Delta_0
=a_p^2-a_p\kappa-2a_p+2\kappa+1-p\equiv0\pmod4.
\]
Consequently every $\Delta_s/2$ has the residue in \eqref{co:cp}.
At a simple prime only $\Delta_0$ is used.
In matching parity the right side of \eqref{co:BU-bound} can vanish
only when $r=1$ and $J$ is split only. The same recurrence, retaining
the character phases, gives
$B_U[Jm^2]\equiv B_U[J]\eta(m)\pmod {2}$.
The corresponding coefficient of $\Theta_J$ has exactly that
residue, and different squarefree $J$ have disjoint supports.
The numerator in \eqref{co:HU} is therefore divisible by the integer
$2$, not just by the maximal ideal of a ramified character field.
The ordinary odd-character unary theta transformation and dilation
by $J$ give the asserted character and level. For the odd source
these calculations use the same sufficiently advanced stage,
uniformly for $w(U)\le W$, with precision paying the fixed
divisions as in Lemma~\ref{co:minimum}.
\end{proof}
\subsection{Trace operators and omission of a whole rational prime}
The next step turns changing a Frobenius label into a difference of
coefficient tests. Fix a coefficient valuation ring $\mathcal O$,
enlarging its field if needed for the finite eigensystem span below,
and write $\mathfrak m$ for its maximal ideal. Divisibility by $2^a$
means divisibility in $2^a\mathcal O$; reduced traces and expansions
mean coefficientwise reduction of Fourier expansions modulo $\mathfrak m$.
These are different operations; no saturation of the integral module
below is assumed.
The normalized squarefree tests are binary because their phase-removed
values are rational $2$-adic integers (integer congruences at an odd
stage), and every $2$-power root of unity reduces to $1$.
Multiply the form under consideration by
$\theta=1+2\sum_{m>0}q^{m^2}$. This gives a weight-two form with the
same reduced expansion. Work in the finite-dimensional sum of its
characteristic-zero good Hecke eigensystems, retaining old
multiplicity spaces and its diamond transforms. The associated
two-dimensional representations are the usual modular Galois
representations for cusp forms \cite{Deligne1969}, and the sums of
the two defining characters for Eisenstein series. Their commuting
trace and determinant operators are denoted by $t(u),d(u)$.
Arithmetic Frobenius at a good prime $q$ has trace $T_q$ and
determinant $q\langle q\rangle$.
Let $\mathcal M$ be the $\mathcal O$-module generated by this
weight-two vector and the undivided terms $\theta B_U[J]\Theta_J$,
together with all their good-Hecke and diamond translates.
\begin{lemma}[Integral traces]\label{co:traces}
The module $\mathcal M$ is an integral Fourier lattice preserved by
the trace and determinant operators. On the coefficientwise reduced
trace orbit of the indicated vector,
\begin{equation}\label{co:trace-identities}
d(u)t(v)=t(v),\qquad
t(u)t(v)=t(uv)+t(uv^{-1}),\qquad t(1)=0.
\end{equation}
Every coefficient test of a finite product of traces can be performed
using distinct fresh primes avoiding the tested index. In particular
testing $t(u_1)\cdots t(u_a)$ at $n$ becomes testing the original
reduced coefficient at $nq_1\cdots q_a$.
\end{lemma}
\begin{proof}
We first check integrality of the generators. Diamonds act by scalars
on $B_U$ and on each unary term. For \eqref{co:HU}, the only apparent
denominator is removed by
\[
\langle u\rangle H_U=\psi_U(u)H_U+
\frac12\sum_J B_U[J]
\bigl(\psi_U(u)-\eta(u)\chi_{-4}(u)^i(-J/u)\bigr)\Theta_J;
\]
the difference of the two characters is zero or twice a unit.
The same formula applies after multiplication by theta.
Good Hecke operators commute with diamonds, and their coefficient
formula preserves integrality. Thus all generators of $\mathcal M$
are integral. Finitely many Fourier coefficients inject the ambient
finite-dimensional space into a finite-dimensional coefficient space,
so $\mathcal M$ embeds in $\mathcal O^r$ for some finite $r$.
It is consequently a finite, closed $\mathcal O$-module.
Continuity and simultaneous Chebotarev approximation on the finitely
many eigensystems approximate $t(u)$ and $d(u)$ by $T_q$ and
$q\langle q\rangle$ at every prescribed precision. These operators
preserve $\mathcal M$, so their limits do as well. They act by scalars
on each good eigensystem and by the identity on its old multiplicity
space, which retains the asserted commutation with level operators.
The formula
$(T_qf)[n]=f[qn]+q(\langle q\rangle f)[n/q]$ has no second term
when $q\nmid n$. Successive distinct fresh primes therefore give
the coefficient assertion. On a single character system,
determinants reduce to $1$ because all characters have $2$-power
order and the cyclotomic character is odd. On a subtracted vector
the displayed diamond formula leaves unary corrections. A fresh
trace kills each correction modulo the maximal ideal: the tested
index $nq$ cannot have its prescribed squarefree part $J$ when
$q\nmid nJ$. This continues to hold after further fresh tests.
Thus $d(u)t(v)=t(v)$ on the reduced trace orbit. Apply the matrix
identity
\[
\operatorname{tr}(A)\operatorname{tr}(B)
=\operatorname{tr}(AB)+\det(B)\operatorname{tr}(AB^{-1})
\]
to each characteristic-zero system and use that determinant
identity. Finally $t(1)=2$ before reduction.
\end{proof}
\begin{lemma}[The level-prime inertia identity]\label{co:inertia}
For $p\in U$, choose inertia $\tau_p$ on which the new ramified
quadratic character is $-1$, and an arithmetic Frobenius lift $\phi_p$
in the same decomposition group. In characteristic zero,
\begin{equation}\label{co:inertia-identity}
U_p^{(2)}\bigl(t(g\tau_p\phi_p)-t(g\phi_p)\bigr)
=d(\phi_p)\bigl(t(g\tau_p)-t(g)\bigr).
\end{equation}
Here $U_p^{(2)}$ extracts $p$-multiples in weight two. The identity
commutes with further good traces.
\end{lemma}
\begin{proof}
A constituent old from a level omitting $p$ has trivial inertia, so
both differences are zero, whatever $U_p^{(2)}$ does on its old
multiplicity space. Every other constituent has conductor exponent
one and primitive ramified quadratic determinant. The conductor
formula excludes monodromy and identifies it as principal series
with one unramified and one ramified character. Local compatibility
\cite[Theorem~(A)]{Carayol1986}, after converting its geometric
Frobenius convention to the arithmetic convention fixed above,
identifies its $U_p^{(2)}$ eigenvalue
with the Frobenius eigenvalue $\lambda$ on the unramified line.
Write
\[
\rho(\tau_p)=\begin{pmatrix}1&0\\0&-1\end{pmatrix},\qquad
\rho(\phi_p)=\begin{pmatrix}\lambda&0\\0&\mu\end{pmatrix}.
\]
If $z$ is the lower-right entry of $\rho(g)$, the two trace
differences are $-2z\mu$ and $-2z$. Both sides of
\eqref{co:inertia-identity} equal $-2z\lambda\mu$.
The Eisenstein character pair has the same calculation.
A unary term is old if $p\notin J$, and has precisely this ramified
quadratic character if $p\in J$. This exhausts the constituents.
\end{proof}
For the interface below a \emph{label} is the restriction of a
Frobenius element to an elementary $2$-extension unramified outside
the indicated support. Its coordinates are quadratic characters.
When $F$ is quadratic, choosing one of the two places at a split prime specifies an
orientation; changing its ramification bit means multiplying by the
image of inertia at that place. Product labels are the restrictions
of products of Frobenius elements, so their quadratic coordinates add.
Simple primes use rational quadratic labels. Section~\ref{gr:section}
will realize these labels simultaneously by primes and interpret the
independent inertia changes as graph edges.
\begin{proposition}[Symbol interface]\label{co:symbol-interface}
Under the arithmetic lower bounds in Lemma~\ref{co:minimum}, the
symbols of either coefficient source have the following properties.
\begin{enumerate}
\item For matching $U$, the trace on split elements factors through
the maximal elementary $2$-extension of $F$ unramified outside
$S\cup U$. For opposite $U$, traces factor through the corresponding
rational elementary extension. Adding a split vertex $q$ is read
by coefficient $1$ of $t(\operatorname{Fr}_q)H_U$; adding a simple
vertex is read on the opposite $B_U$.
\item Let $p\in U$ and $g=\operatorname{Fr}_q$. Changing one
oriented ramification bit at $p$ changes the symbol by
\begin{equation}\label{co:contraction}
A\bigl((U\setminus\{p\})\cup\{q'\}\bigr),\qquad
\operatorname{Fr}_{q'}\sim g\phi_p,
\end{equation}
unless $p,q$ are both simple, in which case the change is zero.
The replacement has the product labels at all retained places.
Its coefficient system has support $S\cup(U\setminus\{p\})$:
the entire rational prime $p$, including both places above it when
it splits in $F$, is omitted.
\item A split vertex with trivial label in the elementary extension
for the other vertices has zero symbol.
\item In matching parity, insertion of two simple primes with equal
rational labels on the fixed support and at the existing split
vertices preserves the symbol if
$\#E(\mathbb F_q)/2$ is odd for that label. Their mutual simple
bits and their incidences with other simple vertices do not affect
this statement.
\end{enumerate}
These are coefficient identities. Simultaneous realization of
their labels by primes is a separate arithmetic construction.
\end{proposition}
\begin{proof}
For an odd source choose the stage in Lemma~\ref{co:operators} with
cutoff $R\ge2$; only these fixed divisibilities and finitely many
bounded-weight coefficient tests are needed below.
Two fresh traces in matching parity, at least one split, add weight
at least $3/2$. The allowed valuation in \eqref{co:BU-bound} is an
integer, so it is at least $2$ before the unary division and the
result vanishes after it. In opposite parity any two fresh traces
add weight at least $1$; subtracting $e_U=1/2$ and using integral
valuation again gives vanishing. The estimates persist under
further tests. Hence respectively
$t(g)t(u)=0$ for split $u$, or $t(g)t(u)=0$ for every $u$.
Applying \eqref{co:trace-identities} to $t(gu)t(u)$ gives
$t(gu^2)=t(g)$. The quotient by the subgroup generated by squares
is elementary abelian, and the Galois systems are unramified off
their level. This proves the stated factorization. At coefficient
$q$ the fresh split weight $1$ pays exactly the division in $H_U$;
the fresh simple weight $1/2$ pays exactly $e_U$ in opposite parity.
No unary term survives. This proves the symbol-reading assertion.
For contraction, apply \eqref{co:inertia-identity} at coefficient
$1$, so that its left side is a coefficient-$p$ test. Replace its
trace arguments by fresh primes. The first-shell calculation in
Lemma~\ref{co:operators} gives, in each of the split--split,
split--simple, and simple--split cases,
\[
c_p(D')A(D'),\qquad D'=(U/p)q',\quad
\operatorname{Fr}_{q'}\sim g\phi_p.
\]
Here the omitted source is exactly $B_V$ or $H_V$, with
$V=U\setminus\{p\}$, constructed from the same original modular form
and the same $\eta,\psi,\delta$;
no new minimum or normalization is chosen. To check the integer
powers explicitly, put $n_U=\lceil\delta+w(U)\rceil$ and let
$m_U$ be $1$ in matching parity and $0$ otherwise. After fresh
tests kill the unary terms, the first shell relative to the
omitted-source test has multiplier
\[
2^{1+n_V-n_U-m_U+m_V}c_p(D').
\]
For split $p$ the exponent is zero. For simple $p$ and matching
$U$ it is also zero. For simple $p$ and opposite $U$ it is one;
this is precisely the simple--simple case. The character change
$\psi_U/\psi_V=(p/\cdot)$ is already the Legendre factor in this
shell calculation. The phase is a unit and disappears on reduction.
Let $q_0',q_1'$ realize $g\phi_p,g\tau_p\phi_p$, respectively,
and put $D_i'=Vq_i'$. Applying inertia changes the Legendre term
in \eqref{co:cp} by a sign, so $c_p(D_1')=c_p(D_0')+1$.
Although these Frobenius elements may first be specified in the full
current elementary extension, their restrictions to the omitted-source
extension agree: inertia
at either place above $p$ is killed in that extension. Its original
characters are unramified at $p$, and its only new character factors
come from $V$. Thus it is unramified at the whole rational prime
$p$, giving $A(D_1')=A(D_0')$. If $\Delta_{\tau_p}A(Uq)$ denotes
the difference of the original tests with Frobenius labels $g\tau_p$
and $g$, the reduced local identity is now explicitly
\[
\Delta_{\tau_p}A(Uq)
=c_p(D_1')A(D_1')+c_p(D_0')A(D_0')=A(D_0'),
\]
which is \eqref{co:contraction}. When both vertices are simple the product
Frobenius is split and the first-shell valuation has a positive
excess; its reduction is zero. The entire-prime assertion follows
from the actual support of the omitted-$p$ coefficient form, rather
than from any convention about a chosen orientation. Isolation
is the same symbol-reading test with $t(1)=0$.
For the final assertion first use two copies of a single simple
Frobenius trace on matching $H_U$. Put
$b=B_U[1]$, $\epsilon=\chi_{-4}(q)$, and
$\kappa=(\psi_U/\eta)(q)$. Substitution of
\eqref{co:recurrence} and of the unary character gives the two terms
in the coefficient-$1$ test of $T_q^2$, modulo $\mathfrak m$:
\begin{align*}
H_U[q^2]&=\frac{b\eta(q)}2
(a_q-\kappa\epsilon-q\epsilon^i),\\
q(\langle q\rangle H_U)[1]&=\frac{b\eta(q)}2
q(\kappa-\epsilon^{i+1}).
\end{align*}
Their sum is
\[
\frac{b\eta(q)}2
\left(a_q+\kappa(q-\epsilon)
-q\epsilon^i(1+\epsilon)\right).
\]
For $q\equiv3\pmod4$ the parenthesized half has residue $a_q/2$;
for $q\equiv1\pmod4$ it has residue $a_q/2+1$.
In both cases it is $\#E(\mathbb F_q)/2$ modulo $2$.
Since $b$ reduces to $A(U)$, the squared trace has the required
coefficient. Fresh-prime testing replaces its two copies by
distinct primes. To pass to any pair with the specified labels,
vary one prime at a time. After that simple prime is omitted the
opposite-parity symbol depends only on rational elementary labels;
simple--simple changes vanish by the proved contraction rule.
Thus only the stated support and split-vertex labels matter.
\end{proof}
==== END A-pointwise-2-converse-for-elliptic-curves-with-rational-two-torsion//coefficients.tex ====
==== BEGIN A-pointwise-2-converse-for-elliptic-curves-with-rational-two-torsion//cyclotomic.tex ====
\section{Cyclotomic interpolation for extensions of trivial residual modules}
\label{cy:section}
Throughout this section, $E/\mathbb Q$ is fixed, its conductor is $N$, and
$E[2]$ is an extension of two trivial $\mathbb F _2$-modules. Write
$T=T_2E$ and $V=T\otimes_{\mathbb Z _2}\mathbb Q _2$. The word
``ordinary'' applied to global cohomology below means that the real-place
modification has not been made. It imposes no ordinary-reduction hypothesis
at~$2$.
The discriminants considered here are products of distinct signed primes
$q^*=(-1/q)q$, with $q\nmid 2N$. Fix their sign and local squareclasses at
$2N$. For such a discriminant $h$, put
\[
t_q=\dim_{\mathbb F _2}E[2]^{\operatorname{Fr}_q},\qquad
2w(h)=\sum_{q\mid h}t_q,\qquad
\mathcal L(h)=\frac{\sqrt{|h|}\,L(E^{(h)},1)}{\Omega_E^{\operatorname{sgn}(h)}}.
\]
Here fixed nonzero periods in the two period lines have been chosen once.
Changing these choices changes the bounds below by a constant depending
only on~$E$. Valuations of algebraic central values mean their $2$-adic
valuations, with $v_2(2)=1$. When $s_2(E^{(h)})=0$, the group
$\operatorname{Sha}(E^{(h)}/\mathbb Q)[2^\infty]$ is finite; set
\[
s(h)=\operatorname{length}_{\mathbb Z _2}
\operatorname{Sha}(E^{(h)}/\mathbb Q)[2^\infty],\qquad
D(h)=v_2(\mathcal L(h))-2w(h)-s(h)
\]
when the central value is nonzero. By Lemma~\ref{pr:ranks}, $s(h)$ is
defined at a corank-zero base even before its central value is known to
be nonzero. Only the definition of $D(h)$ requires that nonvanishing.
We prove the two assertions needed later.
\begin{theorem}[Cyclotomic lower bound]\label{cy:lower-bound}
There is a constant $C_E$ such that, for every allowed $h$ with
$L(E^{(h)},1)\ne0$,
\[
v_2(\mathcal L(h))\ \geq\ 2w(h)+s(h)-C_E.
\]
The constant is independent of the number and sizes of the primes dividing
$h$.
\end{theorem}
\begin{theorem}[Transfer to the missing vertex]\label{cy:missing-vertex}
Let $h_0$ be an allowed discriminant with $s_2(E^{(h_0)})=0$. Suppose that
for arbitrarily large $b$ there are a finite set $Q$ of distinct primes
$q\nmid 2Nh_0$ and linear forms
$\lambda_q:\mathbb F _2^b\longrightarrow\mathbb F _2$ giving
\[
h_x=h_0\prod_{q\in Q}(q^*)^{\lambda_q(x)},\qquad x\in\mathbb F _2^b,
\]
all in the fixed local and sign classes. Suppose that a number $B$,
independent of the family, satisfies
\[
L(E^{(h_x)},1)\ne0,\qquad D(h_x)\leq B\qquad(x\ne0).
\]
Then $L(E^{(h_0)},1)\ne0$ and $D(h_0)\leq B+C_E$.
Here $C_E$ is independent of $h_0$, $Q$, and $b$; the dimension needed in
the proof is allowed to depend on the fixed precision being tested.
\end{theorem}
\subsection{The integral scalar family and two algebraic steps}
Put
\[
\Lambda=\mathbb Z _2[[t]],\qquad
\mathcal O=\widehat{\Lambda_{(2)}},\qquad
\kappa=\mathcal O/2\mathcal O=\mathbb F _2((t)).
\]
The completion defining $\mathcal O$ is $2$-adic. An element of $\mathcal O$
has, modulo any $2^M$, a Laurent expansion with a finite negative part.
There is no evaluation homomorphism $\mathcal O\to\mathbb Q _2$ given by
$t=0$. All central evaluations below are made on elements of
$\Lambda[1/2]$ before extending coefficients to~$\mathcal O$.
For $G=(\mathbb Z/2\mathbb Z)^b$, index its characters by
$x\in\mathbb F _2^b$, and denote character evaluation by a subscript~$x$.
The arithmetic construction below will produce a scalar family
$u\in\Lambda[1/2][G]$ with one uniform denominator:
\[
2^{C_{\mathrm{int}}}u\in\Lambda[G].
\]
At a constituent with $s_2(E^{(h_x)})=0$, its central value $u_x(0)$
will vanish exactly when $L(E^{(h_x)},1)$ does; in the nonvanishing case,
\[
\bigl|v_2(u_x(0))-D(h_x)\bigr|\leq C_{\mathrm{loc}}.
\]
Both constants will be independent of the number and sizes of all varying
primes, including those of the base $h_0$. Integrality gives the lower
bound, while a congruence between the constant coefficients at $0$ and at
some nonzero address gives the missing-vertex theorem.
Two different arguments establish the uniform denominator. Rational
divisibility will show $u_x\in\Lambda[1/2]$ character by character.
A fixed integral Kato construction will instead give
$2^{C_{\mathrm{int}}}u\in\mathcal O[G]$ before taking characters.
The first lemma combines these assertions without a loss depending on
$|G|$; the second supplies the congruence used at the end.
\begin{lemma}[Integral Fourier intersection]\label{cy:fourier}
Suppose $u\in\mathcal O[1/2][G]$, all $u_x$ belong to $\Lambda[1/2]$, and
$2^C u\in\mathcal O[G]$. Then $2^C u\in\Lambda[G]$.
\end{lemma}
\begin{proof}
Fourier inversion in characteristic zero expresses every group-basis
coefficient of $u$ as a sum of the $u_x$ divided by $|G|$. Thus it belongs
to $\Lambda[1/2]$. The corresponding coefficient of $2^C u$ also belongs
to~$\mathcal O$. For a power series with bounded $2$-denominator,
nonnegative $\mathcal O$-valuation means that every coefficient is integral.
Consequently $\mathcal O\cap\Lambda[1/2]=\Lambda$, which proves the assertion
with the original exponent~$C$.
\end{proof}
The common-zero parity step below is the characteristic-two case of the
Chevalley--Warning principle; see
\cite[Section~1, equation~(1)]{HeathBrownChevalleyWarning}.
We retain the elementary proof of this step.
\begin{lemma}[Binary congruences]\label{cy:binary}
Let $z_1,\ldots,z_r\in\mathbb Z _2[G]$. Given $M\geq1$, if
$b>r(2^M-1)$, there is an $x\ne0$ such that
\[
(z_j)_x\equiv(z_j)_0\pmod {2^M}\qquad(1\leq j\leq r).
\]
The bound depends on the number of tests and their precision, and not on
the group-ring coefficients.
\end{lemma}
\begin{proof}
Choose generators of $G$. Every character-value function $z(x)$ has an
expansion
\[
z(x)=a_\varnothing+
\sum_{\varnothing\ne I\subseteq\{1,\ldots,b\}}
2^{|I|}a_I\prod_{i\in I}x_i,
\qquad a_I\in\mathbb Z _2.
\]
This follows by expanding each sign character as a product of factors
$1-2x_i$. The $j$th binary digit of a $2$-adic integer $a$ is
$\binom a{2^j}\bmod2$, equivalently the coefficient of $Y^{2^j}$ in
$(1+Y)^a$ over $\mathbb F _2$. On the Boolean cube, the contribution of
the monomial $x_I=\prod_{i\in I}x_i$ to this generating function is
\[
1+x_I\bigl((1+Y)^{2^{|I|}a_I}-1\bigr).
\]
Its nonconstant terms have $Y$-degree at least $2^{|I|}$. In a contribution
to $Y^{2^j}$, the sum of the quantities $2^{|I|}$ used is at most $2^j$.
Since $|I|\leq2^{|I|}$, the resulting Boolean polynomial has degree at
most~$2^j$. These formulas for $2$-adic exponents follow either by
continuity from nonnegative integers or in each finite coefficient
quotient.
Express all $rM$ digit equalities as Boolean equations $P_{ij}(x)=0$.
Their total degree is at most $r(2^M-1)$. The indicator of their common
zero set is $\prod_{i,j}(1+P_{ij})$. The remaining count is the
Chevalley--Warning degree argument \cite{Chevalley1935,Warning1935}.
A Boolean polynomial of degree less than $b$ has even sum over
$\mathbb F _2^b$: every monomial omits a variable,
and summing over that variable gives zero in $\mathbb F _2$. Hence the
number of common zeros is even. Since $0$ is a common zero, another one
exists.
\end{proof}
\subsection{The positive global complex}
The scalar $u$ will be the coordinate of two cohomology classes in an
integral determinant line. One class comes from the real place and the
other from Kato's construction. We first construct the complex and show
that its localization at $(2)$ is free of rank two in degree one.
Let $\Gamma=\operatorname{Gal}(\mathbb Q_\infty/\mathbb Q)$ be the real
cyclotomic $\mathbb Z _2$-group and identify its generator with $1+t$.
For a binary family, let $\psi:G_{\mathbb Q}\to G$ be its universal
quadratic character: evaluation at $x$ is $\chi_{h_x/h_0}$. Put
\[
R=\Lambda[G],\qquad
\mathbb T=T\otimes_{\mathbb Z _2}R,
\]
with the fixed twist $\chi_{h_0}$, the universal twist $\psi$, and the
inverse cyclotomic action. One may replace $G$ by the actual image of
$\psi$ and subsequently extend scalars. In a family in
Theorem~\ref{cy:missing-vertex}, a nonzero character trivial on that image
already gives the conclusion directly.
Let $S$ contain $2N$, the prime factors of $h_0$, and all the new primes.
Use continuous cochains with their completed coefficient topology, and set
\begin{equation}\label{cy:positive}
C_S^+(\mathbb T)=
\operatorname{Cone}\bigl(R\Gamma(G_{\mathbb Q,S},\mathbb T)
\longrightarrow R\Gamma(\mathbb R,\mathbb T)\bigr)[-1].
\end{equation}
The real complex here uses ordinary cochains in nonnegative degrees.
There are no imposed local conditions at the finite places.
\begin{lemma}[Perfection and the real-place class]\label{cy:perfect}
The complex \eqref{cy:positive} is perfect over $R$ and computes derived
coefficient specializations. There is a class $a\in H^1(C_S^+(\mathbb T))$
whose rational span is the real-place summand. Its integral index in
that summand depends only on $E$ and the fixed real sign.
\end{lemma}
\begin{proof}
Here are the finiteness and coefficient issues in the use of global
cochains. For a finite coefficient quotient of $R$, continuous cochains
are filtered unions of finite products of free coefficient modules.
Refinement of finite partitions makes these modules flat; over an Artin
local coefficient ring they are free. For finite residual coefficients,
global cohomology is finite in each degree, and restriction to the real
places is an isomorphism in degrees greater than two. The latter is the
real-place clause of global duality. Thus the residual fiber has bounded
amplitude and finite-dimensional cohomology. The same amplitude bound
holds for every finite coefficient module, by a composition series and
the cohomology long exact sequence.
This gives a bounded finite free model as follows. Over the residue
field split off the acyclic disks, leaving its finitely many cohomology
spaces. Lift each invertible disk differential over an Artin coefficient
quotient and cancel it. Invertibility of a lifted inverse follows from
nilpotence of the maximal ideal. Equivalently one can perform this on a
free resolution; a minimal free model cannot have an additional term
whose reduction has zero cohomology, because its residual differential
is zero. Hence the remaining ranks are the finite residual ranks and
are bounded independently of the coefficient quotient. Minimal models
identified by a homotopy equivalence over an Artin local ring are
isomorphic. Lifting their changes of basis therefore makes these models
compatible along $R/\mathfrak m^n$. Taking the inverse limit gives a
bounded finite free model over~$R$. The cochain construction with finite
free coefficients and the same cancellation also proves derived base
change. This is a fixed-$S$ argument; it does not assert uniform bounds
on the number of generators as $S$ varies.
All the characters of the family have the same real sign, so the action
at the real place is the fixed action on $T\chi_{h_0}$ extended to $R$.
Choose a nonzero vector in its integral invariant line. Its connecting
image in the fiber is $a$. There are no global invariant vectors in the
rational elliptic representation, and consequently this connecting map
is injective rationally. The choice and its index are fixed. Notice also
that $H^1(\mathbb R,\mathbb T)$ is killed by~$2$; thus twice any ordinary
cohomology class lifts to the positive complex. These observations use
compact global duality, including the real-place extension
\cite[Sections~5.7 and~6.9.1]{Nekovar2006} of the complex formulation
in \cite[Theorem~6.3.4]{Nekovar2006}.
\end{proof}
\begin{lemma}[Residual concentration]\label{cy:residual}
There is an isomorphism in the derived category
\[
C_S^+(\mathbb T)\otimes_R\mathcal O[G]
\simeq \mathcal O[G]^2[-1].
\]
\end{lemma}
\begin{proof}
The ring $\mathcal O[G]$ is local, with residue field $\kappa$ obtained
by setting $2=0$ and every $g=1$. Thus the quadratic characters disappear.
Since $E[2]$ has a filtration with trivial factors, it is enough first to
compute the positive residual complex for the trivial representation.
Let $K_n$ be the real cyclotomic layer of degree $d_n=2^n$. The number
of finite primes of $K_n$ above a fixed $S$ is bounded with $n$: at $2$
there is total ramification, and the Frobenius of each fixed odd prime
has nonzero image in $\Gamma$. The $2$-ranks of the class groups of
$K_n$ are also bounded. Indeed the cyclotomic $\mu$-invariant is zero
for this abelian tower \cite{FerreroWashington1979}; the structure theorem
and class-group control identify their $2$-ranks, up to bounded kernels
and cokernels, with those of the coinvariants of a finitely generated
$\mathbb Z _2$-module. The same bound holds for $S$-class groups.
The signature map of units in $K_n$ is surjective. For $n\geq1$, let
$\zeta$ be a primitive $2^{n+2}$th root and put
$\epsilon=1+\zeta+\zeta^{-1}$. If $f_1(X)=X^2-2$ and
$f_{n+1}(X)=f_n(X^2-2)$, then $f_n$ is the minimal polynomial of
$\zeta+\zeta^{-1}$ and $f_n(-1)=-1$. Hence
$N_{K_n/\mathbb Q}(\epsilon)=-1$. Its signature vector has odd
augmentation. The embeddings form a regular set for the cyclic
$2$-group $\operatorname{Gal}(K_n/\mathbb Q)$, whose group algebra over
$\mathbb F _2$ is local. A vector with odd augmentation is a unit in
this group algebra, so its translates span every signature. For $n=0$
the assertion follows from the unit~$-1$.
We now use Kummer theory to isolate the one dimension that survives
after inverting $t$.
Writing $s_n$ for the number of finite support places and $c_n$ for the
$2$-rank of the $S$-class group, one has
\[
\dim H^1(G_{K_n,S},\mu_2)=d_n+s_n+c_n.
\]
The real degree-one restriction is surjective by the signature result.
In degree two, the Kummer sequence has class-group term of dimension
$c_n$ and Brauer term. The Brauer invariants at the real and finite
support places have their single sum relation. Since a finite support
place above $2$ is present, restriction of the Brauer term to the real
places is surjective and has kernel of dimension $s_n-1$. Thus the
degree-two kernel of restriction to infinity is bounded. Restriction
is an isomorphism in higher degrees by global duality. The degree-zero
restriction is injective. The cohomology of the positive fiber is
therefore bounded in every degree except one; its degree-one dimension
is $d_n+O_S(1)$.
Apply Lemma~\ref{cy:perfect} modulo $2$. Derived specialization to the
$n$th layer is specialization modulo $t^{2^n}$. For a perfect complex
over $\mathbb F _2[[t]]$, the universal-coefficient exact sequence shows
that a rank-$r$ cohomology module contributes $r2^n+O(1)$ to these
dimensions; the torsion terms contribute only $O(1)$. It follows that
after inverting $t$ the trivial-constituent fiber has dimension one in
degree one and vanishes in every other degree. Extensions preserve
this concentration, so the fiber for $E[2]$ has dimension two in degree
one. A minimal finite free complex over the local ring $\mathcal O[G]$
has zero residual differentials. The concentration just proved forces
all its terms except the rank-two term in degree one to vanish.
No denominator is introduced by this exact freeness statement.
\end{proof}
\subsection{Fixed-form zeta classes and the integral coefficient map}
We need a global class whose denominator stays bounded as the tame
support grows. For this purpose every auxiliary trace will land in
the lattice of one fixed modular form. Let $f$ be the fixed weight-two
form of $E$, and choose its fixed integral cohomological lattice, with
the Tate twist making its
rational representation~$V$. Kato's classes (8.1.2)--(8.1.3), for a fixed
symbol $\xi\in\mathrm{SL}_2(\mathbb Z)$, exist for arbitrary $m$ and any
$S$ containing the prime factors of $mN$. The trace in Section~8.9 is
a homomorphism into the lattice
\[
\mathbb Z _2[\operatorname{Gal}(\mathbb Q(\mu_m)/\mathbb Q)]
\otimes V_{2,\mathbb Z _2}(Y_1(N))(1).
\]
Section~8.11 then projects to the lattice of the fixed form.
Proposition~8.12 supplies its norm relations. These assertions hold at
the coefficient prime~$2$ \cite[Sections~5.1, 8.1, 8.9--8.12]{Kato2004}.
In particular, the auxiliary modular-curve level used to define a trace
does not become the level of the projected newform.
Choose an isomorphism of the resulting fixed rational representation with
$V$. A single power $2^{c_E}$ sends its integral lattice into~$T$.
Choose fixed symbols with nonzero projections on the required period
lines; their existence and the construction with both signs are made
explicit in \cite[Sections~13.6 and 13.9]{Kato2004}. Taking a fixed linear
combination, if needed, costs another fixed denominator.
Here is the coefficient map in this application. Let $m$ be odd and
divisible by the conductors of all the tame characters in the family,
including the odd part of $\chi_{h_0}$. At a sufficiently large
cyclotomic level write $A_{m,n}=\operatorname{Gal}(\mathbb Q(\mu_{m2^n})/
\mathbb Q)$. Shapiro identifies the cohomology of the cyclotomic field
with global cohomology with induced group-ring coefficients. Push the
class along the ring homomorphism
\begin{equation}\label{cy:coefficient-map}
\mathbb Z _2[A_{m,n}]\longrightarrow
\mathbb Z _2[\Gamma_n\times G],\qquad
[\sigma]\longmapsto
\chi_{h_0}(\sigma)[\sigma|_{\Gamma_n}][\psi(\sigma)].
\end{equation}
Use inverse actions on both sides if this is required by the chosen
Shapiro convention. This is a homomorphism of integral coefficient
modules and acts on actual cohomology classes. It requires neither
division by $|G|$ nor surjectivity of a cohomological specialization map.
The prime $2$ belongs to the fixed omitted support, so Proposition~8.12
gives exact norm compatibility as $n$ increases. The pushed classes
therefore give an integral smoothed Iwasawa class with coefficients in
$\mathbb T$, up to the single fixed power~$2^{c_E}$.
We next remove smoothing. In weight two and with $r=r'=1$, its factors
are
\begin{equation}\label{cy:smoothing}
(c^2-c\sigma_c)(d^2-d\sigma_d).
\end{equation}
This is the specialization of
\cite[Theorem~6.6 and (4.2.4)]{Kato2004}. Choose a fixed
$e\geq\max(2,v_2(N))$, increasing it to accommodate the finitely many
fixed symbol levels. The Chinese remainder theorem permits $c,d>1$,
prime to $6Nm$, such that
\[
c,d\equiv1\pmod{3N_{\mathrm{odd}}m},\qquad
c,d\equiv1+2^e\pmod{2^{e+1}}.
\]
Their tame actions in \eqref{cy:coefficient-map} are trivial. If
$\sigma_c$ acts in $\Gamma$ as $(1+t)^{\kappa(c)}$, then
$v_2(\kappa(c))=e-2$. Modulo the maximal ideal of $\mathcal O[G]$ its
smoothing factor is
\[
1-(1+t)^{\kappa(c)}\ne0\quad\hbox{in }\mathbb F _2((t)).
\]
It is therefore a unit in $\mathcal O[G]$. Its central value is
$c(c-1)$, of valuation exactly~$e$. The same statements hold for $d$
and for inverse-translate conventions. Thus unsmoothing introduces
neither a $2$-adic denominator at $(2)$ nor a growing central denominator.
The integers $c,d$ vary with $m$; the congruences fix the needed valuations,
not their exact cyclotomic group elements.
Denote the resulting rational ordinary class by $z_S^{(0)}$. Theorem~9.7 of
\cite{Kato2004} applies to $k=2,r=r'=1$. In the character constituent
$x$, its central reciprocity formula reads
\begin{equation}\label{cy:reciprocity}
v_2\bigl(\exp_2^*z_{S,x}^{(0)}(0)\bigr)
=v_2(\mathcal L(h_x))+
\sum_{\substack{q\in Q\,,\ q\nmid h_x}}
v_2\bigl(\#E^{(h_x)}(\mathbb F _q)\bigr)+O_E(1)
\end{equation}
when the value is nonzero, and a zero central value gives a zero dual
exponential. Fixed omitted primes are included in the error.
To check the normalization, the character sum in
\eqref{cy:coefficient-map} is the unnormalized sum, as in Kato's trace.
Its tame quadratic Gauss sum supplies $\sqrt{|h_x|}$ up to a root of
unity. Tame conductors and their Gauss sums are $2$-adic units. The
period vector is the fixed symbol vector of $f$. More explicitly,
in the required sign line write the period represented by
$\delta(f,1,\xi)^{\pm}$ as $c_\xi^{\pm}\Omega_E^{\pm}$.
The nonzero scalar $c_\xi^{\pm}$ is fixed: the character-sum formula
of \cite[Theorem~6.6]{Kato2004} uses this same $\delta(f,1,\xi)^{\pm}$
for every tame character. The tame character enters through its
$L$-value and Gauss sum, not through a newly chosen period vector.
The comparison with the minimal differential of $E^{(h_x)}$ at $2$ has bounded valuation,
because there are only finitely many local twisting classes there.
At a good unused prime the omitted factor is
\[
1-\frac{a_q(E)\chi_{h_x}(\operatorname{Fr}_q)}q+\frac1q
=\frac{\#E^{(h_x)}(\mathbb F _q)}q;
\]
at a good ramified prime the local $L$-factor is~$1$. These account for
every varying factor in \eqref{cy:reciprocity}.
\subsection{Rational divisibility and the local Euler corrections}
The fixed-form construction controls the denominator at $(2)$. We now
prove the complementary assertion: the determinant coordinate has no
pole at any other height-one prime. This step is characterwise and uses
only rational divisibility. Write $A=\Lambda[1/2]$. Theorem~12.5(3) of
\cite{Kato2004} is a height-one divisibility over~$A$; its possible local
$H^2$ correction comes from Galois cohomology over $\mathbb Q_2$.
The global groups there use $j_*$ outside that prime. The stronger integral assertion
12.5(4) is not used. In the non-CM case, the large-image input in Kato's
proof follows from the open image theorem \cite{Serre1972}; see
\cite[Remark~12.8, (12.8.2), Theorem~13.4(v), and Section~13.13]{Kato2004}.
This is not an additional residual-image hypothesis in Theorem~12.5(3).
For CM the required rational statement at every prime, including~$2$, is supplied by
\cite[Theorem~2.6 and Remark~2.7]{BurungaleTian2026}. Neither statement
asserts uniform integrality in a quadratic group ring.
Fix a character constituent $h$ and write $V_h=V\chi_h$. Let $C_{j_*}$
be the rational Iwasawa complex in Kato's formulation and $C_S$ the
rational full global complex. At an odd prime put
\[
W_q=H^1(I_q,V_h).
\]
The localization triangle has local cone
\begin{equation}\label{cy:local-block}
Q_q=R\Gamma\bigl(G_{\mathbb F _q},
W_q\otimes A(\Gamma^\iota)\bigr)[-1]
=\left[W_q\otimes A
\xrightarrow{\ 1-\gamma_q\operatorname{Fr}_{W_q}\ }
W_q\otimes A\right]
\end{equation}
in degrees one and two. Here $\gamma_q$ denotes the scalar cyclotomic
Frobenius in the chosen convention. Its exponent is nonzero. Consequently
$D_q=\det(1-\gamma_q\operatorname{Fr}_{W_q})$ is a nonzero element of~$A$,
and the map in \eqref{cy:local-block} is injective. This remains true
after localization at a height-one prime dividing $D_q$: localization
does not mean reduction modulo that prime. Thus
\[
H^1(Q_q)=0,\qquad
\operatorname{length}_{A_{\mathfrak p}}H^2(Q_q)_{\mathfrak p}
=v_{\mathfrak p}(D_q).
\]
The long exact sequence gives
\begin{equation}\label{cy:local-sequence}
\begin{gathered}
H^1(C_{j_*})\simeq H^1(C_S),\\
0\longrightarrow H^2(C_{j_*})\longrightarrow H^2(C_S)
\longrightarrow\bigoplus_{q\in S\setminus\{2\}}H^2(Q_q)
\longrightarrow H^3(C_{j_*}).
\end{gathered}
\end{equation}
In particular, the length of $H^2(C_S)$ is at most the length of
$H^2(C_{j_*})$ plus the sum of the local lengths. No assertion that
$H^3(C_{j_*})$ vanishes is required here.
At a good unused prime, $W_q=V_h(-1)$, so, on including the quadratic
Frobenius sign in $Z$,
\begin{equation}\label{cy:good-euler}
D_q=P_q(Z),\qquad P_q(Z)=1-\frac{a_q(E)}q Z+\frac1q Z^2.
\end{equation}
At a good active prime, inertia acts by $-1$ on $V_h$. Its rational
inertia cohomology is zero, so $W_q=0$ and $D_q=1$.
At fixed bad primes the general expression is $H^1(I_q,V_h)$ in
\eqref{cy:local-block}. Replacing it indiscriminately by $V_h^{I_q}(-1)$
would be incorrect. For example, at split multiplicative reduction,
\[
V_h^{I_q}=\mathbb Q _2(1),\qquad
H^1(I_q,V_h)=\mathbb Q _2(-1),
\]
and the correct determinant is $1-\gamma_q/q$, with nonzero central
value. The incorrect replacement would give $1-\gamma_q$.
In general $D_q(0)\ne0$ follows from local duality:
$H^2(\mathbb Q _q,V_h)=H^0(\mathbb Q _q,V_h)^*=0$, since elliptic torsion
over a local field is finite. The fixed bad-prime corrections therefore
have nonzero central values for every reduction type.
Let $z_{\mathrm{BK},h}$ denote a generator of the rational Kato zeta
line in the real cyclotomic component with the fixed sign. Compare it
with the fixed-form class only after taking the character constituent.
In the chosen Euler convention the comparison is
\begin{equation}\label{cy:rational-comparison}
z_{S,h}^{(0)}
=c_h\epsilon_h(t)E_h^{\mathrm{fix}}(t)
\prod_{\substack{q\in Q\,,\ q\nmid h}}
P_q\bigl(\chi_h(\operatorname{Fr}_q)\gamma_q\bigr)
z_{\mathrm{BK},h},
\end{equation}
where $c_h\in\mathbb Q_2^\times$ and $\epsilon_h$ is a group-like
unit of~$A$. The factor $E_h^{\mathrm{fix}}$ consists of the omitted
local $L$-polynomials away from $2$ at primes of $N$ and the fixed
auxiliary support. It ranges over a finite collection, because the
local twisting classes there are fixed. The primes of $h_0$ are good
ramified primes and contribute~$1$, regardless of their number.
These factors have nonzero central values: they are good-prime point
counts divided by the prime, multiplicative factors $1\mp1/q$, or~$1$.
They are not being identified with the local cohomology determinants
$D_q$ at bad primes; any discrepancy there will be cleared by a fixed
multiplier below.
Here is the derivation of \eqref{cy:rational-comparison}. Both global
Iwasawa $H^1$ modules have rank one. The reciprocity formulas
\cite[Theorems~6.6, 9.7, 12.5(1), and Lemma~13.10(2)]{Kato2004}
compare the same fixed period vector and the relevant imprimitive
$L$-values. Their quotient supplies the displayed omitted Euler
factors; a good ramified prime contributes~$1$. Coprime Gauss-sum
factorization and the fixed period comparison supply $c_h$ and
$\epsilon_h$; restricting conductor exponents to either fixed parity
removes any factor $\chi_h(2)^n$. Infinitely many of these tests have
nonzero central value by cyclotomic nonvanishing, as used in
\cite[Theorem~13.5(2)]{Kato2004}. After clearing the finitely many
denominators of the scalar ratio, the difference is a power series
vanishing at infinitely many finite-order characters. Weierstrass
preparation forces that difference to be zero. This proves the generic
comparison without identifying the integral lattices of varying
twisted newforms. The scalar $c_h$ need not have a uniformly bounded
$2$-valuation: every such scalar is a unit of $A$ and does not affect
height-one divisibility over~$A$. The uniform integral and central bounds instead concern
the actual fixed-form class and the fixed period vector in
\eqref{cy:reciprocity}.
Equations \eqref{cy:local-sequence}--\eqref{cy:good-euler} now show exactly
how the imprimitive factors enter divisibility. At any height-one
$\mathfrak p$ of $A$, enlarging the global complex increases its
degree-two length by at most $\sum_qv_{\mathfrak p}(D_q)$, whereas
multiplying the rank-one zeta generator by $\prod_qD_q$ adds exactly
that quantity to its index. To cover Kato's local correction at $2$,
multiply the class also by a fixed element $F_2\in\Lambda$ divisible over $A$ by the
characteristic ideal of the relevant local $H^2_{\mathrm{Iw}}$.
It may be chosen with $F_2(0)\ne0$. Indeed derived local control gives
\[
H^2_{\mathrm{Iw}}(\mathbb Q _2,T\chi_h)_\Gamma\otimes\mathbb Q _2
=H^2(\mathbb Q _2,V_h)=0,
\]
so the characteristic ideal has no factor $t$. Local twisting classes
at $2$ are fixed, and one element covers their finite set.
Likewise any additional fixed Euler factors in the rational comparison
can be cleared by a fixed product with nonzero central value, using the
actual blocks \eqref{cy:local-block}. Denote the total fixed multiplier
by~$F_E$. Multiplying by it only changes the fixed constants in
\eqref{cy:reciprocity} and the integrality bound. Set
$z_S=F_Ez_S^{(0)}$. For this corrected class we have at every
height-one $\mathfrak p$ of~$A$
\begin{equation}\label{cy:divisibility}
\operatorname{length}_{A_{\mathfrak p}}H^2(C_S)_{\mathfrak p}
\leq
\operatorname{length}_{A_{\mathfrak p}}
\bigl(H^1(C_S)/Az_{S,h}\bigr)_{\mathfrak p}.
\end{equation}
We finish by checking that the chosen Euler convention can be imposed
integrally before character evaluation. If the original coefficient
map uses inverse Frobenius, the varying unused factors change as follows.
In $\mathcal O[G]$, both $P_q(Z)$ and $Z^2P_q(Z^{-1})$ are units: their
reductions are nonzero polynomials evaluated at the nonconstant
cyclotomic Frobenius. The identity
\[
P_q(Z)-Z^2P_q(Z^{-1})=\frac{q-1}{q}(1-Z^2)
\]
shows that their ratio $r_q$ is $1$ modulo~$2$. If $g_q$ is the universal
inertia element, then
\begin{equation}\label{cy:euler-convention}
U_q=1+\frac{r_q-1}{2}(1+g_q)\in\mathcal O[G]^\times
\end{equation}
is integral. At an active character it is $1$; at an unused character
it is $r_q$. Its unused central value is $1$, since $Z=\pm1$ there.
Thus inverse Frobenius conventions can be reconciled before character
evaluation, without a denominator for each prime. Products of these
units have the same property. The resulting rational character
coordinates have the Euler convention in \eqref{cy:divisibility}.
We use this convention for $z_S$ from now on; these units alter neither
its uniform integrality bound nor its central reciprocity values.
\subsection{The determinant and its central specialization}
Choose a basis $\mathbf e$ of the invertible $R$-module
$\det_R^{-1}C_S^+(\mathbb T)$; $R$ is local, so such a basis exists.
Lift $2z_S$ to positive cohomology and take its wedge with $a$.
The factor $2$ is fixed, and two lifts differ by a real-place class,
which does not change this wedge. In generic cohomology, the inverse
determinant identifies with the exterior square of the rank-two
degree-one space. Define its scalar coordinate $u$ by
\[
a\wedge\widetilde{2z_S}=u\mathbf e.
\]
Here $z_S$ already includes the fixed multiplier $F_E$ above.
The real-place summand introduces no height-one torsion over~$A$. Thus
\eqref{cy:divisibility} says that $u_x$ has no pole at any height-one
prime of $\Lambda$ except possibly $(2)$. Since $\Lambda$ is normal,
\begin{equation}\label{cy:rational-membership}
u_x\in\Lambda[1/2]\qquad\hbox{for every }x.
\end{equation}
On the other hand, the fixed integral trace,
\eqref{cy:coefficient-map}, unsmoothing over $\mathcal O[G]$, and
Lemma~\ref{cy:residual} give
\[
2^{C_{\mathrm{int}}}u\in\mathcal O[G]
\]
with $C_{\mathrm{int}}$ independent of $S$, $h_0$, $G$, and~$b$.
The determinant identification in a free complex introduces no further
index. Applying Lemma~\ref{cy:fourier} yields the crucial uniform statement
\begin{equation}\label{cy:uniform-integrality}
2^{C_{\mathrm{int}}}u\in\Lambda[G].
\end{equation}
\begin{proposition}[Central determinant formula]\label{cy:central}
There is a constant $C_{\mathrm{loc}}$, depending only on the fixed curve
and local classes, with the following property. If $s_2(E^{(h_x)})=0$,
then $u_x(0)=0$ when $L(E^{(h_x)},1)=0$. Otherwise
\begin{equation}\label{cy:central-formula}
\bigl|v_2(u_x(0))-D(h_x)\bigr|\leq C_{\mathrm{loc}}.
\end{equation}
The rational positive complex at $t=0$ has cohomology only in degree one,
of dimension two, so these are specializations of the same determinant.
\end{proposition}
\begin{proof}
Abbreviate $T_h=T_2E^{(h_x)}$ and $A_h=E^{(h_x)}[2^\infty]$.
Use the integral Kummer subgroup
\[
K_v=\operatorname{im}\bigl(E^{(h_x)}(\mathbb Q _v)^{\wedge}_2
\longrightarrow H^1(\mathbb Q _v,T_h)\bigr)
\]
at every finite place. For $v\ne2$ it is all of $H^1(\mathbb Q _v,T_h)$.
One way to see this integrally is the inverse-limit Kummer exact sequence:
its quotient is $T_2H^1(\mathbb Q _v,E^{(h_x)})$, which is zero because
local Tate duality identifies the $2$-primary local Weil--Ch\^atelet group
with the dual of the finite $2$-completion of the local point group.
At $2$ the quotient
\[
L_2=H^1(\mathbb Q _2,T_h)/K_2
\]
is a free rank-one $\mathbb Z _2$-module. The same Kummer sequence and
local Tate duality identify it with the $\mathbb Z _2$-dual of the free
part of $E^{(h_x)}(\mathbb Q _2)^{\wedge}_2$.
The dual exponential identifies $L_2\otimes\mathbb Q _2$ with the
one-dimensional de Rham target. Integrally its index depends only on
the local curve and the chosen differential. This follows from the
compatibility of the Kummer map and local Tate duality with the formal
elliptic logarithm: on a sufficiently small open formal subgroup that
logarithm is an isomorphism, and its adjoint is the dual exponential.
Every elliptic curve over $\mathbb Q _2$ has such a formal subgroup,
irrespective of reduction type. Since only finitely many local twists
occur, the indices are uniformly bounded.
The assumption on the Selmer corank implies that the compact finite
Selmer group is torsion and that the discrete finite Selmer group has
length $s(h_x)$. Rational Poitou--Tate consequently identifies ordinary
global $H^1$ with the singular line at $2$, and gives ordinary global
$H^2=0$ rationally. Adding the real invariant line gives the claimed
dimension two in degree one. This equals the generic rank, so derived
base change of the perfect complex specializes its determinant without
a change of rational cohomological rank.
The integral calculation explains the two subtractions in $D(h_x)$.
The index of global localization will cancel; the remaining local terms
will cancel the unused Euler factors and contribute $2w(h_x)$ at the
active primes. Let $J$ be the image of the free part of global
$H^1(G_{\mathbb Q,S},T_h)$ in $L_2$, and put
$i=\operatorname{length}(L_2/J)$. The compact/discrete Poitou--Tate
sequence for the mutually orthogonal Kummer conditions contains
\[
\begin{aligned}
0\longrightarrow L_2/J
&\longrightarrow \operatorname{Sel}_{2^\infty}(E^{(h_x)}/\mathbb Q)^\vee
\longrightarrow H^2(G_{\mathbb Q,S},T_h)\\
&\longrightarrow \bigoplus_{q\in S}H^2(\mathbb Q _q,T_h)
\longrightarrow H^0(\mathbb Q,A_h)^\vee\longrightarrow0,
\end{aligned}
\]
with the usual real Tate-cohomology terms inserted. Those terms are
killed by $2$ and have dimensions bounded by the fixed dimension of~$T$.
Taking lengths and using local duality gives
\begin{equation}\label{cy:pt-length}
\operatorname{length}H^2(G_{\mathbb Q,S},T_h)
=s(h_x)-i+\sum_{q\in S}\operatorname{length}H^0(\mathbb Q _q,A_h)
+O_E(1).
\end{equation}
This sequence is the cohomology consequence of integral
Poitou--Tate duality with the finite local conditions; see
\cite[Theorem~6.3.4 and Section~6.9.1]{Nekovar2006}, with the real
Tate-cochain convention of \cite[Section~5.7.1.6]{Nekovar2006}.
No unramified condition replaces
the Kummer condition at a bad prime. Global torsion is bounded by its
injection into $E^{(h_x)}(\mathbb Q _2)[2^\infty]$, whose order is bounded
by the fixed local twisting classes. The finite terms at $2$ and at
fixed bad primes are bounded for the same reason. Passing between
ordinary and positive cohomology changes only these bounded real and
global torsion terms.
Suppose first that the central value is nonzero. The coordinate of
$z_{S,x}(0)$ in the free global line has valuation
\[
v_2(\exp_2^*z_{S,x}(0))-i+O_E(1).
\]
For the inverse determinant, torsion in degree two subtracts its length
and torsion in degree one adds its length. This sign can be checked on
the complex $[\mathbb Z _2\xrightarrow{2^a}\mathbb Z _2]$ in degrees one
and two: its inverse determinant lattice maps to $2^a\mathbb Z _2$ in
the rational determinant. A vector's coordinate thus loses~$a$.
Using \eqref{cy:pt-length}, the two appearances of $i$ cancel, leaving
\[
v_2(u_x(0))=
v_2(\exp_2^*z_{S,x}(0))-s(h_x)
-\sum_{q\in S}\operatorname{length}H^0(\mathbb Q _q,A_h)+O_E(1).
\]
At an active good prime, quadratic inertia acts by $-1$ on the Tate
module. Its invariants on $A_h$ are therefore exactly $E[2]$, and taking
Frobenius invariants gives length~$t_q$. At an unused good prime, reduction
identifies the local $2$-primary torsion with
$E^{(h_x)}(\mathbb F _q)[2^\infty]$, of length
$v_2(\#E^{(h_x)}(\mathbb F _q))$. These unused terms cancel exactly
those in \eqref{cy:reciprocity}. The active terms, including the primes
of $h_0$, sum to $2w(h_x)$. All other terms have already been bounded
independently of the growing support. This proves
\eqref{cy:central-formula}.
If the central value is zero, reciprocity makes the dual exponential
zero. The rational ordinary specialization then satisfies the finite
condition at $2$, and it satisfies the finite conditions elsewhere
automatically. It lies in the rational finite Selmer group, which is
zero. Its wedge with the real-place class is therefore zero. The
constant-rank determinant specialization proves $u_x(0)=0$.
\end{proof}
\subsection{Proofs of the interpolation theorems}
\begin{proof}[Proof of Theorem~\ref{cy:lower-bound}]
Use the preceding construction for the single discriminant~$h$.
The forward analytic rank-zero theorem gives $s_2(E^{(h)})=0$ and
finiteness of its Shafarevich--Tate group. By
\eqref{cy:uniform-integrality}, $v_2(u_h(0))\geq-C_{\mathrm{int}}$.
Proposition~\ref{cy:central} then gives
$D(h)\geq-C_{\mathrm{int}}-C_{\mathrm{loc}}$. Every constant in the
construction came from the fixed form, a fixed real period line, or
the finite set of local twists at $2N$.
\end{proof}
\begin{proof}[Proof of Theorem~\ref{cy:missing-vertex}]
At every nonzero address the forward rank-zero theorem gives Selmer
corank zero. Proposition~\ref{cy:central} gives
\[
v_2(u_x(0))\leq B+C_{\mathrm{loc}}\qquad(x\ne0).
\]
Choose an integer $M$ such that
$M-C_{\mathrm{int}}>B+C_{\mathrm{loc}}$. By
\eqref{cy:uniform-integrality}, the constant coefficient of
$2^{C_{\mathrm{int}}}u$ belongs to $\mathbb Z _2[G]$. For sufficiently
large $b$, Lemma~\ref{cy:binary} gives an $x\ne0$ such that
\[
v_2(u_x(0)-u_0(0))\geq M-C_{\mathrm{int}}.
\]
It follows that $u_0(0)$ is nonzero and
$v_2(u_0(0))\leq B+C_{\mathrm{loc}}$: either failure would contradict
the displayed upper bound at this $x$. The corank-zero hypothesis
at the base and Proposition~\ref{cy:central} imply
$L(E^{(h_0)},1)\ne0$ and
\[
D(h_0)\leq B+2C_{\mathrm{loc}}.
\]
Taking $C_E$ large enough proves both theorems with a common constant.
In particular no loss proportional to $b$, $|Q|$, or the prime count of
$h_0$ has entered this argument.
\end{proof}
==== END A-pointwise-2-converse-for-elliptic-curves-with-rational-two-torsion//cyclotomic.tex ====
==== BEGIN A-pointwise-2-converse-for-elliptic-curves-with-rational-two-torsion//graphs.tex ====
\section{Arithmetic graphs and simultaneous addressing}\label{gr:section}
Fix $b\ge1$. An \emph{address} is a point $x\in\mathbf F_2^b$;
a vertex with activation $\lambda\in(\mathbf F_2^b)^*$ is present at $x$
when $\lambda(x)=1$. We will adjoin fresh primes with nonzero activations
to a given collection of primes. The goal is to make one or two binary
coefficient tests equal to $1$ at every $x\ne0$, while retaining all the
original primes and their residue data. No new prime is present at $0$.
The hypothesis is that each test, separately at each nonzero address,
is a nonzero polynomial in the available residue bits.
The arithmetic identities are those of
Proposition~\ref{co:symbol-interface}, conditional on its coefficient
lower bounds. We first describe and realize their graph data, then state
the simultaneous-testing result. Its proof uses contraction operators
and two finite networks that isolate a prescribed derivative at one address.
\subsection{Labels and arithmetic realization}
Fix a finite set $S$ of rational primes, containing $2$ and all the ramified
primes in the fixed data. Let $F$ be either $\mathbf Q$ or a quadratic field.
For the rational two-torsion application $F=\mathbf Q(E[2])$; the construction
with full rational two-torsion also uses a quadratic auxiliary field in place
of $F$. Include in $S$ primes whose prime ideals generate the class group of
$F$, and include the infinite places in $S_F$. Put
\[
V_S(F)=\{\alpha\in F^\times/F^{\times2}:
v_{\mathfrak p}(\alpha)\equiv0\pmod2
\text{ for }\mathfrak p\notin S_F\}.
\]
This is a finite-dimensional $\mathbf F_2$-space: the valuation sequence
expresses it as an extension of the finite squareclass group of $S_F$-units
by a subgroup of the $2$-torsion of the $S_F$-class group. Its Kummer extension
is denoted $F_S^{(2)}=F(\sqrt{V_S(F)})$.
A \emph{split vertex} is a rational prime $p\notin S$ split in $F$, together
with a choice $\mathfrak p\mid p$ if $F$ is quadratic. This choice is its
\emph{orientation}. Choose a generator $\pi_{\mathfrak p}$ with divisor
$\mathfrak p$ outside $S_F$; class-group generation makes this possible.
Conjugation supplies the generator for the other orientation, and
\begin{equation}\label{gr:norm}
N_{F/\mathbf Q}(\pi_{\mathfrak p})=p u_{\mathfrak p},
\qquad u_{\mathfrak p}\text{ a signed rational }S\text{-unit}.
\end{equation}
For $F=\mathbf Q$ use the rational generator $p$. A \emph{simple vertex}
is an inert rational prime in the quadratic case, with rational generator
$p$. The fixed label of a split vertex is its Frobenius on $F_S^{(2)}$;
simple vertices have rational fixed-support labels. A \emph{type} may
additionally record the generators' local squareclasses, valuation parities,
real signs, and prescribed finite ray conditions at $S_F$. Only compatible,
individually realizable types will be prescribed. A \emph{filter} fixes the
total rational labels and sign of the signed-prime product
$\prod_p p^*$, where $p^*=(-1/p)p$.
For a pair of split vertices in the quadratic case there are two independent
forward bits: the residues of $\pi_{\mathfrak p}$ and
$\overline{\pi_{\mathfrak p}}$ at the chosen place of the other vertex.
Write quadratic residues additively in $\mathbf F_2$. Reverse bits are
determined by Hilbert reciprocity and the fixed types. Rational norm bits
are determined by summing the two oriented bits and the fixed correction
in \eqref{gr:norm}. When at least one endpoint is simple, or $F=\mathbf Q$,
there is one rational forward bit. Self-residues are never graph data.
An \emph{arithmetic graph} is this collection of vertices, types, and bits.
A split vertex is \emph{primary} when its generator is a square unit at
every finite place of $S_F$ and positive at all real places.
Different symbols may read compatible projections of the same graph, for
example its rational norm graph. Here compatibility means that the
projection forgets fixed-label coordinates or takes rational norm data,
carries primary vertices to primary vertices, and commutes with the
omitted-vertex product substitution. For the norm projection a single
oriented-bit toggle changes the rational bit once. We use these arithmetic
projections throughout.
\begin{lemma}[Finite realization and product labels]\label{gr:realization}
Fix finitely many old vertices, with all their mutual data, and
individually realizable new types. Every prescription of the independent
forward bits involving at least one new vertex has a realization by
distinct fresh primes, avoiding an arbitrary finite set. The old vertices
and their mutual data remain unchanged. The same holds
with further fixed local tests included in $S$. A contracted split cluster
may be represented by the product of its oriented generators, with
conjugations prescribed along its contraction tree; its support types and
external residue bits are the corresponding sums. A cluster containing
one simple vertex has the corresponding rational product data.
For a symbol satisfying the omitted-vertex dependence in
Proposition~\ref{co:symbol-interface}, its value depends only on the graph
data. Products in the preceding assertion may therefore be replaced by
fresh prime representatives without changing the prescribed contracted
symbol.
\end{lemma}
\begin{proof}
First note that the listed generators account for all the relevant radicals.
If $U$ is a finite set of new primes and a squareclass has even valuations
outside $S_F\cup U_F$, divide by the generators at the primes of $U_F$
where its valuation is odd. The result belongs to $V_S(F)$. At an inert
rational prime its rational generator has valuation one at the unique place
of $F$. Thus fixed-support labels and the displayed residue bits determine
every Frobenius test in an extension with this allowed ramification.
We describe the prime selection carefully. For a new split vertex take a
template $\pi_0$ of its prescribed type, with prime ideal
$\mathfrak p_0$ away from the old vertices. Weak approximation gives
$a\in F^\times$ which is sufficiently close to $1$ at $S_F$, positive at
the prescribed real places, and has any prescribed nonzero residues at the
old prime ideals. Choose those residues so that $a\pi_0$ has the desired
quadratic residues there. Use the ray modulus consisting of sufficiently
deep powers of the primes of $S_F$, all old prime ideals, and the real
places. Choose a prime ideal $\mathfrak q$ in the ray class of
$(a)\mathfrak p_0$. Then
\[
\mathfrak q=(a\beta)\mathfrak p_0,
\qquad \beta\equiv1\pmod{\text{ray modulus}},
\]
and $\pi_{\mathfrak q}=a\beta\pi_0$ is the required generator outside $S_F$.
There are infinitely many such degree-one prime ideals. Indeed, the
prime-ideal theorem in a ray class gives order $X/\log X$ prime ideals of
norm at most $X$, whereas prime ideals of residue degree greater than one
contribute only $O(X^{1/2})$ in a quadratic field. Degree one gives a split
rational prime, and finite exclusions remove only finitely many choices.
At an old inert prime with residue field $\mathbf F_{p^2}$, prescribe the
norm of the nonzero residue. The norm onto $\mathbf F_p^\times$ is
surjective, so either rational residue bit is possible. For a new simple
vertex the requirements are rational congruences, together with the inert
class in $F$. The Chinese remainder theorem and Dirichlet's theorem supply
fresh primes. Successively adjoining vertices proves simultaneous finite
realization. At every step reciprocity determines the reverse bits:
the local Hilbert symbols at the support are fixed, and the only other
contributions are the two endpoint residues. Taking norms gives exactly
\eqref{gr:norm}, so it imposes no additional independent bit.
The same argument applies to a product template. Its valuations and local
squareclasses at $S_F$ are those of the product, and its residues at the
remaining vertices multiply. The ray-class construction replaces that
template by a degree-one prime with those data. In a simple cluster we
instead use rational congruences for the norm labels. No residue at a prime
inside the cluster is imposed on its replacement.
In particular this preserves the rational filter, including its norm
correction. Write $\pi=\prod_i\pi_i$ for the oriented product template
and $\pi'=a\beta\pi$ for its fresh representative; $a\beta$ is a square unit at
$S_F$ and positive at its real places. The support valuations of $\pi'$
and $\pi$ agree exactly. Hence
\[
N\pi=(\prod_i p_i)u,\qquad N\pi'=q u
\]
with the same signed rational $S$-unit $u$: all its valuations and its
sign agree. The quotient $q/\prod_i p_i=N(a\beta)$ is a local square at
the rational support. Its squareclass at $2$ also identifies
$q^*$ with $\prod_i p_i^*$ at those fixed tests. Thus no free choice
of an untracked $S$-unit has entered a product replacement.
Finally, compare two disjoint realizations with identical graph data.
Introduce fresh bridge vertices with matching bits to both realizations,
and replace vertices one at a time. At each replacement the omitted-vertex
elementary Frobenius tests agree, by the generator description in the first
paragraph. The symbol interface gives equal values. A preliminary bridge
removes any overlap between the original realizations. The same reasoning
applies to the product replacements.
\end{proof}
Primary types are realized by the preceding construction, starting with the principal
ray class. Reciprocity makes their fixed radical labels trivial. Their
signed rational contributions are squares at the fixed rational tests.
If stationary vertices are also to have neutral incidences with an
auxiliary, include their primes among these local tests. In particular,
including the support of a fixed quadratic discriminant $k$ ensures that
every such primary auxiliary splits in $\mathbf Q(\sqrt{k})$.
\subsection{The simultaneous-testing statement}
Separate the given vertices into \emph{terminals} and, if present,
stationary external vertices. Terminals may be always present or may
have nonzero linear activations; external vertices have fixed types and
incidences throughout. At each nonzero address the symbol
is a function of the independent bits between the active terminals, with
those types and external data fixed. We call its unique multilinear
polynomial over $\mathbf F_2$ the \emph{terminal polynomial}. Its variables
describe formal changes of the terminal bits, while the prescribed prime
realization supplies a fixed evaluation point.
For a bit $z$ define the finite difference
$\Delta_z f=f|_{z=0}+f|_{z=1}$. On a multilinear polynomial it is the
formal partial derivative. In particular, differences commute and
$\Delta_z^2=0$.
\begin{definition}[Witnesses and permitted partitions]\label{gr:partitionable}
A \emph{witness} is a finite graph on which the
symbol is $1$. A graph is \emph{partitionable} for the symbol if its vertices
can be partitioned into nonempty clusters, each with at most one simple
vertex, with a choice of oriented contraction trees whose contracted graph
is a witness.
This definition includes a specified contraction certificate; it makes no
assertion about a Selmer group.
\end{definition}
\begin{theorem}[Address lemma]\label{gr:address}
Suppose the binary symbols under consideration depend only on the
arithmetic graph data and satisfy the contraction, whole-rational-prime
omission, and primary-isolation rules of
Proposition~\ref{co:symbol-interface}, with the realization freedoms
of Lemma~\ref{gr:realization}. Contraction includes the product labels
and external incidences, and the zero rule for two simple vertices.
These identities are required only on nonempty configurations and for
contractions with nonempty output.
Fix $b\ge1$ and either of the following terminal arrangements.
\begin{enumerate}
\item One symbol, a nonempty set of always active terminals, and a
nonzero polynomial in their independent pair bits.
\item One or two symbols on terminals with nonzero linear activations,
including at least one terminal for every nonzero linear form. At each
$x\ne0$ each symbol separately is a nonzero polynomial in the active
terminal bits. The symbols may read different compatible projections
of a common graph alphabet and may have fixed stationary external data.
\end{enumerate}
Suppose also that at each nonzero address the total active terminal data,
together with any stationary external data, obey the required filter.
Starting from any fixed realization of the terminal and stationary
vertices, one can adjoin finitely many fresh primary auxiliary vertices,
activated by nonzero linear forms, and choose only incidences involving
these new vertices so that every symbol equals $1$ at every $x\ne0$.
All data on the original vertices remain unchanged. No auxiliary is
present at $0$. All auxiliaries may satisfy the fixed splitting and
neutral external-incidence conditions incorporated into the primary type.
It is enough that a terminal polynomial have a nonzero evaluation or
a nonzero iterated edge difference. In particular a partitionability
certificate suffices for the stationary hypothesis.
\end{theorem}
The proof has two algebraic steps. Forests of auxiliary vertices act as
finite-difference operators on the original terminal polynomial. We then
build networks whose degree-one terms select a chosen derivative at a
single address. Applying them to top-degree coefficients will give the
simultaneous value $1$, independently of the fixed terminal bits.
\subsection{From arithmetic contraction to forest operators}
\begin{lemma}[Oriented forests]\label{gr:forest}
Assume the graph-symbol rules stated in Theorem~\ref{gr:address}.
Differentiation in a tree of graph
bits contracts the tree, adding product labels and external incidences;
its value is zero if the tree contains two simple vertices. Differentiation
in any collection of edge variables containing a cycle is zero, even when
different edges use incompatible relative orientations. Two different
oriented variables on the same pair count as a cycle of length two.
\end{lemma}
\begin{proof}
The one-edge assertion is precisely the arithmetic contraction rule,
with product replacements justified by Lemma~\ref{gr:realization}.
Induction proves the tree assertion: a split--split merger is split, a
split--simple merger is simple, and the first merger of two components
containing simple vertices has zero derivative. Along a tree there is
a unique choice of relative orientations once one root orientation is
fixed. Its product template depends on support types and external bits,
not on the internal bits being differentiated.
For clarity, consider first the two variables on a split pair $p,q$.
Differentiation in the inertia bit at one place over $p$ gives the symbol
formed with the \emph{entire rational prime $p$ omitted}. Its elementary
extension is unramified at both places over $p$. Differentiating in the
other oriented bit is therefore zero. Repetition of the same variable
is zero by $\Delta_z^2=0$. This argument would not follow from omission
of only one oriented prime ideal.
For a longer cycle, first contract a spanning path. By the product-label
rule the closing variable has become internal to the merged cluster.
Its change affects neither the product's support data nor any incidence
to a remaining vertex. Equivalently it is an inertia change at an omitted
endpoint, covered by the preceding unramifiedness argument. Thus its
derivative vanishes. Possible conjugation of the product generator only
changes the induced orientation and does not reintroduce that endpoint.
Since there is no self-residue among the inputs to the replacement symbol,
there is no extra closing-edge datum to retain. Additional derivatives
commute with this zero derivative.
For example, contract the aligned edge $12$ in a triangle. With
$\pi_{12}=\pi_1\pi_2$, the two bits toward vertex $3$ are
$t_0=z_{13}^0+z_{23}^0$ and $t_1=z_{13}^1+z_{23}^1$, up to fixed
type corrections. An aligned edge $23$ then differentiates $t_0$;
a conjugately aligned closing edge $31$ differentiates $t_1$.
Their mixed derivative is the omitted-endpoint zero just proved.
If the closing edge is aligned instead, both become the same
difference and its square is zero. Contracting a longer spanning
path reduces to exactly these two alternatives.
\end{proof}
Fix a realizable baseline terminal graph and let $P$ be its terminal
polynomial. A \emph{network} consists of additional primary
vertices and programmed edges, with no direct terminal--terminal edges.
All unprogrammed incidences of its primary vertices are neutral. Within
one network choose one orientation at each terminal and consistently use
the corresponding edge alignment. Networks may choose different
alignments. The \emph{degree} of a contraction is the number of terminal
vertices it removes by merging them.
\begin{lemma}[Network expansion and its first two degrees]\label{gr:network}
A network acts on $P$ by a sum of finite-difference operators. A surviving
term is a forest in which every component containing a primary vertex
meets a terminal. A component meeting $t$ terminals acts by a contraction
of degree $t-1$; a component containing two simple terminals is zero.
Networks on disjoint sets of internal vertices, with no edges between
those sets, multiply as operators on the same terminal polynomial.
Consequently an operator of degree greater than $\deg P$ annihilates $P$.
Let $Q$ be the grounded Laplacian on the internal vertices: $Q_{ij}$ for
$i\ne j$ records the internal edge $ij$, while $Q_{ii}$ is the total
incidence at $i$, including its terminal edges. Let $(c_a)_i$ record the
incidence from internal vertex $i$ to terminal $a$. All entries are in
$\mathbf F_2$. The constant and pair-contraction coefficients of the
operator are respectively
\begin{equation}\label{gr:forest-coefficients}
\det Q,\qquad c_a^{\mathsf t}\operatorname{adj}(Q)c_b\quad(a\ne b).
\end{equation}
The second coefficient multiplies the terminal difference in the alignment
selected by the network. A prohibited pair multiplies the zero operator.
\end{lemma}
\begin{proof}
Expand the evaluation at the programmed incidences from the graph with
all primary incidences zero. For a set $E_0$ of programmed bits this is
the elementary identity
\[
f(\mathbf1)=\sum_{T\subseteq E_0}(\Delta_T f)(\mathbf0).
\]
Lemma~\ref{gr:forest} kills cyclic $T$. A tree component meeting no
terminal contracts to an isolated primary vertex, and the isolation rule
kills it. This includes an untouched primary vertex. In any remaining
component all primary labels are trivial and all external primary bits
at the expansion baseline are zero. Its contracted label and its
incidences are exactly those obtained by contracting its terminal
vertices in the original graph. Choose any terminal tree inducing its
relative orientations; the corresponding terminal differences therefore
give the same value. Its degree is the number of lost terminal vertices,
namely $t-1$. This establishes the operator interpretation, rather than
merely a count of possible contraction degrees.
Expanding two disjoint networks at once gives the composition of these
operators: the product-label rule gives the same final labels in either
order, and any cycles created between networks vanish by
Lemma~\ref{gr:forest}. Ordinary polynomial differentiation now proves
the asserted degree bound.
This argument also explains the compatible projections. Taking norms
commutes with every oriented product in a forest; primary factors have
neutral norm corrections, and each programmed oriented toggle projects
to its one rational toggle. Thus the projected forest acts by the
corresponding differences on the projected \emph{original} terminal
polynomial at every degree. If projected differences repeat an edge
or merge two simple clusters, that term is zero. Forgetting fixed-label
coordinates plainly commutes with the same products. Each projected
network therefore acts on the original projected terminal polynomial.
The determinant counts are instances of the all-minors matrix-tree
theorem \cite[\S1]{Chaiken1982}. We give the Cauchy--Binet proof
in the form needed here, including singular $Q$. Let $I$
be the internal vertex set and $A$ the terminal set. Write the full
Laplacian as $L=DD^{\mathsf t}$, where a column of the incidence matrix
$D$ has ones at its edge's endpoints. Cauchy--Binet applied to
$L_{I,I}=Q$ counts edge sets whose incidence matrix with the terminal
rows deleted is invertible. Such an edge set is a forest with exactly
one terminal in each component: a cycle gives dependent columns; a
component with no deleted row gives dependent rows; and for a tree with
one row deleted, leaf removal proves invertibility. These are exactly
the degree-zero terms.
For the pair coefficient apply Cauchy--Binet with row set $I\cup\{a\}$
and column set $I\cup\{b\}$. Each counted forest has one root from
$A\setminus\{a\}$ and one from $A\setminus\{b\}$ in each component.
This requires that $a,b$ lie together and that every other terminal lie
in its own component. The corresponding Laplacian minor is
\[
\det\begin{pmatrix}Q&c_b\\c_a^{\mathsf t}&0\end{pmatrix}
=c_a^{\mathsf t}\operatorname{adj}(Q)c_b.
\]
The corner is zero because the network has no direct terminal edge;
signs disappear in characteristic two. The determinant identity uses
the adjugate and so remains valid when $Q$ is singular.
\end{proof}
\subsection{Two networks which select one address}
Fix a nonzero target $x_*$ and a terminal bit $z$ joining two terminals
$a_0,a_1$ active at $x_*$. In the
moving case assume that some terminal has each nonzero activation form;
in the stationary case all terminals are always present.
Choose a basis $\lambda_1,\ldots,\lambda_b$ of the dual space with
$\lambda_i(x_*)=1$ for every $i$. To obtain it, choose $\lambda_1(x_*)=1$
and replace a basis $\mu_2,\ldots,\mu_b$ of the annihilator of $x_*$ by
$\lambda_1+\mu_2,\ldots,\lambda_1+\mu_b$.
Make $b$ groups of three primary vertices, group $i$ activated by
$\lambda_i$. Since these forms are a basis, all groups are present
exactly at $x_*$. At every other address at least one group is absent.
In each physical three-dimensional vertex space choose
a basis $(e_i,u_i,v_i)$ with $e_i=(1,1,1)$. Thus the physical coordinate
change is block diagonal by groups.
Prescribe a symmetric bilinear form $Q$ by making the $e_i$ orthonormal
and orthogonal to all $u_j,v_j$, and putting on the remaining coordinates
\begin{equation}\label{gr:H}
H(T,B)=\begin{pmatrix}0&T\\T^{\mathsf t}&B\end{pmatrix}.
\end{equation}
Matrix entries in the physical coordinates are denoted by the same
letter $Q$. For a subset $J$ of active groups, $Q_J$ means its physical
principal restriction. Define the column $h_i=Qe_i$.
It is supported on group $i$: in adapted coordinates it is the covector
dual to $e_i$, and the coordinate change does not mix groups. Its value
is independent of $T$ and $B$.
Attach $h_i$ to an anchor terminal with activation $\lambda_i$. In the
stationary case attach it to any fixed terminal instead. The support
property gives, on every address,
\[
\sum_{i\in J}h_i|_J=Q_J\mathbf1_J.
\]
Program internal edges using the off-diagonal
physical entries of $Q$. A matrix is their grounded Laplacian precisely
when the sum of the terminal incidence columns is its row-sum vector,
so these anchor columns give exactly $Q_J$, including its diagonal.
We shall also attach the same additional column $c$ to $a_0$ and $a_1$.
In the moving case with distinct activation forms $\alpha,\beta$, attach
$c$ also to a terminal of activation $\alpha+\beta$; this terminal is
inactive at $x_*$. If $\alpha=\beta$, no third column is needed. The
sum of the additional active columns is zero at every address, since
$\alpha+\beta+(\alpha+\beta)=0$. In the stationary case the two columns
cancel directly. Coincident terminal roles mean addition of their
columns. This preserves the prescribed grounded matrix after every
restriction.
For a concrete instance, take $b=1$, target $x_*=1$, and two always-active
split terminals $a_0,a_1$ with selected bit $z$. Label the three primary
vertices $1,2,3$, all activated by $x$. Choose
$e=(1,1,1)^{\mathsf t}$, $u=(1,0,0)^{\mathsf t}$,
$v=(0,1,0)^{\mathsf t}$, and $T=(0)$, $B=(1)$.
The form in this adapted basis is $\operatorname{diag}(1,0,1)$.
In physical coordinates $h=Qe=(0,0,1)^{\mathsf t}$ and the covector
dual to $u$ is $c=(1,0,1)^{\mathsf t}$. Using $a_0$ as the anchor gives
\[
Q=\begin{pmatrix}0&0&0\\0&1&1\\0&1&0\end{pmatrix},\qquad
c_{a_0}=h+c=\begin{pmatrix}1\\0\\0\end{pmatrix},\qquad
c_{a_1}=c=\begin{pmatrix}1\\0\\1\end{pmatrix}.
\]
Thus the programmed graph is the path
$a_0\! -\! 1\! -\! a_1\! -\! 3\! -\! 2$.
Here $\det Q=0$ and $\operatorname{adj}(Q)=uu^{\mathsf t}$, so
$c_{a_0}^{\mathsf t}\operatorname{adj}(Q)c_{a_1}=1$.
There are only two terminals, so no higher contraction degree occurs:
the network acts exactly by $\Delta_z$ at $1$. At $0$ all three primary
vertices are absent, and the network acts by the identity.
Write $O_{\ge d}$ for a sum of contraction operators of degrees at least
$d$; it is a filtration notation, not an estimate.
\begin{lemma}[The two address programs]\label{gr:programs}
With the preceding grounding convention one has the following programs
for every $b\ge1$.
\begin{enumerate}
\item A network $\mathcal S$ with
\[
\mathcal S_{x_*}=\Delta_z+O_{\ge2},\qquad
\mathcal S_x=1+O_{\ge1}\quad(x\ne x_*).
\]
\item Two networks on the same activated sites, indexed by
$\varepsilon\in\mathbf F_2$, with
\[
\mathcal N_x^{(\varepsilon)}=1+O_{\ge1},\qquad
\mathcal N_x^{(1)}+\mathcal N_x^{(0)}
=\begin{cases}\Delta_z+O_{\ge2},&x=x_*,\\
O_{\ge2},&x\ne x_*.
\end{cases}
\]
\end{enumerate}
All nonzero-activation sites are absent at $0$. Unspecified incidences,
including those to stationary external vertices, may remain neutral.
\end{lemma}
\begin{proof}
For the first program take $T=I+P$, where $P$ is the cyclic permutation
matrix on $b$ coordinates, and $B$ diagonal with exactly one nonzero
entry. A proper principal group restriction of $P$ is a union of
directed paths, hence nilpotent. Thus $T_J$ is invertible for every
proper $J$, and so is $H(T_J,B_J)$: its kernel equations first give
$v=0$ and then $u=0$. This includes the empty restriction, whose
determinant is $1$.
On the full group set, both $\ker T$ and $\ker T^{\mathsf t}$ are
spanned by the all-one vector. A radical vector of $H$ has
$v=t\mathbf1$; summing the equation $T^{\mathsf t}u+Bv=0$ gives $t=0$
because $\mathbf1^{\mathsf t}B\mathbf1=1$. Thus the full $Q$ has
one-dimensional radical spanned by $r=\sum_i u_i$. In characteristic
two its adjugate is $rr^{\mathsf t}$. Indeed a rank-$n-1$ matrix has a
nonzero rank-one adjugate with image its kernel; symmetry identifies
the two factors, and the only nonzero scalar is $1$.
Choose $c$ to be the covector dual to $u_1$, so $c^{\mathsf t}r=1$.
All anchors pair to zero with $r$. At $x_*$ the only columns pairing
nontrivially with $r$ are consequently those of $a_0,a_1$. Formula
\eqref{gr:forest-coefficients} gives zero constant and exactly
$\Delta_z$ in degree one. At every other address the group subset is
proper and the constant is $1$. When $b=1$, $I+P=0$ and $B=(1)$,
so the same radical calculation applies without an exception.
For the second program let $T=I+L$, where $L$ is the consecutive lower
shift, and take $B_\varepsilon=\varepsilon E_{bb}$. Every group
restriction of $T$ is invertible, and block multiplication gives
\begin{equation}\label{gr:inverse}
H(T,B)^{-1}=
\begin{pmatrix}T^{-\mathsf t}BT^{-1}&T^{-\mathsf t}\\
T^{-1}&0\end{pmatrix}.
\end{equation}
Again take $c$ dual to $u_1$. On a group subset $J$, the change of
$c^{\mathsf t}Q_J^{-1}c$ is
\[
\bigl((T_J^{-1})_{b1}\bigr)^2,
\]
interpreted as zero if $1$ or $b$ is absent. Since
$(I+L_J)^{-1}=I+L_J+L_J^2+\cdots$, that entry is $1$ precisely when
the entire path $1,2,\ldots,b$ is present. This is exactly the target
address. Pairings involving an anchor do not change:
$Q_J^{-1}h_i=e_i$, and the vectors $e_i,h_i,c$ do not depend on
$\varepsilon$. At the target the balancing terminal is absent, so
the only changed pair coefficient is the one for $a_0,a_1$.
Every constant is $1$. Lemma~\ref{gr:network} proves the claims,
including $b=1$, when the path consists of its single vertex.
\end{proof}
\subsection{Proof of the address lemma}
\begin{proof}[Proof of Theorem~\ref{gr:address}]
Use the prescribed realization of the terminal and stationary vertices
as a common baseline for all terminal bits. Regard
their toggles as common variables, omitting variables invisible to a
given symbol. A nonzero iterated difference implies that the polynomial
is nonzero. Conversely, the unique multilinear polynomial of a
nonzero function has a top monomial with nonzero coefficient. These
observations also explain the last assertion using
Lemma~\ref{gr:forest}.
Every nonzero address has an active terminal, and contractions never
remove the last vertex. Thus only nonempty configurations are used.
First consider one symbol. At each $x\ne0$ let $P_x$ have degree $d_x$
and choose a monomial with coefficient $1$ in its top homogeneous part.
For each variable of this monomial add a separate singular program
$\mathcal S$ targeted at $x$. Give all these networks disjoint internal
vertices. At address $x$, its own $d_x$ networks each have zero constant,
so every surviving term spends at least $d_x$ contraction degrees.
An extra degree from any network kills $P_x$. Thus only the designated
degree-one contractions of its own networks and the constant terms
of all other networks survive. Their value is the chosen top
coefficient, namely $1$. A nonzero constant polynomial needs no
network and already has value $1$. This argument applies equally to
always active terminals, since Lemma~\ref{gr:programs} supplies their
stationary anchors.
For two symbols write their polynomials at address $x\ne0$ as
$P_x,R_x$, of degrees $d_x,e_x$, and let $P_x^{\rm top},R_x^{\rm top}$
be their top homogeneous parts. Multiply these parts in the ordinary
polynomial ring
$\mathbf F_2[z_1,\ldots,z_m]$, \emph{before} imposing any Boolean
relations. That ring is an integral domain, so choose a coefficient
equal to $1$ at a monomial
\begin{equation}\label{gr:chosen-monomial}
M_x=\prod_z z^{\nu_{xz}},\qquad
\nu_{xz}\in\{0,1,2\},\qquad
\sum_z\nu_{xz}=d_x+e_x.
\end{equation}
For $\nu_{xz}=2$ add one singular program for $z$ targeted at $x$.
For $\nu_{xz}=1$ add one nonsingular program for $z$ targeted at $x$,
with its own choice $\varepsilon_{xz}\in\mathbf F_2$. All internal
vertices of different programs are disjoint. Let
\[
\Phi(\varepsilon)=\prod_{x\ne0}
\bigl(\text{final evaluation of }P_x\bigr)
\bigl(\text{final evaluation of }R_x\bigr).
\]
We prove that the sum of $\Phi$ over all choices of $\varepsilon$ is $1$.
This proves that some single choice has all the required evaluations
equal to $1$.
Expand every program in contraction degree using
Lemma~\ref{gr:network}. A singular program targeted at $x$ has zero
constant in both factors there, so it costs at least one degree in
$P_x$ and one in $R_x$. A nonsingular program has constant $1$
independent of its choice. In the sum over that choice, a term using
only its constants in all factors cancels. Every remaining term must
therefore spend at least one degree somewhere for this program. The
sum of these minimum costs is exactly
\begin{equation}\label{gr:budget}
D=\sum_{x\ne0}(d_x+e_x),
\end{equation}
by \eqref{gr:chosen-monomial}. A nonzero term cannot exceed this total:
if it did, some factor would receive more differences than its degree.
Hence every surviving term uses every minimum cost exactly.
It follows that a singular program spends exactly one degree in each
factor at its target and zero elsewhere. Its degree-one term there
is precisely $\Delta_z$. Each nonsingular program spends exactly one
degree in exactly one factor. After summing its choice, that
degree-one contribution is zero away from its target, and is
$\Delta_z$ at its target. All other occurrences of these programs
contribute their common constant $1$. Thus no unknown higher forest
coefficient enters the surviving calculation.
At a fixed address, let $Z_2$ be the variables with exponent two and
$Z_1$ those with exponent one. The local allocation sum is
\[
\sum_{J\subseteq Z_1}
[z_{Z_2}z_J]P_x^{\rm top}\,
[z_{Z_2}z_{Z_1\setminus J}]R_x^{\rm top}
=[M_x](P_x^{\rm top}R_x^{\rm top})=1.
\]
Coefficients of the wrong degree in this display are zero. Every
factor must in fact receive its full degree, since the total is $D$
and none can exceed its own degree. Therefore its final derivative
is exactly the displayed top coefficient, independent of the chosen
baseline values. Allocations at different addresses are independent,
so their product is $1$, as asserted. This also covers a constant
factor, whose degree is zero. For example, when both top forms are
$z$, their ordinary product is $z^2$ and the singular program
differentiates each factor once; Boolean reduction would lose this
allocation.
In the two-symbol case, choose the program parameters for which
$\Phi=1$. In either case, apply Lemma~\ref{gr:realization} to adjoin
the resulting finite networks to
the fixed original vertices. Only new vertices and incidences involving
them have been prescribed. The primary types preserve the filters.
All activations are nonzero linear forms,
so no new vertex appears at $0$, and the stipulated neutral and
splitting conditions persist.
\end{proof}
The theorem is finite for each $b$; it does not assert a single infinite
array of primes. For a stabilized symbol, first choose $b$, its finite
terminal data, and all the networks above. Their total weight and the
list of coefficient tests are then finite. A sufficiently advanced
stage at which the symbol interface holds for that weight range
realizes the construction. Required precision and the number of
auxiliaries may grow with $b$. The arithmetic valuation constant
attached to a unit symbol must still be supplied by the coefficient
construction independently of $b$; no determinant or forest count in
this section proves that uniformity.
==== END A-pointwise-2-converse-for-elliptic-curves-with-rational-two-torsion//graphs.tex ====
==== BEGIN A-pointwise-2-converse-for-elliptic-curves-with-rational-two-torsion//introduction.tex ====
\section{Introduction}\label{sec:introduction}
The Birch--Swinnerton-Dyer conjecture predicts that the order of vanishing
of the $L$-function of an elliptic curve equals its Mordell--Weil rank,
and that its Shafarevich--Tate group is finite; its refined form also
specifies the leading coefficient \cite{BirchSwinnertonDyer1965,Tate1966}.
For analytic order zero or one, the rank equality and finiteness are
theorems of Gross--Zagier and Kolyvagin
\cite{GrossZagier1986,Kolyvagin1990}.
The converse problem begins with algebraic information and seeks the
corresponding analytic order.
The appropriate algebraic hypothesis here is the corank of the full
$2$-power Selmer group. It records a possible divisible contribution
from the Shafarevich--Tate group as well as the Mordell--Weil rank.
With the usual local Kummer conditions, put
\[
s_2(E)=\corank_{\Z_2}\Sel_{2^\infty}(E/\Q).
\]
Our main result is the following pointwise statement.
\begin{theorem}[The rational-two-torsion $2$-converse]\label{thm:main}
Let $E/\Q$ be an elliptic curve with $E(\Q)[2]\ne0$ and
$s_2(E)\in\{0,1\}$. Then
\[
\ord_{s=1}L(E,s)=\rank_{\Z}E(\Q)=s_2(E),
\qquad
\Sha(E/\Q)\text{ is finite}.
\]
\end{theorem}
The statement includes one rational two-torsion line and full rational
two-torsion, CM and non-CM curves, every rational isogeny configuration,
and every reduction type at $2$ and at the bad primes. It is a
$2$-converse in this residual range. It does not assert the full
Birch--Swinnerton-Dyer leading-coefficient formula, nor does a bound
$\rank E(\Q)\le1$ alone imply its Selmer hypothesis.
The broader companion \cite[Theorem~1.1]{OpenAIGoldfeldTwoConverse2026}
proves the $2$-converse without the rational-two-torsion hypothesis.
Here we give a separate full treatment of the rational-two-torsion case,
developing its integral interpolation and finite twist constructions
within this paper; the broader converse is not an input to the proof.
The converse with the stronger hypotheses
$\rank E(\Q)=1$ and $\Sha(E/\Q)$ finite is known for every
elliptic curve over $\Q$; see \cite[\S1]{Kim2023SoftConverse}.
The fixed-prime problem asks instead whether a low Selmer corank
alone forces the analytic order. Skinner and Wei Zhang established
general rank-one converses at good ordinary odd primes under residual
and local hypotheses \cite{Skinner2020,Zhang2014Heegner}.
For odd Eisenstein primes, Castella, Grossi, Lee and Skinner prove
the good-reduction converse under a local-character restriction
\cite[Theorem~E]{CGLS2022}; Keller and Yin treat potentially good
ordinary reduction without that restriction
\cite[Theorem~B]{KellerYin2024}. Smith's theorem gives the $50/50$
distribution of $2^\infty$-Selmer coranks in every fixed quadratic-twist
family, counting twist parameters of both signs \cite{Smith2025};
this density statement does not decide a prescribed member.
The CM case includes pointwise results at the prime $2$.
Burungale and Tian prove the rank-zero $p$-converse for CM curves at
every prime \cite{BurungaleTian2026}. In rank one, Burungale and
Skinner treat good ordinary primes and, with a localization hypothesis,
good supersingular primes
\cite[Theorems~A.1 and~A.3, Remark~A.5]{BurungaleSkinner2024Appendix}.
The earlier ordinary CM converse of Burungale, Castella, Skinner and
Tian retains a Hecke-conductor condition
\cite{BurungaleCastellaSkinnerTian2022}.
Kriz's preprint gives rank-zero and rank-one converses for maximal-order
CM curves at primes ramified in the CM field, including the pointwise
$2$-converse for the congruent-number family
\cite[Theorems~1.1 and~10.17(2)]{Kriz2022CM}.
These results already treat individual curves in their stated ranges;
the theorem here includes non-CM curves with rational two-torsion and
imposes no reduction condition at $2$.
The proof brings together two arithmetic interpolation constructions and a
finite construction with quadratic twists. Kato's zeta classes provide
the cyclotomic input \cite{Kato2004}; Ferrero--Washington's
vanishing theorem controls the residual cyclotomic cohomology
\cite{FerreroWashington1979}. Heegner points and their derivative
classes provide the ring-class input, following the Euler-system method
of Kolyvagin and its cohomological formulation by Howard
\cite{Kolyvagin1990,Howard2012}. We retain integral local conditions and
duality maps in the framework of Selmer complexes \cite{Nekovar2006}.
The coefficient tests use the half-integral weight theory of Shimura
and Waldspurger's relation between coefficients and central values
\cite{Shimura1973HalfIntegral,Waldspurger1981}. Their ternary theta
construction lies in the definite quaternion framework developed by
Gross \cite{Gross1987Heights}; we use the Weil representation and the
reduction description of Jetchev and Kane for the weighted Heegner-point
tests \cite{Weil1964,JetchevKane2010}.
Each of these inputs has a precise role; none supplies the pointwise
conclusion by itself.
Here is the common interpolation problem. Fix the curve that is to be
tested, and index a finite family of quadratic twists by
$x\in\mathbb F_2^b$, with $x=0$ denoting that curve. Each auxiliary
prime divides the twist parameter according to a linear form in $x$.
We construct these families so that every nonzero address has the
required analytic order and a uniform normalized valuation bound.
An integral interpolation argument then recovers the value or derivative
at the missing address $0$. The number and sizes of the auxiliary
primes may grow with $b$; the valuation bound may not.
For Selmer corank zero, the required nonzero values are central
$L$-values. Their normalized valuations are measured by coefficients
of half-integral weight forms. The cyclotomic determinant makes these
values part of one integral family, and a congruence on a sufficiently
large binary cube recovers the missing value. This also supplies the
even estimate needed to normalize the rank-one construction.
The rank-one construction requires a further preparation. Fix a negative
quadratic discriminant $h_*$ with $L(E^{(h_*)},1)\ne0$.
For positive $h$ in the prescribed local squareclasses, the
genus-character Heegner sum $P_h$ attached to $\Q(\sqrt{hh_*})$
detects a simple zero of $E^{(h)}$: its height measures
$L'(E^{(h)},1)L(E^{(h_*)},1)$. To normalize the corresponding coefficient tests
uniformly, we first bound the $2$-divisibility of $P_h$ from below
in terms of the prime factors of $h$, with a constant independent
of their number.
For each parameter $h$ of known analytic rank one, a different negative companion
$k'(h)$, with a bounded number of prime factors, combines the even
estimate with a ring-class lower bound to give this uniform bound.
Only after that argument do we normalize the odd coefficients and use
their minimum-depth values as binary nonvanishing tests.
For the return to a prescribed corank-one curve, choose one final
negative discriminant $k$ with $L(E^{(k)},1)\ne0$, and fix
$K=\Q(\sqrt{k})$. This companion does not vary with $h$ or with the
cube dimension. The quadratic corank identity gives $s_2(E/K)=1$.
We construct families with every new prime split in $K$ for which,
at every nonzero address, $E^{(h)}$ has a simple zero and $E^{(hk)}$
has nonzero central value. The two coefficient tests together bound
the normalized index of the ring-class Heegner sum over $K$.
Ring-class interpolation then recovers the derivative at $h=1$.
Thus $h_*$ defines the detector, $k'(h)$ proves its lower bound,
and the fixed $k$ permits the final transfer.
The arithmetic difficulty is uniform integral control at the prime $2$.
Residual irreducibility is unavailable here. On the cyclotomic side we
use Kato classes from one fixed modular-form lattice, separating rational
divisibility from uniform integrality. On the ring-class side we retain
the full integral Kummer conditions at the bad places and actual maps
from finite free models to Galois cochains. Heegner derivative operations
preserve the determinant valuation even when a localized class is
divisible by $2$. These constructions give the missing-vertex
Theorems~\ref{cy:missing-vertex} and~\ref{rd:missing-vertex}.
The finite construction has a different task: it must make the required
coefficient tests nonzero at every nonzero address simultaneously.
The variables are quadratic residue symbols between auxiliary primes.
Changing one such symbol contracts two prime labels; repeated changes
are governed by forests. Finite networks of new primes then isolate
one address at a time. For rank one, the derivative test and its
central-value partner may be separately nonzero without taking nonzero
values on the same assignment of residue symbols. Theorem~\ref{gr:address}
adjoins prime networks making both tests equal to $1$ at every nonzero
address.
Figure~\ref{fig:proof-roadmap} records the order in which the valuation
bounds are obtained and used. Section~\ref{sec:preliminaries} fixes the
Selmer and local conventions. Section~\ref{cy:section} proves the
cyclotomic interpolation statements; Sections~\ref{rl:section}
and~\ref{rd:section} construct the ring-class deformation and prove its
bounds and transfer theorem. Section~\ref{sec:nonvanishing} supplies
auxiliary twists with prescribed local behavior and a bounded number of
prime factors where needed. Sections~\ref{co:section}
and~\ref{gr:section} give the coefficient tests and finite networks.
Section~\ref{pw:section} assembles these results at the prescribed curve.
\input{sections/proof-roadmap-figure}
==== END A-pointwise-2-converse-for-elliptic-curves-with-rational-two-torsion//introduction.tex ====
==== BEGIN A-pointwise-2-converse-for-elliptic-curves-with-rational-two-torsion//nonvanishing.tex ====
\section{Auxiliary twists with prescribed local behavior}\label{sec:nonvanishing}
We need auxiliary twists with specified local behavior and nonzero
central value or derivative. For a fixed witness, no bound on the prime
count is needed. When the curve itself varies, we also need central-value
witnesses with a uniformly bounded number of prime factors. In both cases
their primes can be chosen beyond any previously specified finite set.
\begin{proposition}[Local witnesses]\label{nv:local-witness}
Fix an elliptic curve $A/\Q$ and a nonempty filter of fundamental
discriminants prime to $2N_A$, with specified real sign and unit
squareclasses at finitely many primes. If the root number in the filter
is $+1$, it contains $d$ of arbitrarily large absolute value with
$L(A^{(d)},1)\ne0$.
If the root number is $-1$, it contains such $d$ with
$L'(A^{(d)},1)\ne0$. Any additional finite set of primes may be excluded.
\end{proposition}
\begin{proposition}[Bounded-factor completion]\label{nv:rough}
There is an absolute integer $A_0$ with the following property.
In the positive-root-number case of Proposition~\ref{nv:local-witness},
one can require $\omega(d)\le A_0$ and require every prime divisor of
$d$ to exceed any fixed bound.
The size threshold for the resulting $d$ may depend on the curve, the
filter, and the excluded primes; $A_0$ does not.
\end{proposition}
The use of a nonnegative first moment and a lower linear sieve to obtain
bounded-factor twists goes back to Hoffstein--Luo
\cite[\S2]{HoffsteinLuo1997}. We retain an arbitrary fixed progression
and track an absolute factor-count bound without optimizing it.
The central moment follows the model of
\cite[Proposition~2]{RadziwillSoundararajan2015}, which has a sharper
divisor error; the derivative calculation adapts the first-moment
argument of \cite{Iwaniec1990}. The enlarged modulus and the absolute
exponents needed here are retained in the proof below.
A nonzero first moment alone supplies the derivative witness. For the
central-value completion, positivity allows us to sieve the first
moment, so we also need an error term for each prescribed squarefree
divisor of the twist parameter.
Write $f$ for the weight-two newform of $A$, normalized so that its
central point is $1/2$, and write $\lambda(n)$ for its normalized Hecke
eigenvalues. Thus
\[
L(s,f\otimes\chi)=\sum_{n\ge1}\lambda(n)\chi(n)n^{-s},
\qquad |\lambda(n)|\le d(n).
\]
Choose $H$ divisible by $8N_A$, a reduced residue class $a\bmod H$, and
$\sigma\in\{1,-1\}$ such that $\sigma a\equiv1\bmod4$.
The required twists have discriminant $\sigma n$ with
$n>0$ squarefree and $n\equiv a\bmod H$. A filter is a finite union of
such progressions, so it suffices to use one of them. Fix a nonnegative
smooth nonzero $\Phi$ supported in $(1,2)$.
\begin{lemma}[A fixed-progression moment]\label{nv:moment}
There are absolute constants $\eta>0$ and $B>0$ such that the
following assertions hold. For squarefree $b$ prime to $H$, put
\[
S_{j,b}(X)=
\sum_{\substack{n>0\ {\rm squarefree}\\ n\equiv a\ (H)\\b\mid n}}
L^{(j)}(1/2,f\otimes\chi_{\sigma n})\Phi(n/X).
\]
If the common root number is $+1$, then
\[
S_{0,b}(X)=cX\prod_{p\mid b}\beta_p+
O_{f,H,\Phi}(b^B X^{1-\eta}),
\qquad
c>0,\quad 0<\beta_p<1,\quad
\beta_p=\frac1p+O(p^{-2}).
\]
The last implied constant is absolute. If the common root number is
$-1$, then
\[
S_{1,1}(X)=cX\log X+c_1X+
O_{f,H,\Phi}(X^{1-\eta}),\qquad c>0.
\]
\end{lemma}
\begin{proof}
We first control the square-divisor tail, then use Poisson summation to
separate a positive main term from a power-saving error. Tracking the
dependence on the imposed divisor $b$ will give the absolute exponents.
Let $G(z)=e^{z^2}$ and
\[
V_j(y)=\frac1{2\pi i}\int_{(2)}
G(z)\Gamma(1+z)y^{-z}\frac{dz}{z^{j+1}}.
\]
Writing $Q_A=\sqrt{N_A}/(2\pi)$, the symmetric approximate functional
equation in the required sign is
\begin{equation}\label{nv:afe}
L^{(j)}(1/2,f\otimes\chi_{\sigma n})
=2\sum_{u\ge1}\frac{\lambda(u)\chi_{\sigma n}(u)}{\sqrt u}
V_j\!\left(\frac{u}{Q_A n}\right).
\end{equation}
For $j=1$, differentiating the completion contributes no central-value
term, because that value is zero.
\par\smallskip\noindent\emph{Square-divisor truncation.}
The quadratic large sieve
\cite[Theorem~1 and Corollary~1]{HeathBrown1995}, with the approximate functional equation,
gives for fixed $\delta>0$
\begin{equation}\label{nv:second}
\sum_{\substack{0<t\le Z\ {\rm squarefree}\\
\sigma t\ {\rm fundamental}\\(t,2N_A)=1}}
|L(1/2+\delta+i\tau,f\otimes\chi_{\sigma t})|^2
\ll_{f,\rho,\delta} Z^{1+\rho}(1+|\tau|)^{C_\rho}.
\end{equation}
Here the polynomial exponent can be chosen independently of the level.
To see that square indices cause no extra power, write each coefficient
index as $ur^2$ with $u$ squarefree. Minkowski's inequality and the
quadratic large sieve bound the square root of the moment of a polynomial
of length $U$ by
\[
(ZU)^{O(\rho')}
\sum_{r\le\sqrt U}\frac{\sqrt{Z+U/r^2}}r
\ll (ZU)^{O(\rho')}(\sqrt Z+\sqrt U).
\]
Coprimality restrictions at $r$ are discarded only after taking absolute
squares. To include both parts of the approximate functional equation,
work first with $t\asymp T$. Their lengths are
$U\ll_f T(1+|\tau|)^C$, with rapidly decreasing tails.
The main coefficients have the additional factor $n^{-\delta}$,
which improves the partial-summation bound. The dual coefficients
have $n^{\delta}$ and lose at most $U^{\delta}$, whereas their
functional-equation multiplier contributes $T^{-2\delta}$, up to
a fixed power of $1+|\tau|$. The resulting factor is at most
$T^{-\delta}(1+|\tau|)^{C_\delta}$ and causes no positive power
of $T$. Mellin inversion separates the smooth $t$-weights; the
gamma factors control the resulting integrals and tails.
Taking the auxiliary large-sieve exponents sufficiently small and
summing dyadic intervals proves Equation~\eqref{nv:second}.
Let $\mathcal A_j(n)$ denote the right side of
Equation~\eqref{nv:afe}, also for nonsquarefree $n\equiv a\bmod H$.
Expanding $\mu^2(n)$ and retaining divisors $d\le Y$ leaves an error
\begin{equation}\label{nv:tail}
O_{f,H,\Phi,\rho}(X^{1+\rho}/Y).
\end{equation}
Indeed write $n=tv^2$, with $t$ squarefree. The omitted terms have $v>Y$,
and their multiplicity is at most the divisor count of $v$.
Let $L^{(v)}(s,f\otimes\chi_{\sigma t})$ be the primitive-twist
$L$-function with its Euler factors at primes dividing $v$ removed.
For $0<\delta_{\mathrm M}<\rho/8$, Mellin inversion gives
\[
\mathcal A_j(tv^2)=\frac{2}{2\pi i}
\int_{(\delta_{\mathrm M})}
G(z)\Gamma(1+z)(Q_Atv^2)^z
L^{(v)}(1/2+z,f\otimes\chi_{\sigma t})
\frac{dz}{z^{j+1}}.
\]
Choose the large-sieve and divisor exponents smaller still.
On this line the removed Euler
factors cost an arbitrarily small fixed power of $v$. Cauchy's
inequality and Equation~\eqref{nv:second} give, for suitably small
$\rho'>0$,
\[
\sum_{t\le2X/v^2}^{\!*}
|L(1/2+\delta_{\mathrm M}+i\tau,f\otimes\chi_{\sigma t})|
\ll_{f,\rho',\delta_{\mathrm M}}
(X/v^2)^{1+\rho'}(1+|\tau|)^C,
\]
where the star retains the primitive discriminant conditions.
The gamma factors control the integral. The divisor count, the removed
Euler factors, and $X^{\delta_{\mathrm M}}$ are absorbed in $X^\rho$,
leaving $X^{1+\rho}\sum_{v>Y}v^{-2}\ll X^{1+\rho}/Y$.
The factor $z^{-2}$ changes only the constant.
\par\smallskip\noindent\emph{Poisson summation and the main term.}
Only $(d,H)=1$ occurs, because $n\equiv a\bmod H$ is a reduced
class. For $d\le Y$ put $J=[d^2,b]$ and
$c_J\equiv aJ^{-1}\bmod H$, which is therefore defined.
Write $u=u_1q$ with $u_1\mid H^\infty$ and $(q,H)=1$, and put
$\kappa(u_1)=\chi_{\sigma a}(u_1)$. Define
\[
G_\nu(q)=\sum_{y\bmod q}\left(\frac yq\right)e(\nu y/q),
\quad
W_j(z,R)=\int_0^\infty
\Phi(x)V_j\!\left(\frac z{Q_Ax}\right)e(-Rx)\,dx,
\]
where $e(x)=e^{2\pi ix}$. Put
\[
T_j(J)=\sum_{\substack{n>0,\ n\equiv a\ (H)\\J\mid n}}
\mathcal A_j(n)\Phi(n/X).
\]
Poisson summation in the fixed progression gives the exact identity
\begin{align}\label{nv:poisson}
T_j(J)
&=\frac{2X}{HJ}
\sum_{u_1\mid H^\infty}\frac{\lambda(u_1)\kappa(u_1)}{\sqrt{u_1}}
\sum_{(q,H)=1}\frac{\lambda(q)}{q^{3/2}}
\left(\frac{\sigma JH}{q}\right) \\
&\hspace{14mm}\cdot
\sum_{\nu\in\Z}G_\nu(q)e(\nu c_J\bar q/H)
W_j\!\left(\frac{u_1q}{X},\frac{\nu X}{HJq}\right).
\nonumber
\end{align}
Here $q\bar q\equiv1\bmod H$. The arithmetic factor follows directly by
applying the Chinese remainder theorem to
\[
\sum_{\substack{x\bmod Hq\\x\equiv c_J\ (H)}}
\left(\frac xq\right)e(\nu x/(Hq))
=e(\nu c_J\bar q/H)\left(\frac Hq\right)G_\nu(q).
\]
When $(q,J)>1$, the additional Jacobi symbol is zero.
At zero frequency only square $q$ occur. Put
\[
H_p(z)=1+(1-p^{-1})
\sum_{r\ge1}\lambda(p^{2r})p^{-r(1+2z)},\qquad H_p=H_p(0),
\]
\[
L_H(z)=\sum_{u_1\mid H^\infty}
\lambda(u_1)\kappa(u_1)u_1^{-1/2-z},
\qquad F_J(z)=\prod_{p\nmid HJ}H_p(z).
\]
The zero-frequency term is
\[
\frac{2X}{HJ}\frac1{2\pi i}\int_{(2)}
G(z)\Gamma(1+z)(Q_AX)^z\widehat\Phi_0(z)L_H(z)F_J(z)
\frac{dz}{z^{j+1}},
\quad
\widehat\Phi_0(z)=\int_0^\infty\Phi(x)x^z\,dx .
\]
For the Satake parameters $\alpha_p,\alpha_p^{-1}$, with
$|\alpha_p|=1$, we have
\[
\sum_{r\ge0}\lambda(p^{2r})x^r
=\frac{1+x}{(1-\alpha_p^2x)(1-\alpha_p^{-2}x)}.
\]
Thus $F_J$ factors through $L(1+2z,\operatorname{sym}^2f)$ and a
correction product convergent near $\Re z\ge-1/16$. Omitting a prime
dividing $J$ removes its correction factor and inserts an inverse
symmetric-square Euler polynomial; it creates no pole even when
$H_p(z)$ has zeros. The omitted factors cost at most $J^{B_1}$ for an
absolute $B_1$. Symmetric-square continuation, including the CM case
\cite[Theorem~9.3 and Remark~9.9]{GelbartJacquet1978}, and the
polynomial strip bounds from its functional equation therefore give an error
$O_{f,H,\Phi}(J^{B_1}X^{1-1/16})$ when shifting to $\Re z=-1/16$.
The residues have a positive leading coefficient. Indeed
\[
F_p^{\rm ev}=\sum_{r\ge0}\lambda(p^{2r})p^{-r}
=\frac{1+p^{-1}}{(1-\alpha_p^2/p)(1-\alpha_p^{-2}/p)}>0,\quad
H_p=p^{-1}+(1-p^{-1})F_p^{\rm ev}>p^{-1}.
\]
The fixed Euler factors are positive at the center, and
$L(1,\operatorname{sym}^2f)>0$ by the Rankin--Selberg residue formula.
This also holds for CM forms. Summing the residue over the square
divisor gives, at $p\nmid Hb$, the factor $H_p-p^{-2}$, and at
$p\mid b$ the factor $p^{-1}-p^{-2}$. Consequently
\[
\beta_p=\frac{p-1}{p^2H_p-1}
=\frac1{1+pF_p^{\rm ev}}
=p^{-1}+O(p^{-2}).
\]
For $j=1$, the double pole gives $cX\log X+c_1X$ with the same
positive leading coefficient. Extending the residue sum from $d\le Y$
to all $d$ costs $O(b^{B_2}X^{1+\rho}/Y)$; the $d^{-2+\rho}$ tail
also permits differentiation of that sum.
\par\smallskip\noindent\emph{The nonzero frequencies.}
It remains to obtain a power saving with polynomial dependence on $J$.
Put $\epsilon_q=1$ or $i$ according as $q\equiv1$ or $3\bmod4$
and $\widetilde G_\nu(q)=\epsilon_q^{-1}G_\nu(q)$. This is multiplicative
in $q$. Since $4\mid H$, expand
$\epsilon_q e(\nu c_J\bar q/H)$ in characters $\psi\bmod H$.
For $\nu\ne0$ the resulting Dirichlet series is
\[
D_{\psi,J,\nu}(s)=
\sum_{(q,H)=1}\frac{\lambda(q)}{q^s}
\frac{\widetilde G_\nu(q)}{\sqrt q}
\left(\frac{\sigma JH}{q}\right)\psi(q)
=L(s,f\otimes\xi_{\rm prim})C_{\psi,J,\nu}(s),
\]
where $\xi(q)=\psi(q)(\sigma JH\nu/q)$.
At $p\nmid2HJ\nu$ its local factor is
$1+\lambda(p)\xi(p)p^{-s}$; dividing by the $\mathrm{GL}_2$ Euler
factor leaves a polynomial $1+O(p^{-2\Re s})$.
The bound at primes dividing $\nu$ has an absolute exponent.
Indeed, for an odd prime $p$, put $a=v_p(\nu)$. Directly summing
over the lifts of a residue modulo $p$ gives
\[
G_\nu(p^r)=
\begin{cases}
\varphi(p^r),&r\text{ even},\ r\le a,\\
-p^{r-1},&r\text{ even},\ r=a+1,\\
\epsilon_p p^{r-1/2}
\left(\dfrac{\nu/p^{r-1}}p\right),
&r\text{ odd},\ r=a+1,\\
0,&\text{otherwise}.
\end{cases}
\]
For even $r$ this is the Ramanujan sum; for odd $r$ the inner
sum of lifts vanishes unless $p^{r-1}\mid\nu$, after which
the primitive quadratic Gauss sum gives the displayed value.
In particular
$|\widetilde G_\nu(p^r)/p^{r/2}|\le p^{r/2}$, and the local
polynomial on $\Re s\ge3/4$ is bounded by
$\sum_{r\ge0}(r+1)p^{-r/4}\le C_0$ with $C_0$ absolute.
The inverse degree-two Euler polynomial is bounded by another
absolute constant $C_1$. Primes dividing $J$ simply remove the
local sum, and have the same bound after multiplication by that
inverse Euler polynomial. Thus all primes dividing $J\nu$ cost
at most $(C_0C_1)^{\omega(J\nu)}\le (2J|\nu|)^C$, for an
absolute $C$. The remaining correction product converges
absolutely on $\Re s\ge3/4$, since its local error is
$O(p^{-3/2})$. Hence the correction is holomorphic on
$\Re s>1/2$ and has the asserted polynomial bound with absolute exponent.
The conductor of $\xi_{\rm prim}$ is at most $4HJ|\nu|$, and the
twisted cuspidal $L$-function is entire. Thus, with absolute $C$,
\[
|D_{\psi,J,\nu}(3/4+i\tau)|
\ll_{f,H}(J|\nu|)^C(1+|\tau|)^C.
\]
Integration by parts in both weight variables gives, for fixed $r,s$
and any $K>0$,
\[
|(z\partial_z)^r(R\partial_R)^sW_j(z,R)|
\ll_{f,\Phi,r,s,K}(1+z)^{-K}(1+|R|)^{-K}(1+|\log z|).
\]
The logarithm is needed only for $j=1$ and is $O(\log X)$ on the
actual sums, where $z\ge X^{-1}$.
For an absolute small $\rho>0$ the sums can therefore be restricted to
\[
u_1q\ll X^{1+\rho},\qquad
0<|\nu|\ll_H JX^{2\rho},\qquad
q\gg_H X^{1-\rho}/J,
\]
at a negligible cost with polynomial dependence on $J$.
A trivial bound $|G_\nu(q)|\le q$ gives a polynomial majorant for the
discarded tails. To identify the sum to be shifted, write
$D_{\psi,J,\nu}(s)=\sum_q\gamma(q)q^{-s}$. On a dyadic range
$q\asymp T$, take a fixed smooth cutoff $\vartheta$ and put
\[
w_T(y)=\vartheta(y)
W_j\!\left(\frac{u_1Ty}{X},\frac{\nu X}{HJTy}\right).
\]
Each dyadic $q$-sum after the character expansion is therefore
\[
\sum_q\frac{\gamma(q)}q w_T(q/T)
=\frac1{2\pi i}\int_{(1)}
\widehat w_T(s)T^sD_{\psi,J,\nu}(1+s)\,ds,
\]
where $\widehat w_T(s)=\int_0^\infty w_T(y)y^{s-1}\,dy$.
The weight bounds permit a contour shift to $\Re s=-1/4$, giving
\[
O_{f,H,\Phi}\bigl((J|\nu|)^C X^{C'\rho}T^{-1/4}\bigr).
\]
There is no residue. Summing over $\nu$, over the dyadic ranges,
and over $u_1$ (for which
$\sum_{u_1\mid H^\infty}|\lambda(u_1)|/\sqrt{u_1}<\infty$)
is bounded, up to a further $X^{O(\rho)}$, by
\[
\frac XJ\,J^C(JX^{2\rho})^{C+1}
(X^{1-\rho}/J)^{-1/4}
=J^{2C+1/4}X^{3/4+O(\rho)}.
\]
This gives $O(J^{B_3}X^{3/4+C''\rho})$. Choose $\rho$ so that this is
$O(J^{B_3}X^{7/8})$.
\par\smallskip\noindent\emph{Choosing the truncation.}
Since $J\le bd^2$, summing the two contour errors for $d\le Y$,
and adjoining Equation~\eqref{nv:tail}, gives
\[
O_{f,H,\Phi}\bigl(
b^{B_4}Y^{2B_4+1}X^{1-\kappa}
+b^{B_4}X^{1+\rho}/Y\bigr)
\]
for absolute $\kappa>0$ and $B_4$.
Take $Y=X^\delta$ with $(2B_4+1)\delta<\kappa/2$, and then
$\rho<\delta/4$. Both errors have the form in the statement,
with absolute positive saving. This proves the lemma.
\end{proof}
\begin{proof}[Proof of Propositions~\ref{nv:local-witness} and~\ref{nv:rough}]
For negative root number, Lemma~\ref{nv:moment} gives a nonzero sum of
derivatives in every sufficiently large dyadic interval. One derivative
is nonzero, and its central zero is already forced by the sign.
For positive root number, use the nonnegative sequence
\[
a_n=\mu^2(n)\mathbf1_{n\equiv a\ (H)}
L(1/2,f\otimes\chi_{\sigma n})\Phi(n/X).
\]
Its nonnegativity follows from the Waldspurger formula
\cite{Waldspurger1981}. The lemma gives a dimension-one sieve density
$\beta_p$ and an absolute exponent of distribution:
choose $\theta>0$ with $(B+1)\theta<\eta$ and put $D=X^\theta$.
The total remainder for squarefree $b\le D$ is
\[
O_{f,H,\Phi}(D^{B+1}X^{1-\eta})=o(X/\log X).
\]
The density estimate also gives the interval product condition
\[
\prod_{\substack{w\le p<z\\p\nmid H}}(1-\beta_p)^{-1}
\le \frac{\log z}{\log w}
\left(1+\frac{C_{f,H}}{\log w}\right)
\qquad(2\le w<z).
\]
Indeed, comparison with $(1-p^{-1})^{-1}$ contributes a logarithmic
error $O(\sum_{p\ge w}p^{-2})$, and Mertens' product estimate gives
the displayed bound; finitely many small primes are absorbed in
$C_{f,H}$. The same comparison gives
\[
\prod_{\substack{p<z\\p\nmid H}}(1-\beta_p)\asymp_{f,H}(\log z)^{-1}.
\]
The lower linear sieve \cite[Theorem~11.13]{FriedlanderIwaniec2010},
with $z=D^{1/s}$ and any fixed $s>2$, therefore gives positive
remaining weight. Since $z=X^{\theta/s}$ tends to infinity, its
surviving parameters eventually avoid all primes up to any prescribed bound.
Every surviving parameter is prime to $H$ and has all other prime
factors at least $z$. Since $n<2X$, its number of prime factors is
bounded by any fixed integer greater than $2s/\theta$ for sufficiently
large $X$. This proves Proposition~\ref{nv:rough} and the positive-sign
part of Proposition~\ref{nv:local-witness}. Enlarging $H$ and choosing
a reduced CRT lift excludes any additional finite prime set.
\end{proof}
The constants in the size threshold may change when the curve or
the support is changed. The exponent $\theta$ and the factor-count
bound do not. In particular, a previously chosen twist can be
absorbed into the curve before this proposition is applied.
==== END A-pointwise-2-converse-for-elliptic-curves-with-rational-two-torsion//nonvanishing.tex ====
==== BEGIN A-pointwise-2-converse-for-elliptic-curves-with-rational-two-torsion//paper.tex ====
\documentclass[11pt]{article}
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\hypersetup{pdftitle={A pointwise 2-converse for elliptic curves with rational two-torsion},pdfsubject={Selmer coranks, integral interpolation, and pointwise quadratic-twist constructions},pdfauthor={OpenAI}}
\title{A pointwise $2$-converse\\for elliptic curves with rational two-torsion}
\author{OpenAI}
\date{September 24, 2026}
\begin{document}
\maketitle
\begin{abstract}
We prove a pointwise $2$-converse for elliptic curves over $\mathbf Q$
with nonzero rational two-torsion: if the $2^\infty$-Selmer corank is
zero or one, then the analytic rank and Mordell--Weil rank equal that
corank, and the Shafarevich--Tate group is finite. The result allows
arbitrary reduction at $2$.
\end{abstract}
\setcounter{tocdepth}{1}
\tableofcontents
\clearpage
\input{sections/introduction}
\input{sections/preliminaries}
\input{sections/cyclotomic}
\input{sections/ring-limits}
\input{sections/ring-determinants}
\input{sections/nonvanishing}
\input{sections/coefficients}
\input{sections/graphs}
\input{sections/pointwise}
\bibliographystyle{plain}
\bibliography{references}
\end{document}
==== END A-pointwise-2-converse-for-elliptic-curves-with-rational-two-torsion//paper.tex ====
==== BEGIN A-pointwise-2-converse-for-elliptic-curves-with-rational-two-torsion//pointwise.tex ====
\section{Assembly at a fixed curve}\label{pw:section}
Fix $E/\mathbb Q$ with nonzero rational two-torsion. The even construction
has two uses: it proves the corank-zero case and supplies the uniform
estimate needed to normalize the odd coefficients. After that normalization,
the corank-one argument fixes a nonvanishing companion and constructs the
two simultaneous coefficient tests required for ring-class transfer.
\subsection{Interfaces and conventions}\label{pw:interfaces-section}
We use the fixed-support conventions of Section~\ref{sec:preliminaries}.
Thus all variable prime factors are odd primes of good reduction,
$p^*=(-1/p)p$, and a filter fixes the real sign and unit squareclasses
at a finite support containing $2N$, with finitely many further exclusions.
Recall
\[
t_p=\dim_{\mathbb F_2}E[2]^{\operatorname{Fr}_p},\qquad
w(h)=\frac12\sum_{p\mid h}t_p.
\]
Whenever the two-primary group is finite, we write
\[
s(h)=\log_2|\Sha(E^{(h)}/\mathbb Q)[2^\infty]|.
\]
This includes a base of Selmer corank zero and every parameter of analytic
order zero or one. It is distinct from $s_2(E^{(h)})$, the Selmer corank.
The normalized central value is denoted $L_*(h)$, and $v=v_2$ with
$v(2)=1$.
Here are the preceding results in the form used below. Their roles are
different: coefficient units give upper bounds, interpolation returns those
bounds to an unknown base, and the ring-class lower bound makes the odd
normalization possible. The latter step is carried out in
Proposition~\ref{pw:odd-lower}.
\begin{proposition}[Collected interfaces]\label{pw:interfaces}
The following statements hold for the fixed curve and fixed local data.
\begin{enumerate}
\item The cyclotomic missing-vertex statement of
Theorem~\ref{cy:missing-vertex} applies to any fixed base $h_0$ of Selmer
corank zero. If arbitrarily large binary cubes based at $h_0$, with no old
prime removed, have nonzero central values at every other address and satisfy
\[
v(L_*(h_x))-2w(h_x)-s(h_x)\le B\quad(x\ne0),
\]
then the base central value is nonzero and satisfies the same inequality
with $B$ replaced by $B+C$. The constant $C$ depends only on the fixed curve
and local data, not on $h_0$, its number of prime factors, or the cube
dimension. The dimension required for the argument may depend on the base.
The cyclotomic lower bound at an analytically nonvanishing parameter is also
available.
\item Corollary~\ref{rd:ring-lower} has the following uniform form. If
$k<0$ is an odd fundamental discriminant with at most $A$ prime factors,
$K=\mathbb Q(\sqrt{k})$ is neither an extra-unit field nor the CM field,
$k$ is a square at $2Nh$, and the twists $h,hk$ have respective analytic
orders one and zero, then their ring-class Heegner sum has free-line index
$j_h(k)$ satisfying
\begin{equation}\label{pw:ring-lower-interface}
2j_h(k)\ge 4w(h)+s(h)+s(hk)-C_{E,A}.
\end{equation}
For a fixed such $k$, Theorem~\ref{rd:missing-vertex} applies to
cubes based at $h=1$ whose new primes all split in $K$. If
$s_2(E/K)=1$ and the nonzero addresses have product analytic order one and
uniformly bounded
$2j_h(k)-4w(h)-s(h)-s(hk)$, the base product has analytic order one.
\item The even coefficient system, exact same-filter square comparison,
divided operators, and trace identities of
Proposition~\ref{co:even} and Lemmas~\ref{co:operators},
\ref{co:traces}, and \ref{co:inertia} are available.
After a uniform lower bound for the relevant point indices has been proved,
the moving odd coefficient system and its stabilized minimum are available
as in Proposition~\ref{co:odd} and Lemma~\ref{co:minimum}. The resulting unit symbols
satisfy the contraction, good-twin, and primary-isolation rules of
Proposition~\ref{co:symbol-interface}. An even unit gives
\begin{equation}\label{pw:even-unit-interface}
L_*(h)\ne0,\qquad v(L_*(h))\le 2w(h)+C;
\end{equation}
an odd unit gives analytic order one and
\begin{equation}\label{pw:odd-unit-interface}
j(P_h)\le w(h)+C.
\end{equation}
The constants are independent of the number and sizes of the variable
primes. The fixed genus partner used to define $P_h$ is denoted $h_*$.
\item Arithmetic realization and the forest and address statements of
Lemmas~\ref{gr:realization} and \ref{gr:forest} and
Theorem~\ref{gr:address} hold.
In particular, separate nonzero terminal polynomials may be addressed
simultaneously. Primary auxiliaries preserve all fixed local classes,
have neutral incidence to specified stationary external vertices, and may
be required to split in any compatible fixed fields. They contribute no
prime at address zero.
\end{enumerate}
\end{proposition}
\begin{proof}
These are the cited results, with the arithmetic ring-class hypotheses
verified by Proposition~\ref{rd:assembly}. The even coefficient minimum
uses the cyclotomic lower bound; the odd detector of
Proposition~\ref{co:odd} still needs the arithmetic lower bound for
Lemma~\ref{co:minimum}. Neither interpolation statement nor the even
symbol uses that odd normalization.
\end{proof}
The analytic witnesses and the rough completion used below are supplied by
Propositions~\ref{nv:local-witness} and \ref{nv:rough}; the latter has an absolute
prime-count bound. The bounds in Proposition~\ref{pw:interfaces}, unlike the
eventual sizes of the primes chosen by nonvanishing or Chebotarev, must be
uniform in the parameters indicated there.
\begin{lemma}[Isogeny orientation and coranks]\label{pw:isogeny-corank}
Recall that a rational isogeny preserves analytic rank, Mordell--Weil rank,
and $2^\infty$-Selmer corank. If $k$ is a quadratic discriminant and
$K=\mathbb Q(\sqrt{k})$, then
\begin{equation}\label{pw:quadratic-corank}
s_2(E/K)=s_2(E/\mathbb Q)+s_2(E^{(k)}/\mathbb Q).
\end{equation}
For every $E/\mathbb Q$ one has, separately,
\begin{equation}\label{pw:kummer-corank}
s_2(E)=r(E)+
\operatorname{corank}_{\mathbb Z_2}\Sha(E/\mathbb Q)[2^\infty].
\end{equation}
\end{lemma}
\begin{proof}
These are Lemmas~\ref{pr:invariance} and \ref{pr:ranks}. In particular,
the quadratic identity is an identity of coranks, not an integral
direct-sum decomposition.
\end{proof}
If the isogeny class contains a curve with full rational two-torsion, we use
that curve. Otherwise choose a rational two-isogeny $E\to E'$ and write
$F=\mathbb Q(E[2])$, $F'=\mathbb Q(E'[2])$; both fields are quadratic.
A prime split in $F$ is called split, and one inert in $F$ is called simple.
Their weights are respectively $1$ and $1/2$. A simple prime is good if it
is also inert in $F'$. At such a prime
\begin{equation}\label{pw:good-simple}
\#E(\mathbb F_p)\equiv2\pmod4.
\end{equation}
Indeed there is exactly one nonzero rational two-torsion point in the
reduction. If the group order were divisible by four, its two-primary
subgroup would contain a rational point of order four. Its image under the
two-isogeny would supply on $E'$ a rational two-torsion point different from
the dual kernel, contradicting inertness in $F'$. Thus the good-twin rule
applies. When $F=F'$, every simple prime is good.
A prime split in $F$ and inert in $F'$ is a \emph{donor}: attaching its
label to a bad simple label makes that label good. This terminology
describes finite label operations; the required prime supplies will be
constructed below.
\subsection{Seeds and finite supplies}\label{pw:seeds-section}
The word ``label'' below means an individually realizable fixed-support
elementary Frobenius label, with orientations when required. It does not
include a self-residue of a prime generator. All contractions and
comparisons are those of Proposition~\ref{co:symbol-interface} and
Lemma~\ref{gr:forest}.
\begin{lemma}[Minimum seeds]\label{pw:minimum-seeds}
\begin{enumerate}
\item In an all-split alphabet, suppose a unit symbol has a nonempty
witness. At the least positive vertex count $r$ of such a witness, all
mutual edge bits are immaterial. The symbol is determined by the individual
fixed-support labels. This remains true if the empty configuration is a
unit, provided no contraction to the empty configuration is used.
\item In the quadratic case, suppose the symbol has a nonzero witness,
the alphabet is closed under the allowed contractions, and it contains
the good twin operations. There are
$r\ge0$ and $s_1\ge1$ such that for every $s\ge s_1$ of the permitted simple
parity, the minimum split count of a unit with $s$ simple vertices is $r$.
At these counts all edge bits are immaterial. A chosen minimum seed may be
padded by arbitrarily many good twin pairs.
\end{enumerate}
\end{lemma}
\begin{proof}
For the first assertion, a derivative in an edge of an $r$-vertex graph is a
symbol on $r-1$ vertices. If $r\ge2$, this graph remains nonempty and has
value zero by minimality. If $r=1$, there is no edge. The omitted-vertex
description then leaves exactly the fixed-support labels.
For the second assertion let $r(s)$ be the minimum split count at simple
count $s$, with value $+\infty$ when no witness exists. Once a finite value
occurs, good-twin insertion gives $r(s+2)\le r(s)$. A nonincreasing sequence
of nonnegative integers stabilizes. Choose $s_1\ge1$ in that range. A
derivative of an edge with a split endpoint reduces the split count and
preserves the simple count; it is therefore zero at the minimum. A
simple--simple derivative is zero by the symbol rule. Twin insertion gives
the last assertion.
\end{proof}
Call a list partitionable if some permitted contraction forest has a unit
symbol as its output. A contraction certificate proves that its terminal
polynomial is nonzero: the indicated iterated finite difference is nonzero.
\begin{lemma}[Finite label reserves]\label{pw:reserves}
Fix a minimum seed and a finite label alphabet.
\begin{enumerate}
\item In the all-split rational alphabet, a list with the same total label
as the seed and sufficiently many copies of its labels is partitionable.
\item In the quadratic alphabet, retain $r$ split anchors with the seed
labels, and suppose the list and seed have the same total rational label
on the projection used by the symbol. Suppose there are sufficiently
many simple vertices of the required parity and sufficiently many split
reserves of each realizable rational shift. If $F\ne F'$, also suppose
\begin{equation}\label{pw:donor-balance}
\Delta=\#\{p:\ p\text{ split in }F\text{ and inert in }F'\}
-\#\{p:\ p\text{ inert in }F\text{ and split in }F'\}>C_0.
\end{equation}
Here $C_0$ depends only on the anchors and the reserves. Then all other
split vertices can be attached to simple vertices so that the result is
the minimum seed and good twin pairs. The twins may match a prescribed
finite enrichment of the rational labels. The base simple labels may be
specified on any projection on which the required total agrees.
\end{enumerate}
All reserve requirements depend only on the finite alphabets, seed, and
projection. They do not depend on the original list length.
\end{lemma}
\begin{proof}
In the first case retain the first $r-1$ seed labels and merge all other
vertices to the last cluster. Sufficient copies ensure that this cluster
is nonempty. Its label is forced by the total, hence is the last seed
label. Lemma~\ref{pw:minimum-seeds} makes its unknown edges immaterial.
For the second case, the rational labels of split attachments form the
subgroup with trivial $F$-coordinate. They act transitively on the simple
coset, so every prescribed change of a simple label has a realizable split
shift. Reserve sufficiently many representatives of every such shift.
Remove these reserves and the anchors from the donor pool. If $F\ne F'$,
\eqref{pw:donor-balance}, with $C_0$ exceeding the number removed, leaves a
distinct donor for each bad simple vertex. Attach these donors; every
simple vertex is now good. If $F=F'$, this operation was unnecessary.
Pair equal enriched good labels. By withholding a bounded number of pairs,
leave between $s_1$ and $s_1+C_1$ unpaired vertices, with the correct parity;
$C_1$ depends only on the number of enriched labels. Use reserved shifts
to turn the excess unpaired vertices into good pairs and to set the first
$s_1-1$ base positions to the desired seed labels. The seed labels
themselves need not all be good. Attach every remaining nonanchor split
vertex to the last base position. Its projected label is forced by the
total, the anchors, the other base positions, and the twin pairs. It is
the required last seed label. Each cluster contains one simple vertex,
so every contraction is permitted. Only a bounded number of reserve
shifts was used.
When incidences to a retained split anchor are needed only to remove twins,
they can be prescribed after the contracted labels are fixed, by the
independent terminal-edge realization. Fixed stationary vertices instead
enter the finite label enrichment from the outset. These two uses do not
increase the number of label types with the original list length.
\end{proof}
\begin{lemma}[Neutral packets]\label{pw:packets}
For any of the finite reserve problems above with prescribed rational total
zero, one can choose a finite packet of terminal labels with rational total
zero such that every nonempty union of copies of the packet has the required
supplies. In the quadratic case with \(F\ne F'\), its donor surplus
may be made arbitrarily large. When \(F=F'\), every simple label is
good and no donor surplus is required. Compatible fixed splitting
conditions may be included whenever
the needed individual labels and shifts remain realizable.
\end{lemma}
\begin{proof}
Choose ample copies of every required label, including anchors and shifts.
In the quadratic case with \(F\ne F'\), add enough donors to exceed
all the losses from the
other labels and the prescribed reserve margin. Duplicate every full
label in the resulting list. Duplication gives total zero in every
rational elementary coordinate and preserves any strict donor surplus.
Choose the margins so that after reserving the finitely many anchors in
any one packet, the union of any positive number of packets still meets
the reserve requirements. All labels are realized by distinct fresh
primes; duplication never means repeating an actual prime factor.
\end{proof}
\subsection{The even estimate and the corank-zero case}
\label{pw:lower-section}
\begin{proposition}[Partitionable even estimate]\label{pw:partitionable-even}
Fix an even symbol furnished by Proposition~\ref{pw:interfaces}. For every
partitionable parameter $h$ on its filter with $s_2(E^{(h)})=0$,
\begin{equation}\label{pw:partitionable-bound}
L_*(h)\ne0,\qquad v(L_*(h))-2w(h)-s(h)\le C.
\end{equation}
The constant depends only on the symbol and fixed local data.
\end{proposition}
\begin{proof}
If the parameter has no prime factors, a contraction certificate means
that the empty symbol itself is a unit. Equation~\eqref{pw:even-unit-interface}
then proves the assertion directly. Otherwise the stationary terminal
set is nonempty.
Keep the prime factors of $h$ stationary. Their terminal polynomial is
nonzero by its contraction certificate. The stationary one-symbol case
of Theorem~\ref{gr:address} gives arbitrarily large binary cubes based at
$h$, with only primary new primes, whose other addresses have unit symbol.
Equation~\eqref{pw:even-unit-interface} and $s(h_x)\ge0$ give the uniform
upper bound required by the cyclotomic transfer. The base corank is zero,
so that transfer gives \eqref{pw:partitionable-bound}. A base-dependent
dimension threshold does not affect the asserted constant.
\end{proof}
The original corank-zero base can now be treated without any odd
coefficient system or companion field.
\begin{proposition}[Even original base]\label{pw:even-base}
If $s_2(E)=0$, then $a(E)=0$.
\end{proposition}
\begin{proof}
Lemma~\ref{pr:invariance} gives even functional sign. Fix an even symbol
on the filter of $1$, including all auxiliary class-group tests. If its
value at $1$ is a unit, analytic nonvanishing is already detected.
Otherwise use a nonempty minimum seed in the full-torsion case, or a
stabilized quadratic seed in the other case.
For each $b$, assign a neutral packet from Lemma~\ref{pw:packets} to
every nonzero linear form on $\mathbb F_2^b$, and activate its primes
by that form. At every nonzero address at least one packet is active;
their union satisfies the reserve problem and has a nonzero terminal
polynomial. At zero no terminal is active. The single-symbol case of
Theorem~\ref{gr:address} adds primary auxiliaries and makes the symbol a
unit at every nonzero address. All local classes remain those of $1$.
Equation~\eqref{pw:even-unit-interface}, followed by cyclotomic transfer
at the original corank-zero base, gives $a(E)=0$.
\end{proof}
\subsection{The lower bound needed by the odd coefficients}
\label{pw:odd-lower-section}
For the odd construction, first fix the original coefficient and
class-group support and its unit tests. Then choose the genus partner
$h_*$ as in Section~\ref{co:section}, adjoin its prime factors to the
support, and refine the unit tests there. All variable $h$ avoid this
support. The positive filter below is this prepared filter, fixed before
$h$ varies or the odd coefficients are normalized.
We next use the even estimate at analytically nonvanishing parameters,
not the corank-zero conclusion just proved. The companion $k'=k'(h)$
in this step may vary with $h$, but its number of prime factors will be
bounded. It is distinct from both the detector's fixed genus partner
$h_*$ and the fixed companion used in the final ring-class transfer.
At a parameter already known to have analytic rank zero, the estimate
\eqref{pw:partitionable-bound} transfers between either orientation of a
fixed two-isogeny. The isogeny formula for the BSD quotient gives a bounded
change in $s(h)+2w(h)$. At an active odd good prime the component-group
valuation is $t_p$, and the two active contributions are included in the
two weights. At the fixed support only finitely many local twisting
classes occur. Differential, period, and torsion indices are bounded by
the fixed isogeny and its reverse, and there is no regulator at rank zero.
This uses an isogeny identity for known finite groups, not BSD or converse
finiteness at an unknown parameter.
\begin{lemma}[A bounded companion for a varying odd parameter]
\label{pw:rough-companion}
Fix a positive odd-sign filter, including the coefficient and class-group
support used below, and a negative genus partner $h_*$ with
$L(E^{(h_*)},1)\ne0$, chosen using Proposition~\ref{nv:local-witness} and then
excluded from the variable parameters. There are constants $A,C$ such
that every $h$ on this filter of analytic order one admits a negative odd
fundamental discriminant $k'$ for which
\begin{enumerate}
\item $(k',2Nh)=1$, $k'$ is a square at every place of $2Nh$, and
$\mathbb Q(\sqrt{k'})$ is nonexceptional;
\item $\omega(k')\le A$;
\item $L(E^{(hk')},1)\ne0$ and
\begin{equation}\label{pw:rough-bound}
v(L_*(hk'))\le 2w(hk')+s(hk')+C.
\end{equation}
\end{enumerate}
\end{lemma}
\begin{proof}
At the fixed class-group support and the other fixed label tests, prepare
an even negative-index symbol for each possible total type at this
support, and for both isogeny orientations if needed.
There are finitely many such preparations. The positive index $|hk'|$
has nonzero total rational type: its negative-discriminant parity is
already nontrivial among the tests at $2$. Thus in the full-torsion case
a unit seed has positive count. In the quadratic case use the stabilized
seed of Lemma~\ref{pw:minimum-seeds}.
When $F\ne F'$, interchanging the isogeny orientation interchanges the two
counts in \eqref{pw:donor-balance}. Select the orientation with initial
balance at least zero. The existing donors can then pay for arbitrarily
many existing bad simples. They do not have to be supplied by new primes.
Let $R$ be the absolute prime-count bound in Proposition~\ref{nv:rough}. Insert
in $k'$ a finite dummy supply that meets the anchors, minimum simple count,
and reserve requirements for every one of the finitely many prepared
symbols. When \(F\ne F'\), add donor slack paying for every reserved
loss, the new dummy bad simples, and a further possible loss of $R$
from the later rough factor. When \(F=F'\), all simple primes,
including the later rough primes, are already good; only the simple
and split-reserve supplies are needed. Add a fresh ramified prime if
needed to exclude the exceptional
imaginary fields. The number of dummy primes is bounded independently
of $h$. Their labels and initial incidences may be chosen by
Lemma~\ref{gr:realization}.
Absorb $h$ and the dummy product into the curve. Apply
Proposition~\ref{nv:rough} to complete $k'$ by a squarefree factor coprime to all
primes already used. Prescribe its sign and its local unit classes so
that the final $k'$ is an odd negative fundamental discriminant, is a
square at $2Nh$, and has the required original local classes at $2N$.
These conditions give a nonempty reduced progression by the Chinese
remainder theorem. Its root number is even: since $E^{(h)}$ has odd
sign and $k'$ is negative and square at every prime of $2Nh$,
the quadratic twist formula multiplies that sign by
$\chi_{k'}(-N_{E^{(h)}})=-1$. Thus this is the positive-root-number
case of Proposition~\ref{nv:rough} for the absorbed curve.
Additional fixed primes may be excluded. The proposition makes the
central value at $hk'$ nonzero with at most $R$ new factors. Its size
threshold may depend on $h$ and all these congruences; $R$ does not.
No particular resulting unit class at the auxiliary class-group support
was assumed in advance: dummy supplies were prepared for every possible
total there. For the actual total, Lemma~\ref{pw:reserves} makes $hk'$
partitionable. Analytic nonvanishing gives Selmer corank zero by the
forward theorem. Proposition~\ref{pw:partitionable-even} therefore gives
\eqref{pw:rough-bound}, with the isogeny comparison above if the other
orientation was used. The finite preparation and the bound on dummy and
rough factors give $A,C$ independent of $h$.
\end{proof}
\begin{lemma}[Height comparison]\label{pw:height-comparison}
Here $j_h(k)$ is the index of the conductor-$|h|$ ring-class sum over
$\mathbb Q(\sqrt{k})$, whereas $j(P_h)$ is the index of the conductor-one
genus sum over $\mathbb Q(\sqrt{hh_*})$.
With the fixed modular parametrization and the fixed genus partner $h_*$,
the explicit Gross--Zagier formula gives, whenever the relevant factors
have analytic orders one and zero,
\begin{equation}\label{pw:height-ratio}
2j_h(k)-2j(P_h)=v(L_*(hk))+O(1).
\end{equation}
For $k$ with bounded prime count as in
Lemma~\ref{pw:rough-companion}, the error is uniform in $h,k$; for a fixed
$k$ it is uniform in $h$.
\end{lemma}
\begin{proof}
Use the character-sum normalization of the explicit Gross--Zagier formula
in \cite[Theorem~1.1]{CaiShuTian2014}. The restriction of $\chi_h$ to
$\mathbb Q(\sqrt{k})$ has exact ring-class conductor $|h|$: each odd
prime dividing $h$ splits and retains its ramified quadratic character.
The genus character over $\mathbb Q(\sqrt{hh_*})$ is unramified, so the
sum defining $P_h$ has conductor one. Their canonical
heights are fixed modular factors times, respectively,
\[
|h|\sqrt{|k|}\,L'(E^{(h)},1)L(E^{(hk)},1),
\qquad
\sqrt{|hh_*|}\,L'(E^{(h)},1)L(E^{(h_*)},1).
\]
In that theorem the multiplier is $c\sqrt{|D|}$, the level-discriminant
factor is $2^{\mu(N,D)}$, and the unit factor is
$[\mathcal O_c^\times:\mathbb Z^\times]^2$. Here every level prime
splits, so $(N,D)=1$ and $\mu(N,D)=0$; the excluded extra-unit fields
make the unit factor $1$ in both applications.
Take their ratio. The derivative and the rank-one regulator cancel;
the remaining varying real factor is $\sqrt{|hk|}$, precisely the factor
in $L_*(hk)$. Comparison of the rational free line with its two quadratic
projections has bounded index because their composites are multiplication
by $2$. The modular parametrization and the factor involving $h_*$ are
fixed. The level is split and the discriminant and conductor are prime
to it; no growing conductor-prime product occurs in this normalization.
Local differential and period comparisons have finitely many possibilities
at the fixed support. The extra-unit fields have been excluded, and the
number of additional places in the varying $k$ case is bounded. These
observations give the asserted bounded valuation error.
\end{proof}
\begin{proposition}[Uniform lower bound for the odd detector]
\label{pw:odd-lower}
Every analytically rank-one $h$ in the prepared positive filter of
Lemma~\ref{pw:rough-companion} satisfies
\begin{equation}\label{pw:odd-lower-bound}
2j(P_h)\ge 2w(h)+s(h)-C.
\end{equation}
In particular $j(P_h)\ge w(h)-C$ uniformly in $h$ and its prime count.
\end{proposition}
\begin{proof}
Choose $k'$ by Lemma~\ref{pw:rough-companion}. Combine
\eqref{pw:ring-lower-interface}, \eqref{pw:height-ratio}, and
\eqref{pw:rough-bound}. Since the factors are coprime,
$w(hk')=w(h)+w(k')$, and $w(k')\le\omega(k')\le A$. Hence
\begin{align*}
2j(P_h)
&\ge 4w(h)+s(h)+s(hk')-v(L_*(hk'))-C\\
&\ge 2w(h)+s(h)-2w(k')-C\\
&\ge 2w(h)+s(h)-C'.
\end{align*}
Only even symbols and independently nonvanishing companions were used.
Thus this proof precedes, and does not assume, the odd minimum-depth
normalization.
\end{proof}
\subsection{From the moving odd coefficients to finite graphs}
\label{pw:odd-stages}
Proposition~\ref{pw:odd-lower} now supplies the arithmetic premise of
Lemma~\ref{co:minimum}. We obtain an odd symbol with fixed minimum
normalized depth and the uniform unit bound
\eqref{pw:odd-unit-interface}. To use the finite seed and packet lemmas,
we also need to stabilize its graph labels and values, not just this
numerical depth.
Keep the ultrafilter used in Lemma~\ref{co:minimum}. The intrinsic fixed
support includes the genus partner $h_*$; at an actual stage it also
includes the detecting prime $\ell$. That prime adds at most two places
in any fixed quadratic field. The unit and class-group description of
$V_S(F)$ therefore bounds the dimension of the elementary label space
independently of the stage. At $\ell$ the Schwartz function in
Lemma~\ref{co:theta-bridge} uses only the quadratic residue-line character:
increasing the Kummer precision does not impose a character of growing
order on the variable primes. The old-support characters are fixed.
Choose bases and restrict to an ultrafilter-large set of stages on which
the finite label spaces, their addition tables, conjugation and projection
maps have one abstract form. At every fixed weight bound there are only
finitely many typed graphs and symbol truth tables. Select the table
occurring on an ultrafilter-large set. These selected tables are
compatible under restriction and define one limit symbol. Its contraction
and twin identities follow from the finite-stage identities. The witness
sequence of Lemma~\ref{co:minimum} has bounded weight, hence takes only
finitely many graph types in this alphabet; one type is a witness in the
selected limit table.
If the empty configuration is a unit, a sufficiently advanced actual stage
already detects analytic order one at $h=1$. Otherwise the limit witness
is nonempty, so Lemma~\ref{pw:minimum-seeds} supplies the minimum seeds
and stabilized simple count needed below. Fix them in this remaining
case; Lemmas~\ref{pw:reserves} and \ref{pw:packets} apply to their finite
label alphabets. Every
particular contraction or twin comparison uses only a bounded graph.
Its selected truth table agrees with the actual symbol on an
ultrafilter-large set of stages. Thus any prescribed finite collection
of such comparisons holds at one sufficiently advanced stage.
Later we will add the fixed support of a separate companion and the label
projection read by its even test. Once compatible lifts have been
verified, we may stabilize this finite enrichment by restricting the
\emph{same} ultrafilter. This preserves the intrinsic odd symbol, its
minimum, and its chosen seeds. The actual stage is selected only after
the cube dimension and all its finite networks have been chosen: their
maximum weight gives a finite list of table and precision requirements.
No intersection over all weight bounds, and no single stage working for
every cube dimension, is required.
\subsection{A nonempty split-field symbol with full two-torsion}
\label{pw:split-section}
We give the extra argument required when $E[2]$ is rational. It addresses
the possibility that the usual even minimum is attained only at the empty
configuration after restricting all new primes to split in the companion
field.
\begin{lemma}[Recurrence congruences]\label{pw:recurrence}
Let $p$ be odd, $a\equiv p+1\pmod4$, and let
\[
b_0=1,\quad b_1=a-\kappa,\quad
b_{s+1}=ab_s-pb_{s-1}.
\]
If $\kappa=\pm1$, then $b_{s+1}-b_s$ is even and
\[
\frac{b_{s+1}-b_s}{2}\equiv\frac{a-\kappa-1}{2}\pmod2
\quad(s\ge0).
\]
If $\kappa=0$, put $b_{-1}=0$; then $b_s-b_{s-1}$ is odd for every
$s\ge0$. Finally, for $\kappa=1$, if $a\equiv p+1\pmod{2^m}$ then
$b_s\equiv p^s\pmod{2^m}$ for every $s\ge0$.
\end{lemma}
\begin{proof}
Modulo $2$ the recurrence is $b_{s+1}=b_{s-1}$. Its initial values prove
the parity assertions. For $d_s=b_{s+1}-b_s$, the recurrence for $d_s$
is the same. Directly,
$d_1-d_0\equiv(p-1)(p-\kappa)\equiv0\pmod4$ when $\kappa=\pm1$.
Thus $d_s/2$ is constant modulo $2$. The last assertion follows by
substituting the sequence $p^s$ in the recurrence and checking its first
two terms.
\end{proof}
\begin{lemma}[Nonempty $K$-split witness]\label{pw:split-witness}
Suppose $E[2]$ is rational. Choose a negative unit $k$ of a fixed even
symbol for $E$, prime to all previous fixed support and to the ramification
support of $E[8]$. Put $K=\mathbb Q(\sqrt{k})$, chosen nonexceptional.
Let $S_0$ contain the complete fixed support of the coefficient system,
$2N$, the primes of $k$, and primes generating the class group of $K$.
On the positive filter of $1$ square at $S_0$, there is an
even unit test on nonempty products of primes split in $K$ with these
properties:
\begin{enumerate}
\item it has a nonempty witness;
\item it satisfies the oriented split contraction and primary-isolation
rules over $K$, for contractions with nonempty output;
\item a unit at $D$ implies
\begin{equation}\label{pw:split-unit-bound}
L(E^{(kD)},1)\ne0,\qquad v(L_*(kD))\le 2\omega(D)+C_k.
\end{equation}
\end{enumerate}
This conclusion uses the integral trace and local identities in
Proposition~\ref{pw:interfaces}(3).
\end{lemma}
\begin{proof}
If the ordinary coefficient test already has a nonempty split witness,
there is nothing to construct. Otherwise we first obtain an additional
trace congruence modulo $8$. That congruence makes one further division
of the coefficient form integral. We verify its contraction identities,
then use a prime inert in $K$ to force a nonzero test at a split prime.
\emph{Extra depth if the ordinary test has no witness.}
Use the ordinary even coefficient form for $E^{(k)}$, normalized by
$a(1)=1$. The positive filter is chosen so that $kD$ lies in the original
negative filter. Since $k$ attains its least normalized depth, the exact
ratio comparison of Proposition~\ref{co:even} gives
\begin{equation}\label{pw:split-basic-depth}
v(a(D))\ge\omega(D)
\end{equation}
at every allowed squarefree $D$. The factor $|k|$ cancels from this ratio;
one is not subtracting unrelated bounded-error inequalities.
If equality holds at a nonempty product of $K$-split primes, use the usual
even bit, projected from oriented $K$-labels to rational labels. It has
all the required properties by Proposition~\ref{co:symbol-interface}. We
therefore assume for the rest of the proof that equality never occurs
there. By the rational-phase assertion in Proposition~\ref{co:even},
normalization by $a(1)=1$ makes each nonzero squarefree coefficient a
rational number times a root of unity. Its valuation is an integer,
not just an element of the possibly ramified coefficient field's
discrete value group. Thus strict inequality implies
\begin{equation}\label{pw:split-extra-depth}
v(a(D))\ge\omega(D)+1
\quad\text{for every nonempty allowed all-$K$-split }D.
\end{equation}
Write $g=\sum a(n)q^n$ for the form. Its characters satisfy
$\psi/\eta=\chi_{-4}$, so $\eta$ is odd. For a set $U$ of $K$-split
primes put
\[
B_U=2^{-|U|}\prod_{p\in U}(U_p-\eta(p)V_p)g,
\qquad
\Theta_J=\sum_{m>0,\ (m,S_0)=1}\eta(m)m q^{Jm^2}.
\]
All expansions retain the prescribed support sieves. The ordinary divided
form is
\[
H_U=\frac12\left(B_U-\sum_{J\subseteq U}B_U[J]\Theta_J\right).
\]
Here $B_U[J]$ denotes its coefficient at the squarefree index $J$.
\emph{The resulting trace congruence.}
We first prove
\begin{equation}\label{pw:split-mod8}
a_p(E^{(k)})\equiv p+1\pmod8
\quad\text{for every allowed good }p\text{ split in }K.
\end{equation}
Apply the ordinary $H_{\{p\}}$ trace at a Frobenius lift $\phi_p$,
testing coefficient $p^2$ after theta multiplication. The form
$H_{\{p\}}$ is already integral, and $\theta\equiv1\pmod2$;
hence theta multiplication does not alter any of its reduced
coefficients after these divisions. On the
conductor-one ramified principal-series part the identity is
\[
U_p^2t(\phi_p)=U_p^3+d(\phi_p)U_p.
\]
On the unraised unary part it is the ordinary good-prime coefficient
identity. These are the only two types in this expression. Consequently
\[
(t(\phi_p)f)[p^2]=f[p^3]+(d(\phi_p)f)[p].
\]
Replace the trace by a fresh split prime $p'$ with the same finite data,
including every support-unit test. At each rational quadratic character
unramified outside $S_0$, a Frobenius lift at $p$ has the character value
of $p$. Hence the matched $p'$ has the same value, and $pp'$ has value
$1$. Equivalently, $p'$ has the same unit squareclasses as $p$ at all
places of $S_0$, so their product is square at all of them, including $2$.
Since $\phi_p\in G_K$, the new prime is also split in $K$.
Its coefficient on the left tests $pp'$; by
\eqref{pw:split-extra-depth} it is zero modulo the maximal ideal after
the two divisions. The determinant term on the right is zero: the
unary summand at $p$ has the same character and cancels there.
On the right, the indices $p$ and $p^3$ in $B_{\{p\}}$ have underlying
squarefree index $1$; they involve only $a(1),a(p^2),a(p^4)$. The
$J=\varnothing$ unary term has no coefficient at either nonsquare
index. Thus no other squarefree coefficient contributes there.
For $a=a_p(E^{(k)})$, the phase-free square coefficients are
$1,a-1,a(a-1)-p$. Hence the remaining coefficient is, up to a unit,
\[
H_{\{p\}}[p^3]
=\frac{a^2-2a-p+1-p(a-2)}4
=\frac{(a-1)(a-p-1)}4.
\]
It is zero modulo $2$. Full rational two-torsion makes $a$ even, so
$a-1$ is odd; this proves \eqref{pw:split-mod8}. No individual total
filter condition on $p$ was needed: its matched prime $p'$ makes $pp'$
lie on the filter, and the square-index terms on the right come from $1$.
\emph{One further integral division.}
Now define
\begin{equation}\label{pw:four-divided-form}
\widetilde H_U=\frac14
\left(B_U-\sum_{J\subseteq U}B_U[J]\Theta_J\right).
\end{equation}
We verify integrality at every coefficient. Write a test index uniquely
as $Jrm^2$, where $J\subseteq U$ and $r$ is squarefree and disjoint from
$U$. The underlying squarefree coefficient is at
$D=(U/J)r$. Removing the unit phases, a prime of $U/J$ contributes an odd
factor $b_s-b_{s-1}$ with $\kappa=0$, and a prime of $J$ contributes
$b_{s+1}-b_s$ with $\kappa=(D/p)$. Thus
\begin{equation}\label{pw:shell-valuation}
v(B_U[Jrm^2])\ge\omega(r)
\end{equation}
by \eqref{pw:split-basic-depth}, with one additional unit of valuation
when $D$ is nonempty and all its primes split in $K$.
If $r\ne1$ and all primes of $D$ split in $K$, the last inequality is
at least $\omega(r)+1\ge2$. If $D$ contains an inert prime, then $r$
contains at least two: the total rational $K$-character is trivial on
the filter, and all primes of $U$ split. Then
\eqref{pw:shell-valuation} is again at least $2$. No unary term has
this squarefree part. This proves divisibility by $4$ for $r\ne1$.
Suppose $r=1$ and $J\ne U$. Then $D=U/J$ is nonempty and all split,
so $B_U[J]$ is even. This cancellation can be checked without dividing
by a leading multiplier, which may be zero. Factor from both shell
coefficients the common phase and the scalar
$2^{-|U|+|J|}a(U/J)$, whose valuation is at least $1$ by
\eqref{pw:split-extra-depth}. The remaining factors are the odd
differences at $U/J$, the integral divided differences at $J$, and
the ordinary square-index factors outside $U$. By
Lemma~\ref{pw:recurrence}, their product is congruent modulo $2$ to
the product of their leading values, even when one of those values
is zero modulo $2$. Since $m$ is odd, multiplication by $m$ preserves
this congruence. The common scalar supplies the second factor $2$.
Retaining the phase $\eta(m)$ therefore gives
\[
B_U[Jm^2]\equiv B_U[J]\eta(m)m\pmod4.
\]
This is exactly the cancellation with $B_U[J]\Theta_J$.
Finally let $J=U$, so $D=1$. At a prime outside $U$ the recurrence gives
$b_s\equiv p^s\pmod4$. At a prime of $U$,
\eqref{pw:split-mod8} gives $b_s\equiv p^s\pmod8$, and hence
\[
\frac{b_{s+1}-b_s}{2}
\equiv \frac{a_p-2}{2}\,p^s\pmod4.
\]
Multiplying these congruences, again retaining the phases, gives
$B_U[Um^2]\equiv B_U[U]\eta(m)m\pmod4$. This proves integrality in
the last case and establishes \eqref{pw:four-divided-form}.
For the trace argument we specify the integral module being used.
Write $\psi_U$ for the character of $B_U$. The character of $\Theta_U$
is also $\psi_U$, so its diamond correction is zero. For the proper
subsets the diamond formula is
\[
\langle u\rangle\widetilde H_U
=\psi_U(u)\widetilde H_U+
\sum_{J\subsetneq U}\frac{B_U[J]}2
\frac{\psi_U(u)-\eta(u)(-J/u)}2\Theta_J.
\]
Each $B_U[J]/2$ here is integral, and each divided character difference
is zero or a unit. Let $\widetilde{\mathcal M}_U$ be the $\mathcal O$-module
generated by $\theta\widetilde H_U$ and
$\theta(B_U[J]/2)\Theta_J$ for $J\subsetneq U$, together with all their
good-Hecke and diamond translates. Its generators are integral by this
formula and the good-Hecke coefficient formula. The finite-module and
Chebotarev-continuity argument of Lemma~\ref{co:traces} therefore applies
to $\widetilde{\mathcal M}_U$. A fresh trace kills every displayed unary
correction after coefficientwise reduction, also after further fresh
tests. Thus the same coefficientwise reduced determinant and trace
identities hold on the required orbit.
\emph{The contraction and isolation identities.}
On $\theta\widetilde H_U$, two fresh traces in $G_K$ give zero after
reduction at every coefficient. To see this, start with an arbitrary
test index $Jrm^2$ and choose the two fresh primes outside that index
and the level. The filter imposes $\chi_K(D)=1$; since $U$ is all split,
the number of inert factors in $r$ is either zero or at least two.
The fresh traces replace $r$ by $rq_1q_2$, with both $q_i$ split.
In the first case the valuation after division by $4$ is at least
$\omega(r)+2+1-2=\omega(r)+1\ge1$. In the second case it is at least
$\omega(r)+2-2=\omega(r)\ge2$. The unary terms disappear under these
fresh tests. Additional fresh $G_K$ traces only improve these estimates,
so vanishing holds on the entire reduced trace orbit. The trace identity
therefore gives
\[
t(gu^2)=t(g)\qquad(g,u\in G_K)
\]
on this reduced trace orbit. Thus the tests factor through the elementary
extension of $K$ unramified off the level.
A single fresh $K$-split trace at coefficient $1$ reads the bit
\begin{equation}\label{pw:deeper-bit}
\widetilde A(D)=
\left(2^{-\omega(D)-1}a(D)\text{ with its phase removed}\right)\bmod2
\end{equation}
for a nonempty all-split configuration. The contraction rule follows
from the same integral local identity as in Lemma~\ref{co:inertia}.
For completeness its first shell, with $p\in U$ and a fresh trace $q$,
has underlying configuration $D'=(U/p)q$, which is still nonempty, and
its reduction is
\[
c_p(D')\widetilde A(D'),\qquad
c_p(D')=\frac{a_p-(D'/p)-1}{2}\bmod2.
\]
Changing the oriented inertia bit changes $c_p$ once. The omitted-prime
identity gives the product label and removes the entire rational prime
$p$ from the new ramification support. Consequently its repeated
contraction has precisely the forest interpretation of
Lemma~\ref{gr:forest}. An omitted-vertex label equal to the identity has
trace zero, which is primary isolation. No contraction to the empty
configuration has occurred.
\emph{A nonempty witness for the deeper test.}
It remains to prove that this deeper test has a witness. Fresh
ramification gives $K\not\subseteq\mathbb Q(E[8])$. By Chebotarev there
is a good prime $p$ inert in $K$ and trivial on the original $E[8]$.
Then
\[
p\equiv1\pmod8,\qquad a_p(E^{(k)})\equiv-2\pmod8.
\]
On $\widetilde H_\varnothing$, whose coefficient at $1$ is zero, test
$t(u)^2$ at coefficient $1$, where $u=\operatorname{Fr}_p$. The square
coefficient and matching unary character give, up to a unit,
\[
\frac{a_p(E^{(k)})-1-p}{4}\bmod2=1.
\]
The determinant term at coefficient $1$ cancels. In characteristic two,
$t(u^2)=t(u)^2$, so a single trace in $G_K$ is nonzero. A fresh prime
realizing this finite trace test is split in $K$. Moreover $u^2$ is
trivial on every rational quadratic character, so the prime lies on the
required total rational filter. It is a nonempty witness for
\eqref{pw:deeper-bit}. Finally the exact coefficient ratio gives
\eqref{pw:split-unit-bound}, with the additional depth at most one.
\end{proof}
\begin{lemma}[Orientation correction]\label{pw:coinvariants}
Let $K$ be imaginary quadratic, let $S_0$ contain $2$ and its ramification
support, and let $V_0$ be the elementary label space over $K$ unramified
off $S_0$. Complex conjugation acts on $V_0$ by $c$. If the rational
total of an oriented prime configuration is zero on every rational
quadratic character unramified off $S_0$, then its oriented total belongs
to $(1+c)V_0$. A reserved supply containing every label allows its
orientations to be flipped to realize any prescribed change in
$(1+c)V_0$.
\end{lemma}
\begin{proof}
Complex conjugation has order two outside $G_K$, so
$G_{\mathbb Q}=G_K\rtimes\langle c\rangle$. A $c$-invariant quadratic
character of $G_K$ extends by giving $c$ value zero. At a rational prime
outside $S_0$, inertia lies in $G_K$ and the character is trivial there;
thus its extension remains unramified outside $S_0$. The assumed total
therefore annihilates $(V_0^*)^c$.
For a finite vector space with involution, the annihilator of
$\ker(1+c^*)=(V_0^*)^c$ is $\operatorname{im}(1+c)$. This proves the
first assertion. Flipping a vertex of label $v$ changes its oriented
label from $v$ to $cv$, and hence changes the total by $(1+c)v$.
Reserve one representative for every preimage needed to express the
desired element of the image. Their independent flips give the second
assertion. These flips preserve every rational norm label.
\end{proof}
\subsection{The corank-one original base}\label{pw:assembly-section}
\begin{proposition}[Odd original base]\label{pw:odd-base}
If $s_2(E)=1$, then $a(E)=1$.
\end{proposition}
\begin{proof}
Parity gives odd functional sign. Use the normalized odd limit symbol
on the positive filter of $1$ from Section~\ref{pw:odd-stages}, with
its fixed support, genus partner $h_*$, and chosen ultrafilter. If it
detects the base, a sufficiently advanced actual stage proves the result.
Otherwise choose its nonempty minimum seed. The moving detecting prime
$\ell$ is not an extra fixed support requirement.
\emph{Fixing the companion.}
Choose a separate even symbol for negative discriminants square at $2N$,
excluding the fixed odd support, the relevant torsion ramification, and
the exceptional imaginary fields. Choose a unit $k$ of that symbol.
Then $L(E^{(k)},1)\ne0$. Fix this $k$ and
$K=\mathbb Q(\sqrt{k})$ for the rest of the proof, before choosing any
cube dimension. Its fresh ramification makes it independent of the old
fixed label extensions. At sufficiently advanced stages $\ell$ avoids
$k$. The intrinsic odd label tests continue to omit the primes of $k$.
We construct common terminal packets, individually split in $K$, such
that the odd terminal polynomial and an even terminal polynomial testing
$hk$ are separately nonzero at each nonzero address. They need not have
a common unit evaluation yet; the address lemma will supply that step.
\emph{The quadratic residual case.}
First suppose $F,F'$ are quadratic. Keep the original even symbol
evaluated on $hk$, treating the vertices of $k$ as stationary external
data. Enrich the rational labels of the $h$-vertices by the primes of
$k$. Their total there and at the even support is required to be zero.
Let $V$ be the resulting rational elementary label space, including the
$F,F'$ and $K$ coordinates and all relations among these characters.
The allowed simple labels form the affine set
$\{v\in V:\chi_F(v)=1,\ \chi_K(v)=0\}$, in additive notation.
The difference of any two of its elements belongs to
$\ker\chi_F\cap\ker\chi_K$, precisely the realizable rational labels
of the permitted split attachments. Translation by this subgroup is
therefore transitive. The fields $F,F'$ are unramified outside the old
support, whereas $K$ has a fresh ramified prime; thus $K$ is independent
of their compositum. When $F\ne F'$, this realizes both the good-simple
pattern $(1,1,0)$ and the donor pattern $(0,1,0)$ in their three
coordinates. When $F=F'$, all simple labels are good.
For the old odd projection, take a normal closure of its finite label
extension at any stage with $\ell$ avoiding $k$. It is unramified outside
the old support and $\ell$. Once $\ell$ avoids
$k$, a prime ramified in $K$ remains outside that set. Hence $K$
has trivial intersection with this normal closure, and every old odd
label can be realized together with splitting in $K$. This applies
also to oriented split anchors over $F$. The added stationary $k$
coordinates may have relations, but they are already included in $V$;
they do not impose extra requirements on the omitted odd projection.
These compatible lifts let us stabilize the common alphabet and its
projection maps on the same ultrafilter, as in Section~\ref{pw:odd-stages}.
The intrinsic odd minimum and seed are unchanged. Choose the following
packets in this fixed abstract alphabet, before selecting an actual stage.
Use packets with sufficiently large simple supply and reserves in this
kernel, and with donor slack when \(F\ne F'\). For the even test, take
no split anchors. If the fields are distinct, correct all bad simples
to good ones; if they coincide, every simple is already good. Pair
equal enriched labels, and
retain a nonempty good base pair. Use reserves to match the pair and
the remaining good pairs, and attach every leftover split vertex to the
last base position. Its label is forced by total zero. The output is
a collection of good twin pairs, each matching also at every split
vertex of $k$. Twin insertion into the unit at $k$ proves that this
output has even symbol one. Thus the even terminal polynomial is
nonzero.
For the odd test, project away the newly added tests at $k$. The same
packets contain enough split anchors, simple vertices and reserve shifts
to contract to the stabilized odd seed and good pairs, by
Lemma~\ref{pw:reserves}. Its total on that projection is the prescribed
zero total. Incidences between the retained anchors and the good pairs
can be set after contraction, by independent edge choices. The odd
polynomial is therefore nonzero as well. This odd test does not include
the stationary vertices of $k$.
\emph{Full rational two-torsion.}
Now suppose the curve has full rational two-torsion. Use the even test
of Lemma~\ref{pw:split-witness}, with $K$-oriented labels on its support
$S_0$, and let $v_1,\ldots,v_r$ be a minimum nonempty witness. Its
oriented total $v_1+\cdots+v_r$ lies in $(1+c)V_0$, by
Lemma~\ref{pw:coinvariants}. This even test uses the oriented elementary
extension of $K$, whereas the odd test reads only its rational projection;
the even contraction rule is needed only for nonempty outputs.
Every odd seed label lifts to a $K$-split prime because $K$ was freshly
ramified outside its intrinsic support. Each even label also has a
compatible lift to the common alphabet. Stabilize the finite common
alphabet and projections on the same ultrafilter, preserving the
intrinsic odd symbol as in Section~\ref{pw:odd-stages}.
Choose packets containing ample copies of every needed even label, all
odd seed labels on their rational projection, and orientation-shifting
reserves. Arbitrary pairs of projected labels need not lift: take a
union of sufficiently many compatible lifts for each projection
separately. Duplicate each full label, so the packet has rational
total zero.
For the even contraction, retain anchors with labels
$v_1,\ldots,v_{r-1}$. After reserving these anchors, use orientation
flips to make the total equal to $\sum v_i$; this is possible by
Lemma~\ref{pw:coinvariants}. In a contraction certificate these flips
are choices of conjugations along the forest, as permitted by
Lemma~\ref{gr:realization}; they do not change the common terminal data
separately at different addresses. Merge all remaining vertices into the
last cluster. Its label is $v_r$. The cluster is nonempty, and the
minimum nonempty count makes the output a unit independently of its
mutual edges. For the odd contraction, use its rational projection.
Retain all but the last odd seed label and merge the remainder. Total
zero gives the last label,
so this is also a unit. Orientation corrections for the even test did
not change these rational labels.
\emph{Simultaneous addressing and an actual stage.}
Now fix $b$ and put a sufficient common packet at every nonzero linear
form on $\mathbb F_2^b$. Each nonzero address has two separately
nonzero terminal polynomials. In the fixed abstract alphabet,
Theorem~\ref{gr:address} gives finite primary-auxiliary networks making
both symbols one there. Require their primary types to split in $K$
(and $F$ in the quadratic case) and to be neutral at the stationary
primes of $k$. They are primary at every support coordinate used by
either projection, so both symbol identities apply to the same common
graph data.
Let $W_b$ bound all graph weights required by these finite networks.
A finite intersection of ultrafilter-large sets supplies an actual stage
with the same truth table through $W_b$, all
required divisibilities and trace identities, and sufficient precision
for the prescribed divisions. The moving prime also avoids the fixed
companion $k$. Realize the resulting finite graph at that stage using
Lemma~\ref{gr:realization}. All active products $h$ are positive and
square at the fixed tests, and all their primes split in $K$. No
auxiliary appears at address zero. The depth constant and fixed
companion remain unchanged as $b$ grows.
\emph{Returning to the original curve.}
At every nonzero address the analytic orders of $E^{(h)}$ and $E^{(hk)}$
are respectively one and zero. The odd unit bound, the even unit bound
(or \eqref{pw:split-unit-bound}), and
Lemma~\ref{pw:height-comparison} with fixed $k$ give
\[
2j_h(k)\le 4w(h)+C.
\]
The finite Sha lengths at these addresses are nonnegative, so this also
bounds $2j_h(k)-4w(h)-s(h)-s(hk)$ from above.
At address zero, $L(E^{(k)},1)\ne0$ gives $s_2(E^{(k)})=0$ by the
forward theorem. Lemma~\ref{pw:isogeny-corank} therefore gives
$s_2(E/K)=1$. Ring-class missing-vertex transfer now makes the base
product $L(E,s)L(E^{(k)},s)$ have a simple zero. Its second factor is
nonzero at $1$, and hence $a(E)=1$.
\end{proof}
\begin{proof}[Proof of Theorem~\ref{thm:main}]
Use Lemma~\ref{pw:isogeny-corank} to choose the indicated isogeny
orientation before constructing either coefficient system.
The even coefficient system and cyclotomic transfer give both
Proposition~\ref{pw:partitionable-even} and the corank-zero conclusion
of Proposition~\ref{pw:even-base}. The bounded varying companion,
ring-class lower bound, and height comparison then establish
Proposition~\ref{pw:odd-lower}. This discharges the arithmetic premise
before the odd normalization and finite-graph construction of
Section~\ref{pw:odd-stages}.
For the corank-one case, Proposition~\ref{pw:odd-base} fixes a nonvanishing
even companion before the cube dimension, constructs simultaneous unit
vertices, and applies ring-class transfer at the original base. This gives
$a(E)=s_2(E)$ in both cases. The established Gross--Zagier--Kolyvagin implication in
analytic rank zero or one gives $r(E)=a(E)$ and finiteness of the whole
Shafarevich--Tate group. Equation~\eqref{pw:kummer-corank} remains the
underlying corank identity throughout. In particular, neither a finite
two-Selmer dimension nor Mordell--Weil rank was substituted for the
original Selmer corank hypothesis.
\end{proof}
\begin{remark}[The prescribed base]\label{pw:status}
The final companion has Selmer corank zero because it was chosen with
nonzero central value. Thus \eqref{pw:quadratic-corank} supplies the
required corank over $K$; it is not an additional companion hypothesis.
Only the finite realizations and their precisions change as the cube
grows. The valuation constants remain fixed, so transfer applies to the
prescribed original curve without deleting a density-zero set of bases.
\end{remark}
==== END A-pointwise-2-converse-for-elliptic-curves-with-rational-two-torsion//pointwise.tex ====
==== BEGIN A-pointwise-2-converse-for-elliptic-curves-with-rational-two-torsion//preliminaries.tex ====
\section{Selmer coranks and fixed local data}\label{sec:preliminaries}
For an elliptic curve $A/\Q$, put
\[
r(A)=\rank_{\Z}A(\Q),\qquad
a(A)=\ord_{s=1}L(A,s),\qquad
s_2(A)=\corank_{\Z_2}\Sel_{2^\infty}(A/\Q).
\]
The Selmer group uses the local Kummer image at every place.
The analytic continuation and functional equation are supplied by modularity
\cite{BCDT2001}. Write $\varepsilon(A)\in\{1,-1\}$ for the root number.
\begin{lemma}[The three ranks]\label{pr:ranks}
There is an exact sequence
\[
0\longrightarrow A(\Q)\otimes\Q_2/\Z_2
\longrightarrow \Sel_{2^\infty}(A/\Q)
\longrightarrow \Sha(A/\Q)[2^\infty]\longrightarrow0.
\]
Consequently
\[
s_2(A)=r(A)+\corank_{\Z_2}\Sha(A/\Q)[2^\infty].
\]
If $a(A)\in\{0,1\}$, then $r(A)=a(A)$ and the whole group
$\Sha(A/\Q)$ is finite. In that case $s_2(A)=a(A)$.
\end{lemma}
\begin{proof}
The Kummer sequences for multiplication by $2^n$, with the local
Kummer conditions imposed, give
\[
0\longrightarrow A(\Q)/2^n A(\Q)
\longrightarrow \Sel_{2^n}(A/\Q)
\longrightarrow \Sha(A/\Q)[2^n]\longrightarrow0.
\]
Their direct limit is the displayed sequence. Taking $\Z_2$-coranks proves
the identity. The analytic-rank-zero and analytic-rank-one assertions are
the Gross--Zagier--Kolyvagin theorem
\cite{GrossZagier1986,Kolyvagin1990}; see also the precise unrestricted
forward statement in \cite[\S1]{Skinner2020}.
\end{proof}
No finite $2$-Selmer dimension will be substituted for $s_2$.
In particular, $s_2(A)=1$ does not supply a rational point of infinite order:
the exact sequence allows a contribution from the divisible part of $\Sha$.
\begin{lemma}[Parity and change of curve]\label{pr:invariance}
For every elliptic curve $A/\Q$,
\[
(-1)^{s_2(A)}=\varepsilon(A).
\]
A rational isogeny preserves $a$, $r$, and $s_2$. If
$K=\Q(\sqrt{k})$ is quadratic, then
\[
s_2(A/K)=s_2(A/\Q)+s_2(A^{(k)}/\Q).
\]
\end{lemma}
\begin{proof}
The $2$-parity assertion is due to Monsky \cite{Monsky1996};
see also \cite[Theorems~1.4 and~4.19]{DokchitserDokchitser2010},
which states the Selmer-parity result for every prime and credits
Monsky for the case $2$.
For an isogeny and its dual, both composites are multiplication by the
degree. The corresponding maps on rational points and on Selmer groups
respect the Kummer conditions. After rationalizing their Tate modules,
the maps are inverse up to a nonzero scalar; hence the ranks and coranks
agree. The equality of $L$-functions is isogeny invariance.
Over $K$, identify $A^{(k)}$ with $A$, and let $c$ be the nontrivial
element of $\Gal(K/\Q)$. Restriction and the two conjugation
projections $1+c$ and $1-c$ define maps between the sum of the two
Selmer groups over $\Q$ and the Selmer group over $K$ whose composites are
multiplication by $2$. Functoriality of local Kummer images gives these
maps at every place. Rationalization therefore gives the stated corank
identity. This argument does not assert an integral direct-sum
decomposition of the Selmer groups.
\end{proof}
Fix a curve $E/\Q$ of conductor $N$. A finite support $S$ always contains
the primes dividing $2N$. It may be enlarged by finitely many primes when
the fixed construction is prepared. A \emph{local filter} consists of a
real sign and unit squareclasses at $S$ for the variable fundamental
discriminant. The filter is imposed on every relevant Fourier index,
not just the squarefree indices. Its root number is constant.
All variable primes lie outside $S$.
For an odd prime $p$, set
\[
p^*=\left(\frac{-1}{p}\right)p.
\]
Nonempty products of distinct $p^*$ are odd fundamental discriminants.
We also admit the empty product $h=1$, which denotes the untwisted
curve rather than a quadratic field. For a squarefree product $h$ of
such factors, set
\[
t_p=\dim_{\F_2} E[2]^{\operatorname{Fr}_p},
\qquad
w(h)=\frac12\sum_{p\mid h}t_p.
\]
In the rational-two-torsion range, $t_p$ is $1$ or $2$.
Whenever the $2$-primary group below is finite, write
\[
s(h)=\length_{\Z_2}\Sha(E^{(h)}/\Q)[2^\infty].
\]
Thus $s(h)$ is a finite length, whereas $s_2(E^{(h)})$ is a corank.
In particular $s(h)$ is defined when $s_2(E^{(h)})=0$: the exact
sequence in Lemma~\ref{pr:ranks} and finite generation of the dual
Selmer group make its $2$-primary quotient finite. This conclusion
concerns the $2$-primary group; the low-analytic-rank theorem supplies
finiteness of the whole group after analytic nonvanishing is proved.
Let $v$ be a valuation above $2$, normalized by $v(2)=1$.
After the coefficient field has been fixed, its value group is discrete.
The notation $L_*(h)$ denotes $L(E^{(h)},1)\sqrt{|h|}$ divided by the
fixed real or imaginary period appropriate to the filter.
The notation $\mathcal L(h)$ in Section~\ref{cy:section} denotes
this same normalized value.
Every modular parametrization used to form Heegner points is normalized
to send the cusp $\infty$ to the identity of the elliptic curve.
Changing among fixed nonzero modular parametrizations, isogenous curves,
or period conventions contributes only bounded constants to valuations.
Every such dependence is fixed before a binary cube is allowed to grow.
Dependence on a growing number of prime factors will always be stated
explicitly.
The following ledger records the order of choices used in the proof.
It distinguishes bounds from the sizes and precisions needed to realize
the auxiliary primes.
\begin{center}
\small
\begin{tabular}{@{}>{\raggedright\arraybackslash}p{0.29\textwidth}>{\raggedright\arraybackslash}p{0.65\textwidth}@{}}
\toprule
Data or bound & Allowed dependence and order of choice \\
\midrule
Fixed arithmetic constants & The curve, its parametrization and periods,
the fixed part of the support, and the chosen local classes; these are fixed before
the varying prime support or cube dimension. \\
Varying quadratic companions & The same data and a bound $A$ for
the number of prime factors of the companion discriminant; its prime
sizes do not enter. \\
Fixed-companion transfer & The companion field is fixed before the
cube dimension. Constants may depend on its fixed base Selmer data,
as specified in Section~\ref{rd:section}. \\
Finite realization & Choose the cube dimension and finite networks,
then the required coefficient precision and a sufficiently late
arithmetic stage. Prime sizes and stage thresholds may depend on
these choices. \\
\bottomrule
\end{tabular}
\end{center}
==== END A-pointwise-2-converse-for-elliptic-curves-with-rational-two-torsion//preliminaries.tex ====
==== BEGIN A-pointwise-2-converse-for-elliptic-curves-with-rational-two-torsion//proof-roadmap-figure.tex ====
\begin{figure}[htbp]
\centering
\small
\setlength{\tabcolsep}{6pt}
\begin{tabular}{c}
\fbox{\parbox{0.86\textwidth}{\centering
Even coefficients, finite prime networks and cyclotomic transfer\\
\begin{tabular}{@{}c@{\qquad}c@{}}
$\swarrow$ & $\searrow$\\
Uniform even estimate & Corank-zero conclusion\\
(Proposition~\ref{pw:partitionable-even}) & (Proposition~\ref{pw:even-base})
\end{tabular}}}
\\[5pt]
Even estimate $\Downarrow$\\[5pt]
\fbox{\parbox{0.86\textwidth}{\centering
Fix $h_*$ to define $P_h$; choose a bounded-factor companion $k'(h)$\\
Even estimate + ring-class lower bound\\
$\Downarrow$\\
Uniform lower bound for the $2$-divisibility of $P_h$\\
and odd coefficient normalization}}
\\[5pt]
$\Downarrow$\\[5pt]
\fbox{\parbox{0.86\textwidth}{\centering
Fix $k$ with $L(E^{(k)},1)\ne0$ and put $K=\Q(\sqrt{k})$\\
Odd and even tests + finite prime networks split in $K$\\
$\Downarrow$\\
Simple zeros for $E^{(h_x)}$ and nonzero values for $E^{(h_xk)}$\\
with a common normalized index bound, for every $x\ne0$}}
\\[5pt]
$\Downarrow$\\[5pt]
\fbox{\parbox{0.86\textwidth}{\centering
Fixed-$K$ ring-class transfer and $s_2(E/K)=1$\\
$\Downarrow$\\
Analytic rank one at the original vertex $x=0$}}
\end{tabular}
\caption{The even construction proves the corank-zero case and a uniform
estimate; the latter supplies the bound needed to normalize the odd coefficients.
The detector partner $h_*$ is fixed, the bounded-factor companion $k'(h)$
varies only in the lower-bound step, and the final companion $k$ is fixed
before the simultaneous families are constructed. The families and
auxiliary primes may grow with the cube dimension; the normalized
index bound may not.}
\label{fig:proof-roadmap}
\end{figure}
==== END A-pointwise-2-converse-for-elliptic-curves-with-rational-two-torsion//proof-roadmap-figure.tex ====
==== BEGIN A-pointwise-2-converse-for-elliptic-curves-with-rational-two-torsion//ring-determinants.tex ====
\section{Ring-class switches, determinant lattices, and transfer}
\label{rd:section}
This section constructs the derivative classes and proves the determinant
and transfer statements. The arithmetic models, the character \(\Psi\), and their
evaluation maps are those of Lemmas~\ref{rl:finite-model},
\ref{rl:finite-limits}, \ref{rl:evaluation}, and \ref{rl:kernel},
and Propositions~\ref{rl:characters} and \ref{rl:outer-evaluation}.
There are two outputs. Corollary~\ref{rd:ring-lower} bounds a
Heegner-point index at a known simple zero, uniformly in the conductor
and with dependence only on a bound for the number of companion primes.
Theorem~\ref{rd:missing-vertex} instead fixes the companion field and
recovers an unknown base derivative from a family of known simple zeros.
The first step is an integral switch: adding a derivative prime replaces
a finite local condition by its singular counterpart without changing
the determinant valuation. Repeated switches then give the uniform
lower bound. For the missing vertex we need more: the determinants
must occur in one group-ring family with a uniformly bounded clearing
factor. We construct the arithmetic diagrams before applying these two
arguments, and verify their specialization and family properties where
they are used.
Fix \(E/\mathbb Q\), put \(T=T_2(E)\), and normalize \(v(2)=1\). Let
\[
\Lambda=\mathbb Z_2[[t]],\qquad
\mathcal O=\widehat{\Lambda_{(2)}},\qquad
\Omega=\mathbb F_2((t)).
\]
For a perfect cohomological complex \(C\), use the inverse determinant line
\begin{equation}
\mathcal D(C)=\bigotimes_i(\det C^i)^{(-1)^{i+1}}.
\label{rd:det-convention}
\end{equation}
In particular, for a complex in degrees one and two this is
\(\det C^1\otimes(\det C^2)^{-1}\). A basis of this integral line is fixed
whenever a determinant coordinate is mentioned. Changing it multiplies
the coordinate by an integral unit.
The ring data consist of an imaginary quadratic field \(K=\mathbb Q(\sqrt k)\)
split at \(2N\), a product \(h\) of signed good odd primes split in \(K\),
and \((h,k)=1\). The fields with extra units and the CM field of \(E\)
are excluded. Write
\begin{equation}
t_p=\dim_{\mathbb F_2} E[2]^{\operatorname{Fr}_p},\qquad
w(h)=\frac12\sum_{p\mid h}t_p.
\label{rd:weights}
\end{equation}
Choose the modular parametrization \(\pi:X_0(N)\to E\)
normalized by \(\pi(\infty)=0\), translating it by the rational
point \(-\pi(\infty)\) first if necessary. By the Manin--Drinfeld theorem
\cite{Manin1972,Drinfeld1973}, images of cusp
differences are torsion, so one fixed integer kills every
cuspidal ambiguity. No assertion that an arbitrary translate
has torsion value at \(\infty\) is used.
The scalar deformation is trivial at \(2N\), has unit exponent on the
auxiliary conductor inertia, and odd exponent at the indicated split
conductor Frobenii. Its auxiliary prime \(r\) is allowed to change with
the precision. The trace bound in Proposition~\ref{rl:characters} is part of
the data.
\subsection{The local switch and its integral determinant}
\label{rd:local-subsection}
\begin{lemma}[The split local model]\label{rd:local-model}
Let a sequence of inert good primes \(\ell\) have rational Frobenius
lifts tending on \(T\) to a fixed complex conjugation \(J\).
At the unique prime of \(K\) above \(\ell\), the limiting local complex
has
\begin{equation}
H^0=M,\qquad H^1=F\oplus S=M\oplus M,\qquad H^2=M,
\label{rd:local-cohomology}
\end{equation}
with zero differentials in a free split model. Here \(M=T\otimes\Lambda\)
with the relevant scalar coefficient structure. The two degree-one
coordinates of a cocycle are
\begin{equation}
f(c)=c(\gamma^2),\qquad s(c)=c(\sigma),
\label{rd:coordinates}
\end{equation}
where \(\gamma\) is the chosen rational Frobenius lift and
\(\sigma\) is an oriented pro-\(2\) tame generator. The plane
conditions \(F=\{s=0\}\) and \(S=\{f=0\}\) include \(H^0\) and exclude
\(H^2\). Each is its own orthogonal complement for conjugate-Weil
duality. The strict condition \(H^0\) and the relaxed condition
\(H^0\oplus H^1[-1]\) are mutually dual. All four inclusions can be
equipped with compatible integral duality triangles.
\end{lemma}
\begin{proof}
The prime-to-\(\ell\) local cochains are computed by the tame inertia and
residue-field directions. In these two directions the differentials
are formed from inertia minus one and Frobenius minus one, with the
usual Frobenius action on tame inertia. On \(M\), inertia is trivial at
the derivative prime. The square of a rational Frobenius is trivial
on every ring-class character. In the limit it is also the identity on
\(T\), while \(\ell^2\) tends to one. Thus both differentials become zero,
giving \eqref{rd:local-cohomology}.
Here is a finite-precision check of the pairing, needed at \(2\).
At a retained quotient killed by \(2^a\), impose also
\(q=\ell^2\equiv1\pmod{2^{a+1}}\), as well as trivial
Frobenius-square action on the coefficient module at this quotient.
The tame two-generator complex then has zero differentials,
with the coordinates in \eqref{rd:coordinates}. Its degree-one
scalar generators are the unramified and tame characters.
More explicitly, use the presentation
\(\phi\sigma\phi^{-1}\sigma^{-q}=1\).
After augmentation with trivial coefficient action its Fox
resolution has terms of ranks \(1,2,1\), with
\(d^0=0\) and \(d^1=(0,1-q)\), in the ordered finite and
tame coordinates. The diagonal from the same resolution
has zero pure-finite term, unit mixed terms, and pure-tame
coefficient \(\pm\binom q2\). The latter coefficient is
the augmented second derivative of \(\sigma^{-q}\)
together with its adjacent \(\sigma\) in the relation.
Both \(1-q\) and \(\binom q2\) vanish modulo \(2^a\)
under the imposed congruence. Thus this is an actual
split complex and an actual pure-plane zero pairing
at the retained precision.
The field contains the \(2^{a+1}\)-st roots of unity, so
\(-1\) is a \(2^a\)-th power. The norm-residue identity
\((x,x)=(x,-1)\) for the \(2^a\)-Hilbert symbol therefore
makes the square of each scalar generator zero.
The two generators form a basis; local Tate duality makes
their cross pairing a unit. The same calculation applies
coefficientwise to the free Artin coefficient module, including
its nilpotent \(t\) and quadratic variables. Thus the pure
plane restrictions are already zero at every retained quotient.
All these congruences can be imposed simultaneously with the
derivative Frobenius conditions by increasing their precision.
Taking the compatible limit proves integral vanishing of the
pure squares. It is not deduced from skew symmetry in
characteristic two.
Equivalently, in the limiting tame presentation
\(\phi\sigma\phi^{-1}=\sigma^q\) with \(q\to1\), the
rank-\((1,2,1)\) resolution specializes to the exterior
resolution for two procyclic directions. The finite
Hilbert-symbol calculation specifies the cup product on
its split cohomology and rules out a hidden square term.
Conjugation transports the finite Tate coordinate by \(J\) and the
singular coordinate by \(-J\), with possible scalar units coming from
the orientations. Consequently the cross pairing is
\begin{equation}
\beta(x,y)=\epsilon\,\langle x,Jy\rangle,\qquad
\epsilon\in\Lambda^\times,\quad x\in F,\ y\in S,
\label{rd:cross-pairing}
\end{equation}
where \(\langle\, ,\,\rangle\) is the perfect alternating Weil form.
This proves the assertions about orthogonal complements. Since the
restrictions of this explicit pairing to either pure plane are zero as
maps of the split complexes, their isotropy homotopies may be chosen to
be zero. Only a product of two degree-one terms could map to the
degree-two local invariant target; the degree-zero terms introduce
no additional restriction. These zero homotopies agree on the
strict subcomplex. These are the actual
orthogonality data required in the Selmer duality construction
\cite[Theorem~6.3.4]{Nekovar2006}; the compatibility assertion is therefore
an assertion about these displayed complexes and maps. Lemma~
\ref{rl:finite-limits} is used to retain this finite diagram together
with the unchanged local data.
\end{proof}
We now construct global classes with these local coordinates, using
the CM points \(y_m\in E(H_m)\) fixed in Section~\ref{rl:section}.
For the fixed twist put
\(c=|h|\). If \(I\) is a finite set of derivative primes, put
\(n=\prod_{\ell\in I}\ell\); the points used for this set have
conductor \(crn\).
Let \(C_I\) denote the limiting Selmer complex with coefficients
\(T\chi_h\Psi\). Its local conditions are Kummer at \(2N\), full
at the active primes dividing \(h\) and at \(r\), singular at the
primes in \(I\), and unramified elsewhere. At a new derivative
prime \(\ell\notin I\), the unramified condition is the finite
plane \(F_\ell\) of Lemma~\ref{rd:local-model}. Both plane
conditions retain local \(H^0\) and omit \(H^2\), as in that
lemma. The next result constructs the class \(P_I\) in this
complex; its two displayed localizations are taken at the newly added
prime \(\ell\).
\begin{lemma}[Descent and the switch at \(2\)]\label{rd:derivative}
Perform each finite ring-class construction and its weighted transfer
before retaining its target-side data by the construction following
Lemma~\ref{rl:finite-limits}. There is one
integer \(L\ne0\), depending on the fixed curve, parametrization and
fixed local types, such that the classes obtained from the Heegner
derivatives of \(L\) times the point satisfy, for every finite switched
set \(I\),
\begin{equation}
P_I\in H^1(C_I),\qquad
f(P_{I\cup\{\ell\}})=0,\qquad
s(P_{I\cup\{\ell\}})=J f(P_I).
\label{rd:switch}
\end{equation}
The same \(L\) works for every number of active and derivative primes.
The previously imposed singular conditions are retained.
\end{lemma}
\begin{proof}
We use Kolyvagin's derivative operators \cite{Kolyvagin1990}, keeping
their integral descent explicit at $2$. Choose the orientations
coherently. The relative ring-class group at
an inert prime \(\ell\) is cyclic of order \(\ell+1\). For distinct new
primes these groups form a product: the ring-class exact sequence has
kernel the product of the local residue-unit quotients, and the only
global units here are \(1,-1\), whose images in these quotients are
trivial. Choose a generator \(\sigma_\ell\) compatibly with the
tame orientation in Lemma~\ref{rd:local-model}; its pro-\(2\)
component gives the coordinate \(s\) at the retained precision.
Set
\[
D_\ell=\sum_{i=1}^{\ell} i\sigma_\ell^i,\qquad
D_I=\prod_{\ell\in I}D_\ell.
\]
Direct multiplication gives
\begin{equation}
(\sigma_\ell-1)D_\ell=(\ell+1)-
\sum_{i=0}^{\ell}\sigma_\ell^i.
\label{rd:derivative-identity}
\end{equation}
The Hecke correspondence and reduction of its CM isogenies give, for
every class-group translate,
\begin{equation}
\operatorname{Tr}(y_{c r n})=a_\ell y_{c r n/\ell},
\qquad
\widetilde y_{c r n}=\operatorname{Fr}_\ell
\widetilde y_{c r n/\ell}.
\label{rd:heegner-relations}
\end{equation}
These are the point relations, not an application of a residual
irreducibility theorem. A fixed multiple of the parametrization removes
cuspidal terms.
There is a uniform bound on the \(2\)-primary torsion over the union of
the ring-class fields in use. To see this, choose a fixed nonidentity
scalar homothety in the \(2\)-adic Tate image. Square a lift so that it
belongs to every quadratic subgroup under consideration. Multiply it
by its conjugate under complex conjugation. Its action on each
ring-class field is trivial, because conjugation inverts the abelian
ring-class action, while its action on \(T\) is still a fixed
nonidentity scalar. Write that scalar minus one as \(2^e\) times
a unit. Then \(2^e\) kills the torsion in question. The existence of
this fixed scalar, including the CM case, is the Tate-image input
used in Lemma~\ref{rl:kernel}.
Work first with coefficients modulo \(2^a\). Require \(a_\ell\) and
\(\ell+1\) to vanish to a precision substantially larger than \(a\).
Equations \eqref{rd:derivative-identity} and
\eqref{rd:heegner-relations} then make the derivative Kummer class
invariant under the relative ring-class group. In the
inflation--restriction sequence, both the obstruction to descent and
the ambiguity of descent are killed by \(2^e\), since their
coefficient module is killed by \(2^e\). Multiplication by \(2^{2e}\)
therefore gives a descent independent of this ambiguity: descend
\(2^e\) times the class and multiply its lift by \(2^e\).
There is no repetition of this factor for individual primes; descent
is performed once from the field with the complete conductor.
The target coefficient module is obtained by an integral transfer.
For a finite Artin coefficient ring \(A\), let
\(\Delta=\operatorname{Gal}(H_{cr}/K)\), through which the
finite scalar and quadratic characters factor. Shapiro's
cochain map and the coefficient homomorphism
\[
A[\Delta]\longrightarrow A,\qquad
[\sigma]\longmapsto\chi_h(\sigma)\Psi(\sigma)
\]
send the descended class to coefficients \(T\chi_h\Psi\).
For a universal quadratic family, replace the quadratic
value by its group element in \(A[G]\), also including the
fixed base character. Choose inverse actions on both
sides when using that Shapiro convention. Explicitly,
these maps are finite coset sums with the indicated scalar
weights; there is no division by \(\#\Delta\).
These chain maps, in all degrees, are used at this finite stage.
Naturality of restriction and Kummer corestriction supplies their local
lifts and comparison homotopies. Apply the maps to the classes and
these witnesses before passing to bounded target models. Neither
the complex over \(H_{cr}\) nor its induced coefficient module
\(A[\Delta]\) is retained in the limiting diagram. We check the
required target local conditions next.
At a place over \(2N\) these descent extensions are unramified.
For the identity component of a Neron model, the obstruction to
unramified descent vanishes. Indeed, Lang's theorem gives the assertion
on the connected special fiber, and the successive formal-group
quotients are additive residue-field modules, whose unramified cyclic
cohomology vanishes by the normal-basis and trace calculation.
Passing to the inverse limit of these quotients preserves the assertion.
The remaining obstruction is killed by the geometric component-group
order. Those orders have a uniform bound over unramified extensions
of the finitely many fixed local twisting types. Include their product,
the descent factor, and the parametrization factor in \(L\).
Thus the descended class has the integral Kummer condition at \(2N\).
No good-ordinary or semistability condition was used. Transfer with
the scalar character preserves this condition because \(\Psi=1\)
there.
Here is the local calculation, including the division by two. Apply
the derivatives at all primes other than \(\ell\), and write \(X,Y\)
for the resulting lower and upper points. Put \(F_\ell=\operatorname{Fr}_\ell\).
The lower field splits at the \(K\)-prime \((\ell)\), since that ideal
is principal. Reduction therefore gives \(F_\ell^2\widetilde X=\widetilde X\).
Good reduction identifies the prime-to-\(\ell\) torsion with the
torsion of the reduced curve. In the following coordinate formulas,
\(F_\ell\) acts on reduced points, where it satisfies its
Frobenius polynomial. If \(Q\) is a \(2^a\)-division point
of \(LX\), the finite coordinate
of its Kummer class is
\begin{equation}
f=(F_\ell^2-1)\widetilde Q=
\bigl(a_\ell F_\ell-(\ell+1)\bigr)\widetilde Q.
\label{rd:finite-formula}
\end{equation}
We give the descent-cocycle calculation of the other coordinate.
Choose \(Z\) with \(2^aZ=LD_\ell Y\), and choose the descended
cocycle \(b\) so that on the upper-field Galois group it is
\(b(g)=(g-1)Z\). A change by the already bounded descent ambiguity
has no effect on the eventual limit. The crossed identity gives
\[
U=(\sigma_\ell-1)Z-b(\sigma_\ell)
\in E(H_{crn}).
\]
For \(g\) in the upper-field Galois group, compare the two
expansions of \(b(g\sigma_\ell)\); they give \(gU=U\).
Moreover,
\[
2^aU=L\bigl((\ell+1)Y-a_\ell X\bigr).
\]
Both integers \(\ell+1,a_\ell\) are divisible by \(2^a\).
Consequently
\[
U_0=L\frac{\ell+1}{2^a}Y-
L\frac{a_\ell}{2^a}X
\]
is an actual upper-field point with the same \(2^a\)-multiple
as \(U\). Thus \(U-U_0\) is upper-field torsion of bounded
exponent. This comparison with an actual integer quotient
is what bounds the ambiguity; arbitrary division points
would not give this bound. Reduction makes
\(\sigma_\ell\) act trivially and
\(\widetilde Y=F_\ell\widetilde X\). Therefore
reduction of \(b(\sigma_\ell)=-U\) gives its singular coordinate
\begin{equation}
s=\bigl(a_\ell-(\ell+1)F_\ell\bigr)\widetilde Q
\label{rd:singular-formula}
\end{equation}
up to the uniformly bounded descent torsion already described.
The Frobenius polynomial yields the exact operator identity
\begin{equation}
(F_\ell-a_\ell)
\bigl(a_\ell F_\ell-(\ell+1)\bigr)
=a_\ell-(\ell+1)F_\ell.
\label{rd:exact-frobenius}
\end{equation}
Thus \(s=(F_\ell-a_\ell)f\); when \(a_\ell\) vanishes at the
retained torsion precision, this is \(s=F_\ell f\). Taking the
Frobenius limit gives \(s=Jf\).
For the new finite coordinate, reduction of the derivative point is
\begin{equation}
\sum_{i=1}^{\ell}iF_\ell\widetilde X
=\frac{\ell(\ell+1)}2F_\ell\widetilde X.
\label{rd:half-factor}
\end{equation}
The conductor extension at \(\ell\) is totally ramified, so a
Frobenius lift may be chosen in the upper-field local Galois group.
There the descended cocycle restricts to the Kummer cocycle of the
derivative point. If \(v_2(\ell+1)\ge a+1\), its displayed
reduction is \(2^a\) times an \(F_\ell^2\)-fixed point.
Its finite Kummer coordinate is therefore zero modulo \(2^a\).
The additional \(1\) is essential at
\(2\). It is imposed in the choice of the next precision, and does
not multiply the class by a new denominator. Bounded torsion
ambiguities disappear on passing from \(E[2^a]\) to \(E[2^m]\) with
\(a-m\ge e\). Choose \(a\) and the Frobenius precision successively
faster than \(m\).
The calculation is compatible with every translate in the transfer:
transport the place and its Frobenius before applying that translate.
The scalar weight on the lower and upper class is identical. Applying
the same calculation at each \(\ell\in I\) proves the earlier
singular conditions as well.
Finally complete the weighted transfer at each stage and retain the
resulting target Selmer-cone cycles and their local lifts. An equality
of the transferred local classes gives a boundary witness in the
target local complex; retain that witness after contraction as well.
For instance, retain the degree-zero vectors whose boundaries are the
differences in the two switch identities. Include the split local Fox
complexes, their zero isotropy homotopies, and the target comparison
matrices. The target-side construction in Section~\ref{rl:section} proves
that this is a bounded diagram for each fixed active support and
finite switched set. Its bound is independent of the degrees of
the ring-class fields.
When a further derivative prime is selected, extend the
same original finite-stage diagram and its contraction
maps, using the same inner ultrafilter. Do not construct
an unrelated cohomologically equivalent old complex.
The earlier evaluations, including all degree-two
cocycle identities, are consequently retained by the
same contraction maps after each new switch. Lemma~
\ref{rl:finite-limits} applies to these extended diagrams.
Their limiting classes give
\eqref{rd:switch}. The displayed local formulas agree with the
calculation underlying \cite[Proposition~1.7.4]{Howard2012}; Howard's
odd-prime global hypotheses are not being asserted at \(2\).
\end{proof}
\paragraph{The paired arithmetic determinant.}
All the target classes and comparison maps are constructed in the same
finite diagram.
At each Artin precision, use the character of
Proposition~\ref{rl:characters}, the global Selmer cone, and the
Kummer complexes at \(2N\). At a derivative prime impose enough
Frobenius precision for Lemma~\ref{rd:derivative} and also
\(\ell^2\equiv1\pmod{2^{a+1}}\) at every retained quotient killed
by \(2^a\). Lemma~\ref{rd:local-model} then gives the finite,
singular, strict and relaxed inclusions with their common zero
isotropy homotopies. Applying the actual global duality morphism
\eqref{rl:global-duality} constructs their integral localization
and duality triangles. Perform the descent and weighted transfers of
Lemma~\ref{rd:derivative} at each finite stage. Include their target
cycles, local Kummer lifts and target boundary witnesses in this
diagram before applying Lemmas~\ref{rl:finite-model} and
\ref{rl:finite-limits}, using only the target-side data.
All remaining maps have bounded target models as both source and
target; the ring-class source complexes are not among these objects.
A further switch extends these original finite stages, their
contractions and their fixed inner ultrafilter. Theorem~
\ref{rl:limit-evaluation-package} therefore retains the same
degree-zero, degree-one and degree-two evaluations and the
exact-kernel restrictions. In particular, the old evaluations
are not replaced by maps on an independently chosen equivalent
complex.
Here is how a class produces its paired functional. Work over a
height-one localization where the full local complexes at the active
primes and \(r\) are acyclic. By Lemma~\ref{rl:local-acyclic}
this includes \(\mathcal O\) and every characteristic-zero test
used below. These complexes have integral contractions. They
identify the relaxed Selmer complex with its orthogonal complex;
at switched primes the plane condition is already self-dual, and
at \(2N\) the integral Kummer condition is self-dual. Lift the
class through this comparison and compose it with the retained
global duality map. The result is a closed degree-two functional
\(\lambda_I\), integral up to the single initial multiplier and
any fixed old-support correction. Thus it is not an arbitrarily
chosen rational cohomology dual.
When the generic cohomology has dimension one in degrees one and
two and is zero elsewhere, define \(u_I\) to be the coordinate of
\[
[P_I]\otimes[\lambda_I]\in
H^1(C_I)\otimes H^2(C_I)^*
\simeq\mathcal D(C_I)
\]
over the fraction field, in the fixed integral determinant basis.
For an acyclic constituent set this class-dual coordinate to zero.
The generic functional is determined by the same derived duality
at all localizations; different local contractions represent that
functional, not one cochain asserted to be integral everywhere.
For \(I=\varnothing\) write \(u_h\). The rank-one assertions
needed for the arithmetic applications are verified below by
rational central specialization.
The arithmetic switch must now be compared in these determinant
lattices. A localized class may be divisible by \(2\); the next
lemma shows that such a division does not cost a factor at each switch.
\begin{lemma}[Exact determinant switch over a DVR]\label{rd:det-switch}
Let \(O\) be a DVR with fraction field \(F_O\). Suppose that integral
perfect complexes \(C_{\rm str},C_F,C_S,C_{\rm rel}\), their
localization triangles, and their dualities are the base changes of
the diagram in Lemma~\ref{rd:local-model}. Suppose the generic cohomology of these four global complexes is
concentrated in degrees one and two, that
\(H^1(C_F\otimes F_O)=F_OA\), and
\[
f(A)=v_0\ne0,\qquad s(B)=Jv_0,\qquad f(B)=0
\]
for a class \(B\) in the switched complex. The determinant tensors
are formed from each class and its functional under these same
dualities. Then the switched generic cohomology has rank one,
is spanned by \(B\), and
\begin{equation}
v_O(u_F)=v_O(u_S).
\label{rd:valuation-switch}
\end{equation}
Neither localization is required to be primitive.
\end{lemma}
\begin{proof}
For this proof only, cohomology without a coefficient ring means
cohomology over \(F_O\). Nonzero localization of \(A\) gives
\(H^1(C_{\rm str})=0\). The relaxed localization image is
self-annihilating in \(F\oplus S\), so it has dimension two.
It contains the two independent vectors \((v_0,0)\) and
\((0,Jv_0)\) in the opposite planes. Its intersections with either plane are therefore
exactly one-dimensional. This proves the generic rank assertions.
Put \(V=H^1(C_{\rm rel})\), with ordered basis \(A,B\).
Duality identifies \(H^2(C_{\rm str})\) with \(V^*\); denote the
dual basis by \(A^*,B^*\). The two integral triangles are
\begin{equation}
C_{\rm str}\longrightarrow C_F\longrightarrow F[-1]\longrightarrow,
\qquad
C_{\rm str}\longrightarrow C_S\longrightarrow S[-1]\longrightarrow.
\label{rd:triangles}
\end{equation}
Their fraction-field exact sequences are
\[
0\longrightarrow F_OA\longrightarrow F
\xrightarrow{\delta_F}V^*
\longrightarrow F_OA^*\longrightarrow0
\]
and the same sequence with \(A,F,A^*\) replaced by \(B,S,B^*\).
The boundary is the dual of the relaxed localization. In view of
\eqref{rd:cross-pairing} and \(s(B)=Jv_0\), it satisfies
\begin{equation}
\delta_F(x)=\epsilon_F\langle x,v_0\rangle B^*,
\qquad \epsilon_F\in O^\times.
\label{rd:boundary}
\end{equation}
Choose \(x\in F\otimes F_O\) with \(\delta_F(x)=B^*\).
Then \(v_0\wedge x\) has unit volume in the integral determinant
lattice \(\det_O F\), because the alternating form on \(F\) is
unimodular and \(\langle x,v_0\rangle\) is a unit. Although \(x\)
can be nonintegral, this assertion about its exterior product is
an equality of lattices. Similarly choose \(y\in S\otimes F_O\)
mapping to \(A^*\). The volume of \(Jv_0\wedge y\) is a unit in
\(\det_O S\).
The determinant isomorphisms of the two exact sequences send
\[
A\otimes(A^*)^*
\quad\hbox{and}\quad B\otimes(B^*)^*
\]
respectively, up to units and the irrelevant determinant sign, to
\begin{equation}
(A^*\wedge B^*)^*\otimes(v_0\wedge x),
\qquad
(A^*\wedge B^*)^*\otimes(Jv_0\wedge y).
\label{rd:common-tensor}
\end{equation}
These comparisons take place in
\(\mathcal D(C_{\rm str})\otimes\det F\) and
\(\mathcal D(C_{\rm str})\otimes\det S\).
The two plane determinants are identified by a unimodular map.
Hence the two coordinates have the same valuation.
This also explains exactly where the finite indices go. For a
two-term elementary summand
\([O\xrightarrow{\pi^d}O]\) in degrees one and two, its contribution
to the coordinate in \(\mathcal D\) is \(\pi^{-d}\).
Such summands, including those arising from nonprimitive
localization, occur inside the integral isomorphisms
\eqref{rd:triangles}. They are not discarded when writing
\eqref{rd:common-tensor}. All strict torsion contributions are common
to the two sides. The argument does not ask for an integral
complement to \(v_0\); for example \(v_0=2^de_1\) is paired with
the fractional vector \(2^{-d}e_2\), whose exterior product with
\(v_0\) has unit volume.
\end{proof}
\subsection{Rank reduction and the outer DVR}
\label{rd:outer-subsection}
The local switch preserves a determinant valuation. We now use it to
reduce the dimension of any larger specialized Selmer space until a
nonzero minor controls that valuation. The first argument handles each
characteristic-zero height-one prime. A second, outer limit turns a
hypothetical sequence of unbounded losses at $(2)$ into the same rank
reduction problem.
\begin{lemma}[Selecting a rank-two evaluation]\label{rd:rank-two}
Let \(\mathcal G\) be a group, \(H\) a normal subgroup acting trivially
on an absolutely irreducible representation \(W\) over a field \(F_O\),
and suppose that restriction of the relevant cohomology classes to
\(H\) is injective. For two independent classes, their joint
evaluations on \(H\) span \(W\oplus W\). If the finite localization
is the rank-two projection supplied by \(1+\tau\) on the induced
representation, some actual evaluation has rank two. One of the
two classes may be any prescribed nonzero class.
\end{lemma}
\begin{proof}
On \(H\) a crossed homomorphism is an additive homomorphism, and its
image is equivariant for \(\mathcal G\). A proper submodule of
\(W\oplus W\) dominating both factors would impose a nonzero
scalar linear relation on the two evaluations: the direct sum of
two simple modules is semisimple, and
\(\operatorname{End}_{\mathcal G}(W)=F_O\).
That relation, followed by injectivity of restriction, would
contradict independence of the classes. Thus the joint span is
the whole direct sum. Projecting by \(1+\tau\) gives all pairs
in the two-dimensional finite plane.
It remains to pass from a span to a single element of the additive
evaluation image \(U\). If the determinant quadratic polynomial
\(Q\) vanished on \(U\), then its polar form
\(Q(u+v)-Q(u)-Q(v)\) would vanish for \(u,v\in U\).
Consequently
\[
Q\Bigl(\sum_i a_i u_i\Bigr)
=\sum_i a_i^2 Q(u_i)
+\sum_{i<j}a_i a_j
\bigl(Q(u_i+u_j)-Q(u_i)-Q(u_j)\bigr)
=0.
\]
It would vanish on the entire field span, where it is not the zero
polynomial. This contradiction proves the assertion. This
polarization argument is valid in characteristic two as well.
\end{proof}
\begin{proposition}[Nonnegative valuations away from \((2)\)]
\label{rd:away-two}
Assume the simultaneous diagrams, derivative classes, and exact-kernel
evaluation maps of Lemmas~\ref{rl:evaluation}, \ref{rl:kernel},
\ref{rd:local-model}, and \ref{rd:derivative}. For a primitive
constituent whose generic cohomology has rank at most one in degrees
one and two and zero elsewhere, its class-dual determinant
coordinate belongs to \(\Lambda[1/2]\).
This includes a simple-zero constituent and a base with
\(2^\infty\)-Selmer corank one.
\end{proposition}
\begin{proof}
The zero coordinate needs no argument. Otherwise the generic
rank is one and the class spans it. Localize at a height-one prime
\(\mathfrak p\ne(2)\), and denote the residue field by \(F_O\).
Its characteristic is zero. Lemma~\ref{rl:local-acyclic}, the
degree-zero local conditions, and duality give zero fiber
cohomology in degrees zero and three. Canceling unit blocks
therefore gives a free DVR model in degrees one and two.
A primitive vector in its generic kernel has nonzero image in its
\(\mathfrak p\)-fiber: the kernel of a map of free modules over a
DVR is saturated. Include this vector in each localization test.
The two-sign reduction below has the same pattern as
\cite[Lemmas~1.5.3 and~1.6.4]{Howard2012}; the integral switch
and the evaluation hypotheses needed here at \(2\) have been
constructed above.
Suppose first that \(\Psi^2\ne1\) on \(F_O\). Inertia at \(r\)
distinguishes the two summands of the induced representation.
The Tate representation over \(K\) is absolutely irreducible in
characteristic zero: this follows from the open Tate image, or
from its CM normalizer when \(K\) is not the CM field.
Thus the induced representation is absolutely irreducible.
If the test Selmer dimension is \(s>1\), Lemma~\ref{rd:rank-two}
and the finite Chebotarev realization in Lemma~\ref{rl:chebotarev-evaluation}
give a derivative prime with rank-two finite localization,
nonzero on the chosen primitive vector.
Poitou--Tate duality now determines the effect on the fiber.
The relaxed localization image is a self-annihilating plane.
It contains the whole finite plane, and hence equals that plane.
The strict kernel has dimension \(s-2\), and its replacement by
the singular condition adds no new class. The fiber dimension
therefore drops by two. On the generic field, the localization
is nonzero, so Lemmas~\ref{rd:derivative} and
\ref{rd:det-switch} preserve generic rank one, the nonzero
derivative class, and the determinant valuation.
If \(\Psi^2=1\), use conjugation, including its action
\((1+t)\mapsto(1+t)^{-1}\), to split the fiber into two signs.
All prior plane conditions are stable under this involution in
the split local model. The two Galois summands are nonisomorphic
absolutely irreducible representations; the only CM quadratic
self-twist has been excluded. Each sign has one finite and one
singular coordinate. The local pairing in a sign is a
nondegenerate symmetric hyperbolic pairing, whose only isotropic
lines in characteristic zero are the finite and singular axes.
If both signs occur globally, choose one class from each, including
a nonzero component of the primitive vector, to obtain the same
rank-two reduction.
If only one sign occurs and \(s>1\), choose a rank-one finite
localization nonzero on that vector. In this sign the relaxed
image is the finite line. In the other sign its intersection
with the finite line is zero, and self-duality forces it to be
the singular line. A switch retains the old strict kernel of
dimension \(s-1\ge1\), introduces the other sign, and preserves
the total dimension. The next rank-two switch reduces it.
After finitely many such operations the fiber dimension is one.
It cannot fall below one, since a primitive vector from the
generic rank-one kernel survives after each switch.
At fiber dimension one, a minimal self-dual model over the DVR
has one free term in each of degrees one and two and zero
differential. The class and paired functional are integral at
this height-one prime. Their determinant coordinate is
therefore integral. Reversing
\eqref{rd:valuation-switch} proves nonnegative valuation at
\(\mathfrak p\). Applying this at every height-one prime other
than \((2)\), and using that the normal ring \(\Lambda\) is the
intersection of its height-one valuation rings inside its fraction
field, gives membership in \(\Lambda[1/2]\).
For the stated arithmetic instances, the rational central
specialization has Selmer dimension one and no invariants.
Upper semicontinuity bounds the generic dimension by one.
If the generic class is nonzero, it spans this space. This
uses Selmer corank at an unknown base and does not replace that
corank by Mordell--Weil rank.
\end{proof}
\begin{lemma}[The terminal minor estimate]\label{rd:minor}
Let
\[
C_i=[\,O_i^n\xrightarrow{D_i}O_i^n\,]
\]
be based complexes over DVRs with uniformizer \(2\), with generic
rank-one kernel. Let \(z_i\in\ker D_i\) be integral, and let
\(\lambda_i\in 2^{-a}(O_i^n)^*\) annihilate \(\operatorname{im}D_i\),
where \(a\) is fixed. If an \((n-1)\)-minor \(\Delta_i\) has
\(v(\Delta_i)\le b\), the determinant coordinate \(u_i\) of
\(z_i\otimes\lambda_i\) satisfies
\begin{equation}
v(u_i)\ge-a-b.
\label{rd:minor-bound}
\end{equation}
\end{lemma}
\begin{proof}
Permute rows and columns so that the minor is the upper-left block
\(A_i\). Write \(D_i=\left(\begin{smallmatrix}A_i&b_i\\c_i&d_i
\end{smallmatrix}\right)\). Since its rank is \(n-1\), its Schur
complement is zero. Over the fraction field, use the triangular
basis changes with entries \(A_i^{-1}b_i,c_iA_i^{-1}\) to leave
\(A_i\) and a zero one-by-one block. These triangular changes
have determinant one. If \(\alpha_i\) is the last coordinate
of \(z_i\) and \(\beta_i\) the value of \(\lambda_i\) on the last
target basis vector, cancellation gives exactly
\begin{equation}
u_i=\pm\frac{\alpha_i\beta_i}{\det A_i}.
\label{rd:minor-formula}
\end{equation}
Here \(\alpha_i\in O_i\) and \(\beta_i\in2^{-a}O_i\).
This proves \eqref{rd:minor-bound}. If \(n=1\), the empty
minor is \(1\) and the same formula applies.
\end{proof}
\begin{proposition}[Uniform residual valuation from outer rank reduction]
\label{rd:outer-bound}
Assume all of the following properties of a collection of primitive
ring instances:
\begin{enumerate}
\item after any fixed number of switches, minimal \(\mathcal O\)-models
have size bounded independently of \(h\) and of \(K\) with
\(\omega(k)\le A\);
\item the all-sequence evaluation and exact-kernel restriction of
Proposition~\ref{rl:outer-evaluation}, and the finite Chebotarev
realization of Lemma~\ref{rl:chebotarev-evaluation}, hold for these
models, retaining their degree-two cocycle identities;
\item the integral localization and duality diagrams and the
derivative classes of Lemma~\ref{rd:derivative} survive this same
outer limit; the class and its closed dual functional have a common
bounded clearing exponent.
\end{enumerate}
Then there is \(C(E,A)\) such that
\begin{equation}
v_{(2)}(u)\ge-C(E,A)
\label{rd:residual-bound}
\end{equation}
for every simple-zero ring instance. The hypotheses refer to a
single simultaneous construction. In particular, separate
abstract models realizing its individual cohomology groups do
not establish this proposition's arithmetic applicability.
\end{proposition}
\begin{proof}
The preceding paired-diagram construction supplies the common
clearing exponent in the arithmetic application. The bound on
minimal sizes is Lemma~\ref{rl:residual-bound}; in particular,
active conductor primes do not contribute to that size after
residual localization.
Suppose that \(v_{(2)}(u_i)\to-\infty\). All these coordinates
are nonzero. First put the complexes into square two-term form.
At the residual \(\Omega\)-test, moving-prime inertia excludes
global invariants, and the local degree-zero conditions identify
Selmer invariants with those global invariants. Thus \(H^0=0\).
The retained self-duality gives \(H^3=0\) and equal dimensions
in degrees one and two. A minimal \(\mathcal O\)-model therefore
has only two free terms, of equal rank, in those degrees; all other
terms cancel in integral disks. Fix a nonprincipal ultrafilter
\(\mathcal U\) and fix the bounded rank on an ultrafilter-large
set. Write these models as
\[
C_i=[\,O_i^n\xrightarrow{D_i}O_i^n\,].
\]
Transport the class, the uniformly cleared functional, all evaluation
matrices, and the finitely many retained comparison maps and boundary
witnesses through these integral disk contractions. Their entries
remain integral and their identities remain exact. The individual
active-place complexes, whose number may grow with the outer instance,
are not retained as an outer list of objects. Their effect has already
been incorporated in the target cone and its determinant; only the
bounded minimal models and the fixed number of switch diagrams enter
this limit.
Apply Proposition~\ref{rl:outer-evaluation} to these models and
their retained integral diagrams. It gives a DVR \(O_\infty\)
with uniformizer \(2\), characteristic-zero fraction field
\(F_\infty\), and injective all-sequence evaluation, including
restriction to the product of the exact inner kernels.
The induced representation uses one simultaneous complex
conjugation, so the sequence group over the quadratic fields has
index two in the rational sequence group. Lemma~
\ref{rl:irreducibility} gives absolute irreducibility over
\(F_\infty\): the moving inertia character has infinite order,
and the finite Tate-matrix witness survives the outer quotient.
The outer complex may have more cohomology than any inner generic
complex. Its new cycles can be represented by vectors \(w_i\)
for which \(v(D_iw_i)\to+\infty\), without exact inner lifts.
The retained degree-two evaluation matrices are integral, so the
evaluations of \(D_iw_i\) also tend to zero. This is the part of
Proposition~\ref{rl:outer-evaluation} that makes these new classes
available for the following rank reduction.
Choose in each inner generic kernel a primitive vector \(v_i\).
Some coordinate is a unit on a set in \(\mathcal U\), so its
outer limit is nonzero. Thus the outer kernel has dimension
at least one. If that dimension is \(s>1\), apply
Lemma~\ref{rd:rank-two} to two outer classes, one the surviving
primitive vector. Realize their rank-two evaluation by inner
derivative primes. This realization only asks for finite
quotients at each stage. For an approximate inner cocycle
one chooses the retained precision below the valuation of
its coboundary; no exact inner lift of the extra outer class
is being assumed.
More explicitly, retain the original inner index \(i\)
for outer instance \(j\), its ultrafilter \(\mathcal U_j\),
and the outer ultrafilter \(\mathcal U\). Represent the
selected kernel element by \(u_j=(u_{j,i})_i\).
At stage \((j,i)\), apply finite Chebotarev to this actual
element and the old finite data to choose \(\ell_{j,i}\).
Adjoin its local complexes, the derivative cocycle and
the pairing homotopies to the same original stage
diagram with its old contractions. Take the same
\(\mathcal U_j\) and then the same \(\mathcal U\).
The retained coefficient precision may increase more
slowly, but cofinally; the earlier maps and their
arbitrary evaluations are not reconstructed by an
independent choice of subsequence or model.
The localization of \(v_i\) is generically nonzero on a set
in \(\mathcal U\): its outer image is nonzero. Therefore
the original inner spanning class also localizes nontrivially,
even if its multiple of \(v_i\) has large valuation.
Lemma~\ref{rd:det-switch} preserves each inner determinant
valuation exactly. On the outer field, the relaxed image
is the finite plane and the singular switch lowers \(s\)
by two, by the same localization exact sequence as in
Proposition~\ref{rd:away-two}. The new inner generic kernels
again supply primitive vectors, so the outer rank remains
at least one.
Only finitely many switches are necessary: the initial outer
rank is bounded and drops by two. After each fixed number
of switches the required model-size bound remains available.
At the terminal outer rank one, the outer differential has
a nonzero maximal boundary minor. Its inner determinants
\(\Delta_i\) have one finite valuation on a set in
\(\mathcal U\), by the definition of \(O_\infty\).
Lemma~\ref{rd:minor} then gives a fixed lower bound on
\(v(u_i)\), a contradiction.
This proof uses the minor only after rank reduction. Bounded
matrix size alone would not suffice: the matrices \([2^i]\)
have bounded size and unbounded torsion length. In that
example the entire matrix becomes zero in the outer quotient;
the rank-reduction and evaluation hypotheses would have to
supply additional arithmetic operations before a nonzero
terminal minor could be used.
\end{proof}
\subsection{Central specialization and the local indices}
\label{rd:specialization-subsection}
The preceding bounds concern the deformed determinant. We next identify
its value at $t=0$ with the index of the original Heegner point and the
finite Shafarevich--Tate lengths. The comparison retains the whole
integral complex; in particular, every active split conductor prime has
an explicit local contribution.
\begin{lemma}[Derived specialization at constant rational rank]
\label{rd:derived-specialization}
Let \(C\) be a perfect \(\Lambda\)-complex and put
\(C_0=C\otimes_\Lambda^{\mathbf L}\mathbb Z_2\), where \(t=0\).
Suppose its generic rational cohomology and its central rational
cohomology both have dimension one in degrees one and two and
are zero in other degrees. Put \(A=\Lambda_{(t)}\), with residue field \(\mathbb Q_2\).
Let \(z\) be a degree-one cycle and \(\lambda\) a closed
degree-two functional in a finite free model of \(C\otimes_\Lambda A\),
and let \(z_0,\lambda_0\) be their images under \(A\to\mathbb Q_2\).
Then the generic determinant tensor \([z]\otimes[\lambda]\) is
regular at \((t)\) and specializes, under the determinant base-change
isomorphism, to \([z_0]\otimes[\lambda_0]\). Its central valuation is
computed from the entire integral complex \(C_0\), including
its torsion cohomology. It cannot be computed by substituting
the naive quotient of \(H^1(C)\) by \(t\).
\end{lemma}
\begin{proof}
Represent \(C\) by a bounded free complex. Derived reduction
is its termwise reduction, so the determinant line itself
commutes with specialization. For clarity, the cohomology
base-change sequence is
\begin{equation}
0\longrightarrow H^i(C)/tH^i(C)
\longrightarrow H^i(C_0)
\longrightarrow H^{i+1}(C)[t]\longrightarrow0.
\label{rd:tor-basechange}
\end{equation}
These last terms are the relevant \(\operatorname{Tor}_1\)
terms. Diagonalize the complex over the DVR \(A\), or over its
completion \(\mathbb Q_2[[t]]\). A nonunit pivot
would create an additional central cohomology class in its
two adjacent degrees. The assumed constant dimensions
exclude such pivots. Thus over \(\mathbb Q_2[[t]]\) there
are only invertible disks and the two free cohomology lines.
The rational class-dual tensor consequently specializes
to the stated central tensor. Over \(\mathbb Z_2\), all
elementary divisors of \(C_0\), including the torsion supplied
by \eqref{rd:tor-basechange}, remain in its determinant
isomorphism. This proves the assertion.
\end{proof}
\begin{lemma}[A ramified good odd place]\label{rd:active-local}
Let \(F_v/\mathbb Q_p\) be unramified, \(p\ne2\), and let
\(T'=T_2(E)\otimes\chi\), where \(E\) has good reduction
and \(\chi\) is ramified quadratic. Then
\begin{equation}
H^0(F_v,T')=0,\qquad
H^1(F_v,T')=E^\chi(F_v)^{\wedge}_2,\qquad
\operatorname{length}_{\mathbb Z_2}H^2(F_v,T')
=\dim_{\mathbb F_2}E[2]^{\operatorname{Fr}_v}.
\label{rd:local-length}
\end{equation}
The quotient of the full local complex by the integral Kummer
condition therefore contributes precisely this last length
in degree two.
\end{lemma}
\begin{proof}
The pro-\(2\) tame inertia generator acts by \(-1\).
Its complex has differential \(-2\), so its cohomology is
zero in degree zero and \(T'/2T'\) in degree one.
Taking residue-field cohomology gives
\[
H^1(F_v,T')\simeq E[2]^{\operatorname{Fr}_v},\qquad
H^2(F_v,T')\simeq E[2]/(\operatorname{Fr}_v-1)E[2].
\]
The possible residue-cardinality factor on inertia cohomology
is one modulo \(2\). Kernel and cokernel of an endomorphism
of a finite vector space have the same dimension, proving
the length assertion.
The integral local Kummer exact sequence identifies the
\(2\)-adic completion of the points with all of \(H^1(F_v,T')\).
Indeed the inverse Tate module of the local Weil--Chatelet
group is zero here: local duality identifies its
\(2\)-primary part with the dual of the finite completion
of the point group. The point completion is finite because
the residue characteristic is odd. Equivalently, inertia
\(-1\) implies that the local \(2\)-primary torsion is
exactly the Frobenius-fixed \(2\)-torsion. The Kummer
complex includes the derived invariants of this finite
module on reduction, as in Lemma~\ref{rl:kummer}.
The quotient therefore has exactly the degree-two
cohomology displayed above.
\end{proof}
\begin{proposition}[The central determinant formula]
\label{rd:central-formula}
Suppose the product
\[
L(E^{(h)},s)L(E^{(hk)},s)
\]
has a simple zero at \(s=1\).
Use the primitive Heegner sum \(P_h\) and its free-line index
\(j_h\) defined at the start of Section~\ref{rl:section}, and put
\[
s(h)=\operatorname{length}_{\mathbb Z_2}
\Sha(E^{(h)}/\mathbb Q)[2^\infty].
\]
Use the primitive Selmer determinant coordinate \(u_h\)
with the fixed class multiplier of Lemma~\ref{rd:derivative}.
For the primitive Selmer diagrams constructed above, we have
\begin{equation}
\begin{split}
v(u_h(0))
&=2j_h-
\operatorname{length}_{\mathbb Z_2}
\Sha(E^{(h)}/K)[2^\infty]
-2\sum_{p\mid h}t_p+O_E(1)\\
&=2j_h-s(h)-s(hk)-4w(h)+O_{E,A}(1),
\qquad \omega(k)\le A .
\end{split}
\label{rd:central-value}
\end{equation}
The first error includes the auxiliary prime and the fixed
parametrization multiplier. The second has no dependence
on the number of primes dividing \(h\).
\end{proposition}
\begin{proof}
The two rational twists have analytic ranks zero and one.
The forward low-analytic-rank theorems give finiteness of
their whole Shafarevich--Tate groups; equivalently the
Heegner simple-zero theorem over \(K\) gives rank one
and finite Shafarevich--Tate group there. Thus all finite
lengths in this proof refer to known finite groups.
The specialization comparison is made at each finite stage, before
the target data are retained. Put \(c=|h|\). At the split auxiliary
prime \(r_i\), the point distribution relation is
\[
\operatorname{Tr}_{H_{cr_i}/H_c}(y_{cr_i})
=(a_{r_i}-\sigma_{r_i}-\sigma_{\bar r_i})y_c,
\]
where the two Artin symbols have inverse ring-class actions. At
\(t=0\), the scalar character is trivial, while the rational quadratic
character has value \(\chi_h(r_i)\) on both symbols. The completed
weighted sum therefore gives exactly
\(L m_i P_h\), where \(m_i=a_{r_i}-2\chi_h(r_i)\).
This point identity is retained on bounded target complexes as follows.
Let \(C_h^{\rm Kum}\) be a fixed free model of the ordinary Kummer Selmer complex
over \(K\), and let \(b_h\) represent the Kummer class of \(P_h\).
At the retained coefficient precision, enlargement of the ramification
set and relaxation at the active and auxiliary primes give a target
comparison \(j_i:C_h^{\rm Kum}/2^i\to C_i/(t)\). Each retained local
condition is an isomorphism on degree zero and an injection on degree
one, so the Selmer-cone exact sequence injects its degree-one
cohomology into global cohomology. Kummer naturality and the point
relation therefore give, after contraction, a target degree-zero vector
\(v_i\) with
\[
z_i(0)-L m_i j_i(b_h)=d v_i.
\]
Include this vector, the comparison matrix, and the associated local
localization triangles in the same target diagram. Their ranks are
bounded for this fixed \(h,K\), independently of \(r_i\) and of
\([H_{cr_i}:K]\). All statements are reduced to the same slower
cofinal precision if needed. Lemma~\ref{rl:finite-limits} now retains
the displayed boundary identity. The trace bound gives
\(v_2(m_i)\le c_E\); hence their limit \(m\in\mathbf Z_2\) on the
same ultrafilter is nonzero and has \(v_2(m)\le c_E\).
The active local quotients specialize
to Lemma~\ref{rd:active-local}, and the Kummer conditions at
\(2N\), including their differential lattices, are unchanged.
The central point is therefore nonzero, so generic
and central rational cohomology both have rank one.
To specialize the paired functional, choose its representative
over \(\Lambda_{(t)}\). At an active ramified prime, inertia
minus one is \(-2\), a unit in this ring. At \(r\), the
central Frobenius determinants are nonzero by the trace bound.
Their local complexes thus contract over \(\Lambda_{(t)}\).
Composing these contractions with retained global duality gives
the same generic functional as the contractions over
\(\mathcal O\), and specializes to rational central Kummer
duality. Lemma~\ref{rd:derived-specialization} applies to this
representative; no regularity is claimed for an arbitrary
\(\mathcal O\)-cochain representing the functional.
The ordinary integral Kummer complex has compact Selmer
cohomology in degree one. Its degree-two torsion has
length equal to that of the finite quotient of the
discrete Selmer group by its maximal divisible subgroup.
This follows by derived base change to
\(\mathbb Q_2/\mathbb Z_2\) and the exact local Kummer
conditions; the base-change terms must be retained.
Here that quotient is \(\Sha(E^{(h)}/K)[2^\infty]\).
The free parts of degrees one and two are paired
unimodularly by integral duality. Torsion in degree
one and its dual contribution in degree three have
bounded length, by the fixed-scalar bound on global
torsion.
For completeness, an elementary divisor of size \(2^d\)
whose cokernel is in degree \(i\) contributes
\((-1)^{i+1}d\) to the coordinate in the inverse
determinant. Consequently the degree-two
Shafarevich--Tate torsion subtracts its length. On
the free line, a point of index \(2^{j_h}\) and its
functional each contribute \(j_h\), giving \(2j_h\).
At an active prime \(p\mid h\), both primes of \(K\)
above \(p\) are governed by
Lemma~\ref{rd:active-local}, each with length \(t_p\).
Relaxation therefore subtracts exactly \(2t_p\).
There is no bound hiding an error at every such prime.
At \(r\), the central scalar character is trivial;
the local degree-two determinant has the valuation
of \(r+1-a_r\chi_h(r)\) at each split place.
The trace condition
\[
v_2(2-a_r),\ v_2(2+a_r)\le c_E,\qquad
v_2(r-1)>c_E
\]
makes both valuations at most \(c_E\). It also bounds
the trace multiplier
\(a_r-2\chi_h(r)\) on the central point.
At primes over \(2N\) the integral Kummer condition
is retained. The local twisting types form a fixed
finite set and \(K\) is split there; all comparison
indices, including those of Neron differential
lattices at \(2\), are therefore bounded in terms
of \(E\). This proves the first equality, using
Lemma~\ref{rd:derived-specialization} to compare
the generic tensor with the central tensor.
We spell out the quadratic comparison needed for
the second equality. Set
\[
A_h=\operatorname{Res}_{K/\mathbb Q}(E^{(h)}_K),
\qquad B_h=E^{(h)}\times E^{(hk)}.
\]
There are isogenies in both directions whose
composites are multiplication by \(2\).
The isogeny-invariance identity for the BSD quotient,
with finite Shafarevich--Tate groups, is
\begin{equation}
\frac{\#\Sha(A_h)\operatorname{Reg}(A_h)
\operatorname{Per}(A_h)\prod_p c_p(A_h)}
{\#A_h(\mathbb Q)_{\rm tors}
\#A_h^\vee(\mathbb Q)_{\rm tors}}
=
\frac{\#\Sha(B_h)\operatorname{Reg}(B_h)
\operatorname{Per}(B_h)\prod_p c_p(B_h)}
{\#B_h(\mathbb Q)_{\rm tors}
\#B_h^\vee(\mathbb Q)_{\rm tors}} .
\label{rd:isogeny-identity}
\end{equation}
Regulators and periods are taken with compatible
dual and Neron normalizations. This is the
Cassels--Tate isogeny identity. The elliptic-curve local-measure
formula is due to Cassels \cite[Theorem~1.1]{Cassels1965}.
For the general abelian-variety identity used here, see the proof of
\cite[I, Theorem~7.3, especially (7.3.1), pp.~97--100]{MilneADT2006}
and \cite[Section~4.1]{DokchitserDokchitser2010}. This identity requires
finite Shafarevich--Tate groups, as already established at these
noncentral twists, but does not assume the BSD conjecture.
The fixed-degree composites bound the rank-one
regulator indices, the torsion ratios, and the
differential and period indices by fixed powers
of \(2\). The identity is essential: a statement
that the maps on Shafarevich--Tate groups have
kernel and cokernel killed by \(2\) would not
bound their lengths uniformly.
At every split prime of \(K/\mathbb Q\) the component
factors of \(A_h\) and \(B_h\) agree. In particular
this cancels all primes dividing \(h\).
At an odd prime \(p\mid k\) outside \(2N\), the curve
over \(K\) has good reduction. Its Weil restriction
has connected special fiber: the reduction of
Weil restriction over the ramified quadratic
extension is an extension of the good special
fiber by a connected additive group. Thus its
component factor is \(1\). Of the two rational
twists, one is good and the other is ramified
quadratic with good original curve. The latter
has \(2\)-part of its component number equal to
\(2^{t_p}\). This follows as well from the
ramified tame description in
Lemma~\ref{rd:active-local}: its connected
reduction is unipotent, and its local
\(2\)-primary torsion is the fixed
\(2\)-torsion. At other good nonsplit places
the component factors are \(1\); at \(2N\)
there are only the fixed local comparisons.
Taking valuations in \eqref{rd:isogeny-identity}
therefore gives
\begin{equation}
\operatorname{length}\Sha(E^{(h)}/K)[2^\infty]
=s(h)+s(hk)+\sum_{p\mid k}t_p+O_E(1).
\label{rd:sha-comparison}
\end{equation}
Since \(0\le t_p\le2\), the sum over \(k\) is at
most \(2A\). Substituting this equality and
\(2\sum_{p\mid h}t_p=4w(h)\) proves the result.
\end{proof}
\begin{proof}[Proof of Corollary~\ref{rd:ring-lower}]
The arithmetic diagrams constructed with Lemma~\ref{rd:derivative},
their paired functionals, and the model bound of
Lemma~\ref{rl:residual-bound} give the hypotheses of
Propositions~\ref{rd:away-two} and \ref{rd:outer-bound}.
Those propositions give nonnegative height-one valuations
away from \((2)\) and a lower bound \(-C(E,A)\) at \((2)\).
Normality of \(\Lambda\) gives \(2^{C(E,A)}u_h\in\Lambda\).
Its constant coefficient has valuation at least \(-C(E,A)\).
Apply \eqref{rd:central-value}.
\end{proof}
\subsection{The universal correction and bounded clearing}
\label{rd:clearing-subsection}
For transfer to a missing vertex, the individual determinant coordinates
must belong to one quadratic group-ring family. We first remove the
Euler factors at primes unused by a character. We then clear the
remaining denominators with a factor whose central valuation and
negative Laurent support are bounded independently of the cube size.
The base curve and the companion field are fixed in this argument.
Fix \(K\) and a binary family \eqref{rd:binary-data}, based at
\(h_0=1\). Put
\[
G=(\mathbb Z/2\mathbb Z)^b,\qquad R=\Lambda[G].
\]
Let \(g_q\in G\) be the inertia bit:
\(g_q(x)=(-1)^{\lambda_q(x)}\).
Augmentation is denoted by subscript \(0\).
The universal class is constructed before making any character
evaluation. Fix this finite family and \(G\). At precision \(i\)
take the Heegner point of conductor \(r_i\prod_{q\in Q}q\).
Let \(\varepsilon:G_K\to G\) be the universal quadratic character.
In the finite Shapiro transfer of Lemma~\ref{rd:derivative}, use
the coefficient map
\[
A_i[\operatorname{Gal}(H_{r_i\prod q}/K)]
\longrightarrow A_i[G],\qquad
[\sigma]\longmapsto\Psi_i(\sigma)[\varepsilon(\sigma)],
\]
with the same inverse-action convention on both sides. Complete
this weighted coset sum to obtain a target Selmer-cone cycle over
\(A_i[G]\), with local lifts and boundary witnesses, and then
contract these target data. The target-side rank bound applies
for this fixed family and gives one trace class over \(\Lambda[G]\).
Evaluation at \(x\) commutes with the finite coefficient map and
every target identity, so gives the full trace for that character.
If scalar models have already been chosen, compare the evaluated
model with them through the original target cochains and retain the
resulting comparison matrices and contraction homotopies. There are
only finitely many characters for this fixed \(G\). At \(t=0\), the
universal split-prime multiplier is
\[
a_{r_i}-2[\varepsilon(\operatorname{Fr}_{\mathfrak r_i})].
\]
The conjugate Frobenius values agree because the character values
have exponent two. Its evaluation at \(x\) is
\(a_{r_i}-2\chi_{h_x}(r_i)\), so the central boundary comparison
used in Proposition~\ref{rd:central-formula} is retained in this
same group-ring diagram before character evaluation.
It includes Euler factors at primes unused by \(x\), which the next
lemma removes. No inverse limit over growing \(G\) or growing active
support is used in this construction.
The accompanying dual functional is constructed over
\(\mathcal O[G]\), not separately at its characters. At a new
prime and at \(r\), the full local complex is acyclic modulo
the maximal ideal by Lemma~\ref{rl:local-acyclic}, hence has an
integral contraction over this local ring. Use these contractions
to lift the class to the orthogonal Selmer complex, and then apply
the retained integral global duality. This gives a closed integral
functional, up to one fixed old-support power of \(2\) when such
a support is included. The present base is \(h_0=1\).
\begin{lemma}[Removing unused primes in the group ring]
\label{rd:unused}
Suppose the universal relaxed trace class and its duality
diagram have been constructed over \(R\).
There is \(\eta\in\mathcal O[G]\) such that
\(\eta u_{\rm rel}\) has each character coordinate
equal to the primitive determinant coordinate, up
to a \(\Lambda\)-unit. The correction has no
denominator accumulating with \(\#Q\).
\end{lemma}
\begin{proof}
At an unused split prime \(q\), write \(z\) for the
full scalar Frobenius at one orientation. Conjugation
gives \(z^{-1}\) at the other orientation. The
point trace contributes
\[
e_q=a_q-z-z^{-1},
\]
whereas relaxation of the singular quotient contributes
\begin{equation}
l_q=
\det(1-q^{-1}z\operatorname{Fr}_q\mid T)
\det(1-q^{-1}z^{-1}\operatorname{Fr}_q\mid T).
\label{rd:unused-factors}
\end{equation}
Using reciprocal Frobenius conventions changes only
the corresponding unit conventions in the paired
determinant. The localization triangle gives
\(u_{\rm rel}=e_q^2u_{\rm prim}/l_q\) at this
unused prime. The square is required because the
point and its dual both occur.
Both factors are units in the local ring
\(\mathcal O[G]\). To check this it suffices to
reduce modulo its maximal ideal, generated by \(2\)
and the augmentation ideal. There the scalar
is \((1+t)^m\) with \(m\) odd, and the residual
Frobenius polynomial is nonzero at this nonconstant
scalar. More precisely, before character
evaluation the following identity holds modulo \(2\):
\begin{equation}
(1+a_qz+z^2)(1+a_qz^{-1}+z^{-2})
=a_q^2+z^2+z^{-2}
=(a_q+z+z^{-1})^2 .
\label{rd:residual-euler}
\end{equation}
Thus \(l_q/e_q^2-1\in2\mathcal O[G]\), and
\begin{equation}
\eta=
\prod_{q\in Q}
\left(1+
\frac{l_q/e_q^2-1}{2}(1+g_q)\right)
\in\mathcal O[G].
\label{rd:eta}
\end{equation}
Its factor evaluates to \(l_q/e_q^2\) when \(q\)
is unused and to \(1\) when \(q\) is active.
This proves the claim. The apparent inertia
projector denominator is canceled inside each
factor before taking the product.
\end{proof}
\begin{lemma}[Laurent expansion and simultaneous elimination]
\label{rd:schur}
There is a natural identification
\begin{equation}
\mathcal O/2^M\mathcal O
=(\mathbb Z/2^M\mathbb Z)((t)).
\label{rd:laurent}
\end{equation}
Suppose \(A=\operatorname{diag}(A_1,\ldots,A_m)\)
has entries in \(\Lambda\), each residual determinant
is nonzero, and
\(\operatorname{ord}_t(\det\overline A_j)\le h\).
If \(B=A+2E\) with \(E\) over \(\Lambda\), then
every entry of \(B^{-1}\bmod2^M\) has no term below
\(t^{-Mh}\). The same bound holds for a Schur
complement \(P-QB^{-1}S\), where \(P,Q,S\)
have power-series entries. The bound is independent
of \(m\).
\end{lemma}
\begin{proof}
For a denominator \(s\in\Lambda\setminus(2)\),
write \(s=t^d v+2w\), with \(v\) a power-series
unit. Its inverse modulo \(2^M\) is its finite
geometric expansion about \(t^dv\), which has
finite negative support. Conversely all Laurent
series of finite negative support occur after
inverting \(t\). This proves \eqref{rd:laurent}.
It also proves that an element of \(\mathcal O\)
has a unique series \(\sum a_nt^n\), \(a_n\in\mathbb Z_2\),
with \(v(a_n)\to+\infty\) as \(n\to-\infty\),
and that its valuation is \(\min_n v(a_n)\).
Write \(\det A_j=t^{d_j}v_j+2w_j\), with
\(d_j\le h\), and put
\[
C_j=\frac{\operatorname{adj}(A_j)}{t^{d_j}v_j},
\qquad C=\operatorname{diag}(C_j).
\]
Each entry of \(C\) has pole at most \(h\), and
\(BC=1+2F\), where entries of \(F\) have the same
pole bound. Therefore
\begin{equation}
B^{-1}\equiv C\sum_{j=0}^{M-1}(-2F)^j
\pmod{2^M}.
\label{rd:inverse-expansion}
\end{equation}
The \(j\)-th summand has pole at most \((j+1)h\).
Adding more entries in a matrix product cannot
worsen this pole bound. Multiplication by
power-series matrices proves the Schur assertion.
For a block over \(R\) whose augmentation is \(A\),
the block is invertible over \(\mathcal O[G]\):
its residual determinant in the local ring is a
unit. Each sign evaluation is \(A+2E_x\) with
power-series entries, so the same estimate holds
at all characters. If \(S_0=0\), its Schur
complement has augmentation exactly \(P_0\).
\end{proof}
\begin{lemma}[Clearing a bounded complex]\label{rd:clearing}
Let \(D\) be a based complex of finite free
\(\mathcal O[G]\)-modules of total graded rank
at most \(r_0\), with \(D_0\) defined over
\(\Lambda\). Suppose \(D_0\otimes_\Lambda^{\mathbf L}\mathbb Z_2\)
has rational cohomology of dimension one in degrees
one and two, zero elsewhere, and total integral
torsion length at most \(h_1\).
Let \(z\) be an integral cycle and let
\(\lambda\in2^{-a_0}(D^2)^*\) be a closed functional.
Assume every character constituent is either
generically acyclic or has the same generic rank-one pattern;
in the generically acyclic case its determinant coordinate
is defined to be zero. Multiplication of these
coordinates by any \(\eta\in\mathcal O[G]\)
is permitted. Then there are \(f,d\in\mathcal O[G]\)
such that
\begin{equation}
f_xu_x=d_x,\qquad
f_0\in\Lambda,\qquad
v(f_0(0))\le C(r_0,h_1,a_0)<\infty.
\label{rd:clearing-equations}
\end{equation}
If all entries of \(D_x\) have bounded negative
Laurent support at each precision, uniformly
in \(b\), the same is true of \(f_x\).
No such bound on \(d_x,z,\lambda,\eta\) is required.
\end{lemma}
\begin{proof}
Put the central integral complex into elementary-divisor
bases over \(\mathbb Z_2\). Its positive pivot
valuations sum to at most \(h_1\). Lift these
constant integral basis changes. Formally cancel
the same pivot positions upstairs, recording the
pivot numerators and all denominators in the
resulting rational basis changes and determinant
comparison. There are at most \(r_0\) such
operations. A product \(P\) of sufficiently
large, but bounded, powers of their numerators
clears every denominator. Its degree in the
differential entries is bounded in terms of
\(r_0\); its augmentation belongs to \(\Lambda\)
and has central valuation bounded in terms of
\(r_0,h_1\).
On constituents where \(P_x\ne0\), the remaining
complex is one-by-one in degrees one and two.
If its differential is nonzero, the cycle
identity forces the projected cycle to be zero,
so the same rational determinant expression
gives the stipulated zero coordinate. Otherwise
the expression is the product of the projected
cycle and functional, with the canceled-pivot
factor. Clearing denominators gives
\[
u_x=2^{-a_0}\eta_x Q_x/P_x,\qquad
Q\in\mathcal O[G].
\]
Take \(f=2^{a_0}P^2\) and \(d=\eta P Q\).
On a constituent where a pivot first vanishes,
\(P_x=0\), so both sides of \(f_xu_x=d_x\)
are zero. Thus the identity holds on every
character. Since \(f\) is a polynomial of
bounded degree in the bounded differential
entries, its central valuation and pole
properties follow. All unused-prime
corrections and eliminated-block units occur
in \(d\); none is inserted into \(f\).
\end{proof}
\begin{proposition}[The bounded clearing factor for the family]
\label{rd:family-clearing}
Assume the compatible universal Selmer and Heegner
diagrams and their primitive specializations.
Suppose every character constituent of the universal Selmer complex
is generically acyclic or has generic
cohomology of dimension one in degrees one and two and zero
elsewhere. If the fixed base has \(2^\infty\)-Selmer corank
one over \(K\), the corrected coordinates of
Lemma~\ref{rd:unused} admit the factorization
\eqref{rd:clearing-equations}, with central
valuation and fixed-precision pole bounds
independent of \(b\) and \(\#Q\).
\end{proposition}
\begin{proof}
Use the integral universal class and paired functional just
constructed. To apply Lemma~\ref{rd:clearing}, it remains to
reduce the complex to bounded size without losing uniform control
of its negative Laurent powers.
At augmentation, the localization triangles
present the complex as the old complex extended
by a direct sum of singular local blocks in
degrees one and two, one block for each new
prime. Its differential is upper triangular,
with the old complex a subcomplex. This graded
model lifts over \(R\): split disks over the
augmentation local ring, identify the minimal
parts, and lift the invertible basis changes.
This is an application of the actual finite
diagram construction, rather than just an
equality of Euler characteristics.
Each new block has determinant \(l_{q,0}\).
Its residual \(t\)-order is uniformly bounded.
In the reducible residual range the residual
Frobenius polynomial is \((X+1)^2\), and an
odd scalar exponent gives order two at each
split place; thus order four suffices for the
two-place block. Eliminate the entire new
block over \(\mathcal O[G]\). By
Lemma~\ref{rd:schur}, the resulting complex
\(D\) has the bounded old size, has augmentation
equal to the old complex, and has a uniform
pole bound at every fixed precision.
The central cohomology of the old complex is
the fixed base Kummer cohomology, modified only
by the bounded auxiliary-prime terms. Its
rational dimension is one in degrees one and
two. The quotient of the discrete Selmer
group by its maximal divisible subgroup is
finite because its Pontryagin dual is a
finitely generated \(\mathbb Z_2\)-module.
For this fixed \(K,E\), its length is a finite
constant, even when the Shafarevich--Tate
group itself is not known to be finite.
Together with global torsion and the trace
bound at \(r\), this bounds all central
torsion lengths of \(D_0\).
Apply Lemma~\ref{rd:clearing}. The determinant
of the eliminated block is a unit over
\(\mathcal O[G]\), so its inverse is an
allowed integral multiplier. Include it
and \(\eta\) from \eqref{rd:eta} in the
numerator multiplier of that lemma.
This proves the assertion.
The dependence here is on fixed \(K,E\);
it is different from the uniform
\(C(E,A)\) of Proposition~\ref{rd:outer-bound}.
No uniform bound on the finite Selmer quotient
as \(K\) varies is asserted or needed for
the missing-vertex argument.
\end{proof}
\begin{proposition}[Assembly of the arithmetic diagrams]
\label{rd:assembly}
For the ring data stated at the start of this section, the actual
finite Heegner and Selmer constructions satisfy the simultaneous
diagram hypotheses of Propositions~\ref{rd:away-two},
\ref{rd:outer-bound}, and \ref{rd:central-formula}.
The bounds needed for the lower-bound argument depend only on
\(E\), the fixed local data, and a bound \(A\) on \(\omega(k)\).
For the binary data \eqref{rd:binary-data} with fixed \(E,K\)
and base Selmer corank one, assume in addition that every
character constituent of the universal Selmer complex is generically acyclic
or has generic cohomology of
dimension one in degrees one and two and zero elsewhere.
Then the constructions satisfy all the hypotheses of
Proposition~\ref{rd:family-clearing}, with
bounds independent of the cube dimension and of its
new-prime support.
\end{proposition}
\begin{proof}
The paired arithmetic diagram constructed after
Lemma~\ref{rd:derivative} contains the actual weighted target classes,
local Kummer lifts, target comparison and boundary witnesses, and
simultaneous duality triangles. The target-side construction
gives its inner rank bound for each fixed finite conductor support;
no ring-class source complex is passed to the limit.
Theorem~\ref{rl:limit-evaluation-package} retains
their evaluations and exact-kernel restrictions. The same-stage
extension in Proposition~\ref{rd:outer-bound} shows explicitly
that these data remain compatible after each further switch.
Lemma~\ref{rl:residual-bound} bounds the minimal models in terms
of \(E,A\) and the number of retained derivative primes, not the
active conductor support. Outer rank reduction uses only a bounded
number of switches, hence only finitely many of these bounds.
The paired functional has the common clearing exponent established
above. These are the three hypotheses of
Proposition~\ref{rd:outer-bound}; the characteristic-zero
evaluations and local acyclicity give those of
Proposition~\ref{rd:away-two}. The actual central trace and
\(\Lambda_{(t)}\)-representative, together with the local
quotients of Lemma~\ref{rd:active-local}, were verified in the
proof of Proposition~\ref{rd:central-formula}.
For a binary family, Section~\ref{rd:clearing-subsection}
constructs the universal trace and its dual functional over the
group ring before evaluating characters. Lemma~\ref{rd:unused}
then removes the unused-prime factors integrally. The bounded
complex in Proposition~\ref{rd:family-clearing} comes from the
actual localization triangles and simultaneous Schur elimination,
with the order-four residual local bound. Its central torsion is
controlled by the finite quotient of the fixed base Selmer group
and the bounded auxiliary-prime terms. Together with the stated
constituent-rank assumption, these give all the hypotheses of
Lemma~\ref{rd:clearing}.
For varying \(K\), only its bounded ramification count enters
the residual-model and local-comparison constants. For the final
binary argument \(K\) is fixed, so its finite base Selmer
quotient may enter the clearing constants. At \(2N\) the
original Kummer conditions have been retained throughout; the
construction imposes no ordinary or semistable hypothesis.
\end{proof}
\subsection{Coefficient comparison and the missing vertex}
\label{rd:transfer-subsection}
The bounded clearing factor reduces the last step to finitely many
coefficient congruences. A parity argument on a sufficiently large
binary cube makes a nonzero address agree with the base to the required
precision. If the base derivative were zero, that congruence would
contradict the upper bound already supplied at every nonzero address.
The common-zero count in the following proof is the characteristic-two
Chevalley--Warning principle
\cite[Section~1, equation~(1)]{HeathBrownChevalleyWarning}; its short parity
proof is included here.
\begin{lemma}[A nonzero address with prescribed coefficient bits]
\label{rd:bits}
Let \(A_1(x),\ldots,A_q(x)\) be specified Laurent
coefficients of sign evaluations of elements of
\(\mathcal O[G]\). For \(M\ge1\), if
\begin{equation}
b>q(2^M-1),
\label{rd:bit-threshold}
\end{equation}
there is \(x\ne0\) with
\(A_i(x)\equiv A_i(0)\pmod{2^M}\) for every \(i\).
\end{lemma}
\begin{proof}
Write the element supplying the \(i\)-th test as
\(a_i=\sum_{g\in G}a_{i,g}(t)g\), and let \(n_i\) be the Laurent
degree being tested. Lemma~\ref{rd:schur} gives integral Laurent
coefficients, so
\[
z_i=\sum_{g\in G}[t^{n_i}]a_{i,g}(t)\,g\in\mathbb Z_2[G].
\]
The group \(G\) is finite, hence coefficient extraction commutes
with character evaluation: \((z_i)_x=A_i(x)\). Apply
Lemma~\ref{cy:binary} to these \(q\) integral group-ring elements.
Its bound is exactly \eqref{rd:bit-threshold}; no bound on the
Laurent degrees or on the coefficients enters this argument.
\end{proof}
\begin{lemma}[Transfer through the bounded factor]
\label{rd:binary-transfer}
Suppose \(f,d\in\mathcal O[G]\),
\(u_x\in\Lambda[1/2]\), and \(f_xu_x=d_x\)
for every character. Suppose
\(f_0\in\Lambda\) and
\(c=v(f_0(0))<\infty\).
If every \(f_x\bmod2^M\) has no powers below
\(-L(M)\), then for
\[
M>c,\qquad
b>(L(M)+2)(2^M-1)
\]
there is \(x\ne0\) satisfying
\begin{equation}
v\bigl(u_x(0)-u_0(0)\bigr)\ge M-2c.
\label{rd:binary-congruence}
\end{equation}
\end{lemma}
\begin{proof}
Apply Lemma~\ref{rd:bits} to the coefficients
of \(f_x\) in degrees \(-L(M),\ldots,0\)
and the constant coefficient of \(d_x\).
At its nonzero solution, all negative
coefficients of \(f_x\) are divisible by
\(2^M\), and its constant coefficient is
congruent to \(f_0(0)\). Hence the latter
has valuation \(c\), and
\(v_{\mathcal O}(f_x)\le c\).
Since \(d_x\) is integral over \(\mathcal O\),
\(v_{\mathcal O}(u_x)\ge-c\), and the same
holds for \(u_0\). The Laurent expansion
in Lemma~\ref{rd:schur} then shows that all
their coefficients lie in \(2^{-c}\mathbb Z_2\).
They have no negative coefficients because
they belong to \(\Lambda[1/2]\).
The constant-term convolution converges
\(2\)-adically, and gives
\[
[t^0]d_x\equiv [t^0]f_x\,u_x(0)
\pmod{2^{M-c}},
\qquad
[t^0]d_0=f_0(0)u_0(0).
\]
Use the imposed congruences and divide by
\(f_0(0)\). This proves
\eqref{rd:binary-congruence}. The possibly
large negative support of \(d_x\) was never
tested; only its constant coefficient was used.
\end{proof}
\begin{proof}[Proof of Theorem~\ref{rd:missing-vertex}]
For every character the primitive generic
cohomology has rank at most one. At nonzero
vertices this follows from the simple-zero
theorem; at zero it follows from
\eqref{rd:base-corank} and rational
specialization. There are no generic invariants, and duality
pairs degrees one and two; thus each primitive constituent is acyclic
or has exactly the rank-one pattern. The unused-prime local blocks
are acyclic over the fraction field, by the nonzero factors
\eqref{rd:unused-factors}. The universal Selmer complex has the
same generic pattern, as required by
Proposition~\ref{rd:family-clearing}. With this hypothesis
verified, Proposition~\ref{rd:assembly} supplies the compatible
universal arithmetic construction. By Proposition~\ref{rd:away-two}
and Lemma~\ref{rd:unused}, the corrected
coordinates belong to \(\Lambda[1/2]\).
Proposition~\ref{rd:central-formula} gives
\[
v(u_x(0))\le B_1\quad(x\ne0)
\]
for a constant \(B_1\) depending only on
\(B,E,K\) and the fixed local data.
If the base derivative vanished, the
Gross--Zagier formula would make the
central Heegner sum torsion: its canonical
height is a fixed nonzero multiple of
that derivative. Here only the conductor-one formula at
the fixed base \(h=1\) is used, with the corrected primitive
sum and the split Heegner data of
\cite[Theorem~1.1]{CaiShuTian2014}.
At the base this implies \(u_0(0)=0\).
Indeed, if the generic rank is zero the
coordinate was defined to be zero. If the
generic rank is one, its central rank is
also one by \eqref{rd:base-corank}, so
Lemma~\ref{rd:derived-specialization}
identifies its specialized class with
the zero rational Kummer class of that
torsion point.
Proposition~\ref{rd:family-clearing} supplies
\(f,d\) with a common bound \(c_*\) on
\(v(f_0(0))\) and common pole bounds
\(L(M)\). Choose
\[
M>\max\{c_*,B_1+2c_*\}
\]
and then take \(b\) above the threshold
of Lemma~\ref{rd:binary-transfer}.
That lemma gives a nonzero vertex with
\(v(u_x(0))\ge M-2c_*>B_1\), a
contradiction. Thus the base derivative
is nonzero. The classical split Heegner
sign is odd, so the base product has
a simple zero.
\end{proof}
\begin{remark}[The simultaneous arithmetic interface]
\label{rd:interface-status}
The determinant argument can be reused only with
the simultaneous arithmetic construction verified
in Proposition~\ref{rd:assembly} and
Theorem~\ref{rl:limit-evaluation-package}.
Neither the published duality theorem nor the
classical Heegner formulas alone supplies that
interface.
\end{remark}
==== END A-pointwise-2-converse-for-elliptic-curves-with-rational-two-torsion//ring-determinants.tex ====
==== BEGIN A-pointwise-2-converse-for-elliptic-curves-with-rational-two-torsion//ring-limits.tex ====
\section{Ring-class interpolation and integral deformations}
\label{rl:section}
The ring-class argument has two outputs: a lower bound for the index of
a Heegner point at a known simple zero, and a transfer of nonvanishing to
an unknown base. The first is uniform as the companion field varies
with bounded ramification; the second fixes that field before a binary
family grows. We state both outputs before constructing their cochain
and evaluation maps. Their determinant proofs occupy
Section~\ref{rd:section}.
Fix $E/\mathbf Q$ of conductor $N$, and write $T=T_2E$ and
$\Lambda=\mathbf Z_2[[t]]$. The imaginary quadratic field
$K=\mathbf Q(\sqrt k)$ satisfies the split Heegner hypothesis at $2N$.
We omit the two fields with extra units and, when $E$ has complex
multiplication, its CM field. Let $h$ be a product of distinct signed
good odd primes, all split in $K$, with $(h,k)=1$; $h=1$ is allowed.
The primes dividing $h$ are called the \emph{active conductor primes}.
The character $\chi=\chi_h$ is the corresponding quadratic character
over $\mathbf Q$.
Fix a modular parametrization $\pi:X_0(N)\to E$ with $\pi(\infty)=0$.
Write $H_c$ for the ring-class field of $K$ of conductor $c$, and $y_c$
for the image under $\pi$ of a conductor-$c$ CM point, with fixed CM
orientations. The restriction of $\chi_h$ to $G_K$ factors through
$\operatorname{Gal}(H_{|h|}/K)$. The primitive weighted Heegner sum is
\[
P_h=\sum_{\sigma\in\operatorname{Gal}(H_{|h|}/K)}
\chi_h(\sigma)\,\sigma(y_{|h|}).
\]
Through the quadratic-twist identification, it is a point of
$E^{(h)}(K)$. When the product of $L(E^{(h)},s)$ and $L(E^{(hk)},s)$
has a simple zero,
the Heegner-point theorem makes $P_h$ nontorsion and
$E^{(h)}(K)$ has rank one. Let $j_h$ be the $2$-adic index valuation of $P_h$
in the free rank-one lattice
$E^{(h)}(K)^\wedge_2/\mathrm{torsion}$. The quantities $w(h)$,
$s(h)$, and $s(hk)$ have the meanings fixed in
Section~\ref{sec:preliminaries}; the two Shafarevich--Tate lengths are
finite in this simple-zero case.
\begin{corollary}[Ring-class lower bound]
\label{rd:ring-lower}
Every simple-zero ring instance in the stated data satisfies
\begin{equation}
2j_h\ge 4w(h)+s(h)+s(hk)-C(E,A),
\qquad \omega(k)\le A.
\label{rd:ring-lower-equation}
\end{equation}
\end{corollary}
For the second output, fix $K$ and the base $h_0=1$. A binary family
has the form
\begin{equation}
h_x=\prod_{q\in Q}(q^*)^{\lambda_q(x)},\qquad
\lambda_q:\mathbb F_2^b\longrightarrow\mathbb F_2\text{ linear},
\quad x\in\mathbb F_2^b.
\label{rd:binary-data}
\end{equation}
All primes in $Q$ are good, odd, disjoint from the fixed support, and
split in $K$. All vertices have the same local squareclasses at $2N$
and the same required real sign.
\begin{theorem}[The fixed-\(K\) missing-vertex implication]
\label{rd:missing-vertex}
Fix the ring data \(E,K\), with
\begin{equation}
\operatorname{corank}_{\mathbb Z_2}
\operatorname{Sel}_{2^\infty}(E/K)=1.
\label{rd:base-corank}
\end{equation}
Suppose binary families \eqref{rd:binary-data}
exist for arbitrarily large \(b\), and for
every nonzero \(x\) the product
\(L(E^{(h_x)},s)L(E^{(h_xk)},s)\) has a simple zero and
\begin{equation}
2j_{h_x}-4w(h_x)-s(h_x)-s(h_xk)\le B
\label{rd:nonzero-bound}
\end{equation}
with \(B\) independent of the family. Then
\begin{equation}
\operatorname{ord}_{s=1}
\bigl(L(E,s)L(E^{(k)},s)\bigr)=1.
\label{rd:missing-conclusion}
\end{equation}
This implication does not assume Mordell--Weil
rank one or finite Shafarevich--Tate group
at the base.
\end{theorem}
To prove these assertions, we compare the Heegner-point indices with
integral Selmer determinants. The present section constructs the
limiting complexes and the maps needed to test their classes at new
primes. Taking limits of differential matrices alone would not suffice:
a cycle in the limit may lift only to approximate cycles at finite
precision, and its localization must still be evaluation of a crossed
cocycle.
There are two passages to a limit. In the inner construction $K$ and a
finite active support are fixed while an auxiliary scalar-conductor
prime $r_i$ and the coefficient precision vary. Additional inert
\emph{derivative primes} will realize evaluations and switch local
conditions. In the outer construction $K$ and the active support may
vary, but $\omega(k)$ and the number of retained derivative primes are
bounded. This second passage gives the uniform lower bound; the final
fixed-$K$ transfer instead uses the bounded clearing factor of
Proposition~\ref{rd:family-clearing}.
We use
\[
A_i=(\mathbf Z/2^i\mathbf Z)[t]/(t^i).
\]
The same constructions work over $A_i[G]$ and $\Lambda[G]$, where $G$ is a
fixed finite elementary abelian $2$-group and the additional character has
values in $G$. Statements uniform in the number of generators of $G$ will
be identified separately; finite-diagram compactness by itself is a statement
for fixed $G$.
\subsection{Free models with maps to the original cochains}
\begin{lemma}[Finite Artin models]\label{rl:finite-model}
Let $(A,\mathfrak m,k)$ be an Artin local ring, and let $D$ be a complex of
flat $A$-modules. Suppose $H^*(D\otimes_A k)$ is finite dimensional and
supported in $[a,b]$. There is a bounded finite free complex $C$, with
\[
\operatorname{rank}_A C^j=\dim_k H^j(D\otimes_A k),
\]
and a strong deformation retraction
\[
i:C\longrightarrow D,\qquad p:D\longrightarrow C,\qquad
pi=1,\qquad 1-ip=dh+hd.
\]
Thus maps to the original complex, as well as maps between finite models,
are available. Any fixed finite collection of chain maps, homotopies,
triangles, and perfect pairings can be transported to these models.
\end{lemma}
\begin{proof}
Flat modules over an Artin local ring are free, including modules of infinite
rank. Here is a useful version of the argument. Lift a basis of $M/\mathfrak
mM$ to a flat module $M$, and map the corresponding free module to $M$.
Its cokernel is zero because $\mathfrak m$ is nilpotent. Flatness shows that
the kernel has zero reduction, and nilpotence again makes the kernel zero.
Split the residual complex as its cohomology, with zero differential, plus
contractible two-term disks. Lift all the graded bases. These lifts are
graded isomorphisms: an inverse modulo $\mathfrak m$ lifts to an inverse by
a finite geometric series. Let $d_0$ be the lifted disk differential and
write $d=d_0+\epsilon$. All coefficients of $\epsilon$ belong to
$\mathfrak m$. For the disk contraction $(i_0,p_0,h_0)$, arrange
$h_0^2=h_0i_0=p_0h_0=0$ and
$1-i_0p_0=d_0h_0+h_0d_0$. The operators
$1+h_0\epsilon$ and $1+\epsilon h_0$ have finite geometric inverses.
The elementary perturbation identities give the differential
\[
d_C=p_0\epsilon(1+h_0\epsilon)^{-1}i_0
\]
on the lifted residual cohomology, and give
\[
i=(1+h_0\epsilon)^{-1}i_0,\quad
p=p_0(1+\epsilon h_0)^{-1},\quad
h=h_0(1+\epsilon h_0)^{-1}.
\]
Substitution, using $(d_0+\epsilon)^2=0$, proves the asserted contraction
identities. These calculations are valid for infinite disk sums: each
operator is a module homomorphism and every geometric series terminates
at the nilpotence exponent of $\mathfrak m$.
For a map between original complexes use $p'fi$; insert the contraction
homotopies when composing maps. A quasi-isomorphism between bounded free
complexes has a contractible cone and hence a homotopy inverse. This gives
the stated finite-diagram assertion, including the homotopies expressing
triangles and perfect duality.
\end{proof}
For a profinite group and a finite free coefficient module over $A$,
continuous inhomogeneous cochains are flat: locally constant functions are
the filtered union of the modules of functions on finite clopen partitions,
and these modules are finite free. Their reduction is the cochain complex
with reduced coefficients. The same observations apply to restrictions,
mapping cones, and local conditions given by finite free complexes. In the
applications below the groups are Galois groups of totally imaginary number
fields or nonarchimedean local fields. Their $2$-cohomological dimension and
the usual finiteness theorems give the bounded residual cohomology required
in Lemma~\ref{rl:finite-model}.
\begin{lemma}[Limits of specified diagrams]\label{rl:finite-limits}
Suppose diagrams over $A_i$, or over $A_i[G]$, have uniformly bounded cohomological amplitudes and graded free
ranks, and include the chain homotopies and inverses for every asserted
identity or equivalence. After taking a cofinal subsequence, or a
nonprincipal ultrafilter limit, their matrices define the same diagram of
perfect complexes over $\Lambda$, or $\Lambda[G]$.
Moreover, the contractions into the original cochains may be retained for
the evaluation construction in Lemma~\ref{rl:evaluation}. Adding finitely
many new objects or maps does not require changing already chosen models
for the old objects.
\end{lemma}
\begin{proof}
There are finitely many possible graded ranks. Fix them on an
ultrafilter-large set. At a fixed precision $j$, all matrix entries belong
to the finite ring $A_j$ (or $A_j[G]$), so have unique ultrafilter limits.
These limits are compatible as $j$ varies. Matrix identities, including
the homotopy-inverse identities, are finite polynomial identities and pass
to the limit. New models and maps may be mapped to old models by their
already fixed contractions. Passing to any slower cofinal precision has
the same old limits. For the all-sequence evaluation statements below,
fix this ultrafilter once and for all, and keep the original contraction
at every stage. Later arithmetic diagrams must extend these same
stages. Restriction to an ultrafilter-large set and passage to slower
cofinal coefficient precision preserve every old evaluation on every
old group sequence. An arbitrary later subsequence or a different
ultrafilter is not allowed in this assertion. Notice that this argument preserves specified
arithmetic maps; it does not create an arithmetic map from a formal
description of its expected properties.
\end{proof}
\paragraph{Target-side data.}
The ring-class fields used to construct a class need not have bounded
degree. Their cochain complexes are not objects of the limiting diagram.
At each finite stage, first perform descent, Shapiro transfer, the
coefficient weighting, and every coset sum. These operations give
cochains on the fixed target field \(K\), with their local lifts. Only
then apply Lemma~\ref{rl:finite-model} to the target global and local
complexes and their Selmer cones.
Here is the required inner rank bound. Fix \(K\), the active support,
a finite set of derivative slots, and, when used, \(G\). The permitted
ramification set \(S_i\) consists of this fixed support and a bounded
number of moving primes. Modulo the maximal ideal of \(A_i\) or
\(A_i[G]\), all scalar characters are trivial and the representation is
\(E[2]\). Put \(F=K(E[2])\), a fixed field. Kummer theory bounds
\(H^1(G_{F,S_i},\mathbf F_2)\) by the rank of the \(S_i\)-unit group
and the dimension of the fixed group \(\operatorname{Cl}(F)[2]\).
Both bounds are uniform: the number of primes above \(S_i\) is bounded
and passing to the \(S_i\)-class group only decreases the required
class-group bound. The fixed finite group
\(\operatorname{Gal}(F/K)\) and inflation--restriction then bound
\(H^1(G_{K,S_i},E[2])\). Degree zero is bounded by two, and the global
Euler characteristic bounds degree two; higher degrees vanish because
\(K\) is totally imaginary. Local cohomology has uniform bounds as
well: the local degrees over \(\mathbf Q\) are at most two, there are
only finitely many fixed local conditions at \(2N\), and the moving
good-prime and derivative complexes have fixed ranks. The Selmer-cone
exact sequence gives the same conclusion for every target local
condition used below. Thus Lemma~\ref{rl:finite-model} gives uniformly
bounded target ranks. This inner bound may depend on the fixed active
support and is not a bound for a sequence of growing binary families.
The separate uniform bound after completion at \((2)\) is
Lemma~\ref{rl:residual-bound}; binary-family uniformity is obtained only
after the elimination in Proposition~\ref{rd:family-clearing}.
For clarity, let \(D_i\) be a target cochain complex and write its
contraction as \((\iota_i,p_i,h_i)\). A transferred cycle \(z_i\)
is retained as the vector \(p_i z_i\). Its representative in the
original target cochains differs from \(z_i\) by the boundary
\(d h_i z_i\). If an arithmetic identity is
\(z_i-z'_i=d b_i\), retain the target vectors
\(p_i z_i,p_i z'_i,p_i b_i\); their identity is exact. For a target
comparison map \(f_i:D_i\to D'_i\), retain
\(p'_i f_i\iota_i\), together with the contraction homotopy. For
example,
\[
p'_i f_i z_i-p'_i f_i\iota_i p_i z_i
=d\bigl(p'_i f_i h_i z_i\bigr).
\]
Apply this also to local lifts in the Selmer cone, specialization
comparisons, and the homotopies expressing duality and local switches.
Each specified cycle or boundary witness is a vector in a bounded-rank
target term, so adds only finitely many coordinates. Boundary witnesses
are compressed after transfer and have target rank; no list indexed by
the ring-class cosets is retained. The lemma therefore retains all
these identities without retaining a source cochain complex or a
growing transfer matrix. The original target contractions remain
available in degrees zero, one, and two for all-sequence evaluation.
All operations use the same original stages and ultrafilter; a loss of
precision is accommodated by one common slower cofinal precision for
the entire finite diagram. Adding derivative slots means extending
each old stage by finitely many target objects, not taking a limit with
an unbounded number of objects.
\subsection{Kummer conditions, including their derived degree zero}
\begin{lemma}[Integral Kummer local condition]\label{rl:kummer}
Let $F$ be a finite extension of a nonarchimedean local field of
characteristic zero, and let $B/F$ be any elliptic curve. Put
$M_F=B(F)^\wedge_2=\varprojlim_n B(F)/2^nB(F)$. There is a perfect local
condition
\[
U_F=M_F[-1]\longrightarrow R\Gamma(F,T_2B).
\]
For every $n\geq1$, derived reduction has
\[
H^0(U_F\otimes^{\mathbf L}\mathbf Z/2^n)=B(F)[2^n],\qquad
H^1(U_F\otimes^{\mathbf L}\mathbf Z/2^n)=B(F)/2^nB(F).
\]
Its map in degree zero is the isomorphism onto
$H^0(F,B[2^n])$, and its map in degree one is the finite Kummer injection.
The condition is its own exact orthogonal complement under local Tate
duality. These assertions hold at $F\mid2$, for all reduction types, and
without a condition on $B[2]$.
\end{lemma}
\begin{proof}
The local analytic description of $B(F)$ shows that $M_F$ is a finitely
generated $\mathbf Z_2$-module. Its torsion is the finite group
$B(F)[2^\infty]$. Consequently $M_F$ has a free resolution of length at
most one. Also $H^0(F,T_2B)=0$, since $B(F)[2^\infty]$ has bounded
exponent. The integral Kummer injection into $H^1(F,T_2B)$ therefore
defines the displayed morphism using the truncation in degrees at most
one of $R\Gamma(F,T_2B)$.
Tensor the free resolution of $M_F$ with $\mathbf Z/2^n$. Its two
cohomology groups are $M_F[2^n]$ and $M_F/2^nM_F$. The inverse limits of
the ordinary Kummer exact sequences identify their maps with the two
maps in the statement. In particular finite torsion invariants have not
been discarded.
For orthogonality, compose cup product and the Weil pairing with the local
invariant map. A morphism
\[
M_F[-1]\otimes^{\mathbf L}M_F[-1]
\longrightarrow\mathbf Z_2[-2]
\]
is determined by its homomorphism $M_F\otimes M_F\to\mathbf Z_2$:
the possible $\operatorname{Tor}_1$ is in degree one and gives no extra
degree-zero morphism to this target. The indicated homomorphism vanishes,
because the finite Kummer image annihilates itself at every level by local
Tate duality \cite[I, Theorem~3.2 and Corollary~3.4]{MilneADT2006}.
Choose a nullhomotopy in free models. The induced morphism
\[
R\Gamma(F,T_2B)/U_F
\longrightarrow R\operatorname{Hom}_{\mathbf Z_2}
(U_F,\mathbf Z_2)[-2]
\]
is a quasi-isomorphism. This can be checked after reduction modulo $2$:
degree one is the perfect pairing of the finite Kummer quotient with the
Kummer image, and degree two is dual to $B(F)[2]$. Its perfect cone is
therefore zero by derived Nakayama. This proves exact orthogonality and
also its compatibility with derived reduction. The derived duality of
the ambient local cochains is \cite[Proposition~5.2.4]{Nekovar2006};
the Kummer condition and its orthogonality have been constructed above.
No part of the proof uses an ordinary filtration or a classification of
reduction types.
\end{proof}
Where the scalar deformation is trivial, extend this local condition by
derived scalar extension to $A_i$ or $A_i[G]$. For a finite free
coefficient representation $M$ unramified at a good prime $v$, use the
inflated residue-field complex
\[
U_v=[\,M\xrightarrow{\operatorname{Fr}_v-1}M\,]
\quad\text{in degrees }0,1.
\]
Its quotient in full local cochains is the singular local complex.
The full local condition is the identity on local cochains, and its
orthogonal strict condition is zero. The conditions at a derivative
prime are different: once its local cochains have the split model
constructed in Lemma~\ref{rd:local-model}, they are the truncations
specified below. Write that model as
\[
L_v=D^0\oplus(F_v\oplus S_v)[-1]\oplus D^2[-2],
\]
where the differential is zero, $D^0$ is dual to $D^2$, and the finite
and transverse planes $F_v,S_v$ are each isotropic. Put
\[
U_{\rm str}=D^0,\qquad
U_{\rm rel}=D^0\oplus(F_v\oplus S_v)[-1],\qquad
U_F=D^0\oplus F_v[-1],\qquad U_S=D^0\oplus S_v[-1].
\]
Then $U_{\rm str}^{\perp}=U_{\rm rel}$ and each plane condition is
its own orthogonal complement. Every displayed condition maps
isomorphically onto local $H^0$ and injectively into local $H^1$.
Thus they satisfy precisely the detection hypotheses used below.
Lemma~\ref{rd:local-model} proves their finite arithmetic realization
and supplies the isotropic-plane nullhomotopies. These truncated
conditions are not identified with the full local condition.
For a finite permitted set $S$ the Selmer complex is, by definition,
\begin{equation}\label{rl:selmer-definition}
C_S=\operatorname{Cone}\left(
R\Gamma(G_{K,S},M)\oplus\bigoplus_{v\in S}U_v
\longrightarrow\bigoplus_{v\in S}R\Gamma(K_v,M)
\right)[-1].
\end{equation}
At $2N$ take the Kummer conditions just constructed. Restriction and
localization triangles are the triangles of this cone. The finite rings
$A_i$ and $A_i[G]$ are Frobenius coefficient rings; alternatively the
duality maps may be checked after reduction to their residue field.
Exact local orthogonality and global duality therefore give
\begin{equation}\label{rl:global-duality}
C_S\simeq R\operatorname{Hom}(C_S^{\perp},A_i)[-3],
\end{equation}
with the appropriate dual representation. This application uses the
orthogonality hypotheses of
\cite[Theorem~6.3.4]{Nekovar2006}; the theorem is not being used to assert
the existence of a varying-prime limit. Complex places contribute zero
Tate complexes. If the character is inverted by the nontrivial
automorphism of $K/\mathbf Q$, conjugate transport followed by the Weil
pairing identifies the two representations over the same scalar ring.
All maps and nullhomotopies in this paragraph are finite diagrams to
which Lemma~\ref{rl:finite-limits} applies.
\subsection{The moving conductor character}
\begin{proposition}[Finite-precision ring-class characters]\label{rl:characters}
Fix finitely many split primes of $K$ above odd rational primes, outside
$2N$, and orient one prime in each conjugate pair. There are distinct
split rational primes $r_i$ avoiding all the fixed data and characters
\[
\xi_i:\operatorname{Gal}(H_{r_i}/K)\longrightarrow
\mathbf Z/2^{n_i}\mathbf Z,
\qquad n_i\longrightarrow\infty,
\]
such that $\xi_i$ is surjective on the inertia quotient at $r_i$, is zero
at the primes above $2N$, and has odd value at each prescribed oriented
prime. We can require $r_i\to1$ in $\mathbf Z_2$ and
\begin{equation}\label{rl:trace-bound}
v_2(2+a_{r_i}(E)),\quad v_2(2-a_{r_i}(E)),\quad
v_2\det(1\pm\operatorname{Fr}_{r_i}\mid T)\ \leq c_E.
\end{equation}
The constant $c_E$ is independent of the list and of $K$.
\end{proposition}
\begin{proof}
We give the compatibility argument before imposing the Tate condition.
Choose prime ideals $\mathfrak p_1,\ldots,\mathfrak p_s$ with pairwise
disjoint conjugate pairs, including the oriented prescribed primes and
the primes above $2N$, and enough additional split prime ideals to
generate $\operatorname{Cl}(K)$. Such additional ideals may avoid any
fixed finite set. Put weight $w_j=1$ at a prescribed odd prime and
$w_j=0$ at every additional prime and every prime above $2N$.
The conjugate weight is $-w_j$. Let
\[
R=\ker\bigl(\mathbf Z^s\longrightarrow\operatorname{Cl}(K)\bigr),
\]
choose a basis $m^{(a)}$ of $R$, and choose $\beta_a\in K^\times$ with
$(\beta_a)=\prod_j\mathfrak p_j^{m_j^{(a)}}$.
Work first over $F_\infty=K(\mu_{2^\infty})$ and adjoin $2^n$-th roots
of the finitely many $\beta_a/\overline\beta_a$. Prescribe on these
roots the multipliers
\begin{equation}\label{rl:kummer-prescription}
\zeta_{2^n}^{\,-\sum_j m_j^{(a)}w_j}.
\end{equation}
The minus sign fixes the choice of connecting map in the ring-class
sequence and could be reversed throughout. These prescriptions respect
every Kummer relation. Indeed, if
\[
\prod_a(\beta_a/\overline\beta_a)^{b_a}
\in(F_\infty^\times)^{2^n},
\]
take its valuation at a prime above each $\mathfrak p_j$ of nonzero
weight. These odd primes are unramified in $F_\infty/K$. Thus
$2^n$ divides $\sum_a b_am_j^{(a)}$ for each such $j$, proving the
required compatibility of~\eqref{rl:kummer-prescription}. Primes over
$2$ have weight zero and impose no division by a ramification index.
Units impose no additional weighted relation. Kummer duality supplies
the prescribed automorphism of the radical extension over $F_\infty$.
At a rational prime $r$ split in a sufficiently large finite cyclotomic
subfield, the power-residue character of $\mathbf F_r^\times$ is
surjective onto $\mu_{2^n}$. The split ring-class exact sequence is
\[
1\longrightarrow
(\mathcal O_K/r\mathcal O_K)^\times/(\mathbf Z/r\mathbf Z)^\times
\longrightarrow\operatorname{Pic}(\mathbf Z+r\mathcal O_K)
\longrightarrow\operatorname{Cl}(K)\longrightarrow1.
\]
There is no unit correction under our exclusion of the extra-unit
fields. The first group is $\mathbf F_r^\times$, with a principal
relation measured by $\beta/\overline\beta$ at the oriented prime over
$r$. Consequently~\eqref{rl:kummer-prescription} says exactly that its
primitive power-residue character extends to the class group with the
specified values $w_j$. Since the ring-class extension is unramified at
$2N$, value zero on Frobenius means that the resulting local character
is trivial there. Conjugation acts by inversion.
We now show that imposing these finitely many radical conditions cannot
force either forbidden Tate trace. In the non-CM case the image of
$G_{K(\mu_{2^\infty})}$ is an open subgroup of
$\operatorname{SL}_2(\mathbf Z_2)$, by the open-image theorem
\cite{Serre1972}. Its index in the corresponding fixed-$E$ image changes
by at most two when $K$ changes. Its commutator subgroup contains a
fixed open subgroup $U_E$ of $\operatorname{SL}_2(\mathbf Z_2)$.
One may see uniformity by starting with a principal congruence subgroup
in the fixed image: every index-two subgroup contains its squares, and
the commutators of the resulting fixed congruence subgroup are open.
Because the radical extension over $F_\infty$ is abelian, these
commutators act trivially on it. Hence every permitted Tate coset
contains a translate of $U_E$.
For completeness, in the CM case let $M$ be the CM field. The image over
$KM(\mu_{2^\infty})$ contains a fixed open subgroup of the norm-one
Cartan; the index changes by a bounded factor, as $[KM:\mathbf Q]\leq4$.
Choose $\gamma\in G_{KM}$ with
$\chi_{\rm cyc}(\gamma)=5^4=625$. This is possible uniformly: the
cyclotomic image of $G_{KM}$ has index at most four in
$\mathbf Z_2^\times$, so contains fourth powers. On the radical Galois
group, conjugation by $\gamma$ is raising to $625$, because all the
radicands lie in $K$. On the Tate image over $KM$, conjugation is
trivial. Therefore
\[
\gamma u\gamma^{-1}u^{-625},\qquad
u\in G_{KM(\mu_{2^\infty})},
\]
fixes every prescribed radical and acts on $T$ as $\rho_T(u)^{-624}$.
Raising to the fixed nonzero power $624$ carries an open subgroup of the
one-dimensional norm-one Cartan onto an open subgroup. This proves the
required uniform kernel statement also in the CM case. The open Cartan
image used here is the CM part of the usual $\ell$-adic image theorem
\cite[Chapter~IV, \S2.2, Remark]{Serre1968}.
On a coset of an open subgroup of $\operatorname{SL}_2$, trace is not
identically $2$ or $-2$. The same holds on a coset of an open norm-one
Cartan subgroup; on the other Cartan-normalizer component trace is zero.
The relevant determinant-one sets are compact and have finitely many
cosets modulo the fixed open subgroup. Choose in each coset an element
whose trace is neither forbidden value. The largest of the finitely
many resulting valuations is a constant depending only on $E$.
Finally restrict the prescribed radical automorphism and such a Tate
lift to a finite Galois extension, including as many roots of unity as
required. Chebotarev gives infinitely many rational primes with this
Frobenius class; choose the place over the prime to realize the chosen
representative. They split in $K$ and have arbitrarily high prescribed
congruence to $1$. Since
$\det(1\pm\operatorname{Fr}_r)=1\pm a_r+r$, a congruence precision
larger than the fixed trace bound gives all of~\eqref{rl:trace-bound}.
This is a finite Chebotarev condition at each stage, not a demand that
one prime split in an infinite extension.
\end{proof}
Choose $n_i$ large enough that $1+t$ has order dividing $2^{n_i}$ in
$A_i^\times$, and put $\Psi_i=(1+t)^{\xi_i}$. For example $n_i\geq2i$
is sufficient. After the matrix limits, the inertia exponent at the
moving prime is a unit of $\mathbf Z_2$, and a prescribed oriented
Frobenius exponent is odd. We can orient the generator of inertia so
its action is exactly $1+t$.
\begin{lemma}[Local height-one tests]\label{rl:local-acyclic}
For the individual coefficient system $T\chi_h\Psi$, use full local
conditions at the moving prime $r$ and the active primes dividing $h$,
without adding primes from a larger binary family. The limiting full
local complexes are acyclic over every height-one residue field of
$\Lambda$ of characteristic zero. At $(2)$, with residue field
$\Omega=\mathbf F_2((t))$, the $r$-complex is acyclic, and so are the
complexes at active split primes whose prescribed exponent is odd.
These full conditions therefore agree with their orthogonal conditions
on the stated tests.
\end{lemma}
\begin{proof}
Use tame inertia and unramified Frobenius, or the Hochschild--Serre
complex for these two procyclic directions. At $r$, inertia minus one
is multiplication by $t$ times a unit. It is invertible on all the
indicated fields except the characteristic-zero fiber $t=0$. There,
$r\to1$, and the two remaining Frobenius differentials have nonzero
determinant by~\eqref{rl:trace-bound}, including either sign of the
quadratic twist. The bound also ensures that the integral local
torsion at this central fiber has bounded length.
At an active odd conductor prime, inertia of the quadratic character
acts by $-1$. Its differential is thus $-2$ and is invertible in
characteristic zero. In characteristic two the quadratic character
disappears. At a split prime the Frobenius action is $zB$, where
$B\in\operatorname{GL}_2(\mathbf F_2)$ and
$z=(1+t)^a$ has odd $a\in\mathbf Z_2$. Its coefficient of $t$ is one,
so $z$ is nonconstant. No nonzero polynomial over the finite constant
field vanishes at $z$. Thus both Frobenius complexes, with operators
$zB-1$ and $q^{-1}zB-1$, are invertible over $\Omega$; here $q=1$ in
the residue field. This argument applies also to a reducible residual
representation.
The orthogonal full and strict conditions agree when full local cochains
are acyclic. Equation~\eqref{rl:global-duality} then gives self-duality.
The global degree-zero assertion will follow from the evaluation maps
in Lemma~\ref{rl:evaluation}.
\end{proof}
\begin{lemma}[Uniform residual sizes]\label{rl:residual-bound}
Suppose $\omega(\operatorname{disc}K)\leq A$, the local twisting classes
at $2N$ belong to a fixed finite list, and all active conductor primes
are split with odd exponent. After localization and completion at
$(2)$, the minimal free ranks of the Selmer complexes are bounded in
terms of $E,A$, and the number of retained derivative primes. They do
not depend on the number of active conductor primes.
\end{lemma}
\begin{proof}
Reduce first over $\mathbf F_2[[t]]$. The quadratic character is now
trivial. Removing the active conductor primes from the permitted
ramification set changes the global complex by their singular local
complexes. These, and their unramified local complexes, are acyclic
after inverting $t$, by Lemma~\ref{rl:local-acyclic}. The resulting
smaller-support complex, including its comparison maps, can be
constructed at the original Artin stages before taking the limit.
Its support consists of $2N,r$, and the retained derivative primes.
Its $\mathbf F_2((t))$ cohomology dimensions are bounded by those at
$t=0$: choose a minimal finite free model over $\mathbf F_2[[t]]$.
Here is a uniform bound for the latter residual cohomology. Put
$F=\mathbf Q(E[2])$, a fixed field. Then $FK/F$ is at most quadratic.
Enlarge the allowed set of primes of $F$ by those ramified in $K$.
Its cardinality is bounded by a constant depending on $E,A$ and the
number of derivative primes. Kummer theory bounds the number of
generators of the maximal pro-$2$ Galois group over $F$ unramified
outside this set, using the fixed unit group, the fixed $2$-class group,
and the allowed valuations. The subgroup defining $FK$ has index at
most two; the elementary Schreier bound bounds its generator number.
Every finite Galois $2$-extension of $FK$ with this ramification has
Galois closure of $2$-power degree over $F$, so this subgroup indeed
controls the required pro-$2$ extensions of $FK$.
Over $FK$ the residual Tate module is trivial. Restriction and
inflation--restriction therefore bound the residual $H^1$ over $K$.
The $H^0$ bound is immediate, and the global Euler characteristic or
global duality bounds $H^2$. The local terms at the remaining bounded
set have bounded dimensions, with degrees of the local fields at most
two over the fixed rational local fields. At $2N$ there are only
finitely many local quadratic twisting classes. The same bounds
apply to the Selmer cone. Lemma~\ref{rl:finite-model} now bounds the
minimal ranks. No bound on a torsion exponent is inferred from these
rank bounds.
\end{proof}
\subsection{Evaluation on every sequence, rather than only a chosen list}
\begin{lemma}[All-sequence evaluation]\label{rl:evaluation}
Fix $K$ and a sequence of the finite cochain constructions above. Keep
the contraction maps into the original global cochains. Let
$\Gamma_K=\prod_i G_K$ and
$\Gamma=\Gamma_K\rtimes\langle\tau\rangle\subset\prod_iG_{\mathbf Q}$,
with simultaneous complex conjugation $\tau$. Thus $\Gamma_K$ has
index two in $\Gamma$; the larger unrestricted product of rational
Galois groups is not used for induction. The matrix limit defines a
coefficient action of $\Gamma_K$ on $\Lambda^2$. For every field $L$
receiving a map from $\Lambda$, there is a natural injection
\[
H^1(C\otimes_\Lambda L)\longrightarrow
H^1_{\rm abs}(\Gamma_K,L^2),
\]
where the right side denotes abstract crossed-homomorphism classes.
Degree-zero evaluation likewise injects
$H^0(C\otimes_\Lambda L)$ into $(L^2)^{\Gamma_K}$.
Here $C$ may be the global complex or a Selmer complex whose local
condition maps induce isomorphisms on local $H^0$ and injections on
local $H^1$. The same statement holds for the induced representation
of $\Gamma$.
There are evaluation maps on arbitrary sequences, not just on the
finite list used to detect cohomology. Their degree-two identities
remain valid after arbitrary coefficient extension.
\end{lemma}
\begin{proof}
Let $C_i$ be a finite model and let $\iota_i$ be its chosen map to the
original global cochains, using the projection from the Selmer cone
when appropriate. For every $g=(g_i)$ and $h=(h_i)$ define matrices
\[
e^0_i:C_i^0\to A_i^2,\quad
e^1_{g,i}:C_i^1\to A_i^2,\quad
e^2_{g,h,i}:C_i^2\to A_i^2
\]
by evaluating $\iota_i$ in degrees zero, one, and two. Their source
ranks are fixed and their entries belong to $A_i$. Thus each has a
coefficientwise ultrafilter limit, for every choice of $g,h$.
No diagonal selection over the set of group elements is necessary:
one fixed ultrafilter defines all these limits. Likewise
$\rho(g)=\lim_i\rho_i(g_i)$ is a representation. The original
cochain identities imply
\begin{align}
e^1_g d&=(\rho(g)-1)e^0,\label{rl:eval-boundary}\\
e^2_{g,h}d&=\rho(g)e^1_h-e^1_{gh}+e^1_g.
\label{rl:eval-cocycle}
\end{align}
All identities pass to the limit because their matrix expressions
have finitely many terms. A cycle $x\in C^1\otimes L$ consequently
gives the crossed homomorphism $g\mapsto e^1_g(x)$, and a boundary
gives a coboundary.
We prove injectivity in a way that also controls base change. At each
finite stage let $k=\mathbf F_2$ be the residual field. Choose a
space of representatives for residual $H^1$. Evaluate pairs $(c,m)$,
where $c$ is in this space and $m\in k^2$, by
$b_c(g)-(\rho(g)-1)m$. The intersection of the kernels of all
these evaluations consists precisely of pairs $(0,m)$ with $m$ an
invariant. Since the vector space is finite dimensional, a list of
at most $\dim H^1+2$ elements already has this intersection. For
these elements evaluation is a chain map
\[
C_i\longrightarrow D_i=
[\,A_i^2\xrightarrow{((\rho_i(g_j)-1))_j}(A_i^2)^d\,]
\]
in degrees zero and one, inducing an isomorphism on residual $H^0$
and an injection on residual $H^1$. For Selmer complexes this uses
the exact sequence of~\eqref{rl:selmer-definition}: the indicated
local degree-zero isomorphisms give $H^0(C_i)=H^0_{\rm glob}$ and
$H^1(C_i)\hookrightarrow H^1_{\rm glob}$ residually.
Retain these matrices and their chosen group-element sequences. The
cone of the limiting map $C\to D$ has zero residual cohomology in
degrees at most zero. Successively cancel its unit differential
blocks. The resulting bounded free cone starts in degree one.
After every coefficient extension it still starts in degree one;
the long exact sequence therefore makes $H^1(C\otimes L)\to
H^1(D\otimes L)$ injective. A crossed cocycle that is a coboundary
on all of $\Gamma_K$ is in particular a coboundary on the detecting
list, proving the desired injection. The same cone gives an
isomorphism on $H^0$ after coefficient extension. Since $H^0(D\otimes L)$
is a subspace of $L^2$, degree-zero evaluation is injective; by
\eqref{rl:eval-boundary}, its image is fixed by every element of
$\Gamma_K$, not merely by the detecting list. For induction the same proof
has coefficient rank four and the detecting bound is $\dim H^1+4$.
\end{proof}
We can now exclude global invariants on the tests of
Lemma~\ref{rl:local-acyclic}. In characteristic two the nontrivial
$r$-inertia scalar excludes coefficient invariants. In characteristic
zero the same argument works unless that scalar is trivial. In the
latter case $t=0$, and $T\otimes\mathbf Q_2$ over $G_K$ is absolutely
irreducible: for non-CM curves this follows from the open image, and
for CM curves from the open Cartan and its nontrivial normalizer
component, since $K$ is not the CM field. Twisting by $\chi_h$ does
not alter its invariant subspaces. The degree-zero injection just
proved therefore gives $H^0=0$ for the limiting Selmer complex.
Together with \eqref{rl:global-duality}, this leaves degrees one and
two paired and kills degree three as well.
\begin{lemma}[The exact common kernel]\label{rl:kernel}
In the notation of Lemma~\ref{rl:evaluation}, let $H$ be the group of
sequences each of whose entries acts trivially on the full Tate module,
on all ring-class fields of $K$, and on the quadratic characters in
use. Restriction from the crossed classes in that Lemma to $H$ is
injective at every characteristic-zero test. It is also injective
at the residual $(2)$ test. The characteristic-zero annihilator
needed for this assertion can be chosen with a fixed finite
$2$-valuation depending only on $E$, also as $K$ varies.
\end{lemma}
\begin{proof}
The Tate image contains an open subgroup of scalar homotheties, both
in the CM and non-CM cases
\cite[\S2, Theorem~3 and paragraph~7]{Bogomolov1981Points}. Choose once and for all
$g\in G_{\mathbf Q}$ with Tate action $u\,1$, where
$u\in1+4\mathbf Z_2$ and $u\ne1$. Its square belongs to every
quadratic subgroup. Put
\[
z=g^2\tau g^2\tau^{-1}.
\]
The generalized dihedral property of every ring-class extension
implies that $z$ is ring-class trivial. It is also trivial on every
quadratic character, and its Tate action is the fixed scalar $u^4$.
For every sequence in $\Gamma_K$, its commutator with the constant
sequence $z$ is in the exact group $H$, since its Tate commutator and
all its abelian character images are trivial. Thus $z$ is central in
$\Gamma_K/H$ and acts on the coefficients by $u^4$.
Here the elementary central-scalar argument suffices. If a crossed
cocycle $b$ is defined on a group with such a central element, comparison
of $b(zg)$ and $b(gz)$ gives
\[
(u^4-1)b(g)=(g-1)b(z).
\]
In characteristic zero this makes $b$ a coboundary. Inflation--restriction
then gives injectivity on $H$. The valuation of $u^4-1$ is fixed.
In characteristic two use instead, at the $i$-th stage, an inertia
element at $r_i$ mapping to the chosen generator of $\xi_i$. Its
Tate action is trivial and its ring-class image commutes with all
ring-class images. The associated sequence is therefore central
modulo $H$, and its coefficient action is $1+t$. Since $t$ is
invertible in $\mathbf F_2((t))$, the same calculation applies.
It is essential here that $H$ is defined by the exact actions at each
stage, not merely by an action converging to the identity.
\end{proof}
\subsection{The outer characteristic-zero test}
\begin{proposition}[Integral evaluation through the outer limit]
\label{rl:outer-evaluation}
Let $C_j$ be any family of the preceding limiting complexes, completed
over $\mathcal O_j=\widehat{\Lambda_{(2)}}$, possibly for different
quadratic fields $K_j$. Suppose their minimal ranks are bounded.
Keep their actual evaluation matrices, and transport them through the
integral contractions to these minimal models. For a nonprincipal
ultrafilter $\mathcal U$, put
\[
\mathcal O_\infty=
\left(\prod_{\mathcal U}\mathcal O_j\right)/\mathfrak a,
\qquad
\mathfrak a=\{(x_j):
\{j:v_2(x_j)\geq n\}\in\mathcal U\text{ for every }n\}.
\]
Then $\mathcal O_\infty$ is a DVR with uniformizer $2$ and a
characteristic-zero fraction field $L_\infty$. The outer complex has
an injection of its $H^1$ over $L_\infty$ into crossed classes on the
product of the inner sequence groups. Its restriction to the product
of their exact common kernels is injective. All arbitrary-sequence
degree-two evaluation identities survive. These conclusions also
hold after retaining any fixed finite collection of additional
integral diagrams.
\end{proposition}
\begin{proof}
A nonzero element of the quotient has a representative whose
valuations are bounded on an ultrafilter-large set. A finite
partition then makes its valuation equal to a single integer $n$
on such a set. It is $2^n$ times a unit. Products add these finite
valuations, so the quotient is a domain, and every nonzero ideal has
an element of smallest valuation. It is consequently a DVR. The
integer $2^n$ is nonzero for every $n$, so its fraction field has
characteristic zero.
For each $j$, all-sequence evaluation maps originally have entries
in $\Lambda$. Extension to $\mathcal O_j$ and composition with an
integral contraction still give integral matrices. Thus evaluation
on any sequence of inner group elements has an entrywise image in
$\mathcal O_\infty$. Equations~\eqref{rl:eval-boundary} and
\eqref{rl:eval-cocycle} survive for all such sequences. In particular,
if an outer cycle is represented by vectors $x_j$ for which
$d_jx_j$ has valuation tending to infinity, the evaluation of
$d_jx_j$ does too: the degree-two evaluation matrix is integral.
This proves the crossed identity for an outer cycle without replacing
the $x_j$ by genuine inner cycles.
Over the inner residue field $\Omega_j$, choose a detecting list as
in Lemma~\ref{rl:evaluation}; its length is at most
$\dim_{\Omega_j}H^1(C_j\otimes\Omega_j)+\operatorname{rank}M$.
These lengths are bounded. Include in the integral diagrams the
evaluation map to its two-term complex and a contraction of its cone
to a complex starting in degree one. Such a contraction uses only
unit pivots over $\mathcal O_j$ and has bounded graded sizes.
All its matrices, including inverses and homotopies, are integral
and pass to the quotient. Its cone still starts in degree one
after extension to $L_\infty$. This proves injectivity just as in
Lemma~\ref{rl:evaluation}.
Let $H_j$ be the inner exact kernel. The product of the central
elements of Lemma~\ref{rl:kernel} is central modulo $\prod_jH_j$.
Its scalar is the same fixed $u^4$ in every component, and
$u^4-1$ remains nonzero in $\mathcal O_\infty$. The explicit
central-scalar calculation and inflation--restriction give the
claimed restriction injection. Additional fixed finite diagrams
are treated by the same entrywise operations.
\end{proof}
\begin{lemma}[Irreducibility on the characteristic-zero tests]
\label{rl:irreducibility}
On an inner or outer characteristic-zero test where $\Psi^2\ne1$,
the four-dimensional representation induced from $T\chi\Psi$ is
absolutely irreducible. If $\Psi^2=1$ on an inner test, it splits
as two inequivalent absolutely irreducible quadratic twists. In
the outer test $\Psi$ has infinite order.
\end{lemma}
\begin{proof}
The two induced summands are distinguished by $r$-inertia, whose
scalars are $\Psi$ and $\Psi^{-1}$. On either summand, multiplying
the Tate matrices by scalars does not alter invariant subspaces.
The Tate representation restricted to $G_K$ is absolutely
irreducible under the exclusions already specified. These facts
give absolute irreducibility of the induction when the two inertia
characters differ.
For the outer argument this assertion requires a uniform finite
witness. The images of the quadratic subgroups have index at most
two in the fixed compact $2$-adic Tate image. A compact $2$-adic
analytic group is topologically finitely generated and hence has
only finitely many closed subgroups of index at most two. Choose
one on an ultrafilter-large set. Its absolutely irreducible
representation has finitely many matrices spanning
$M_2(\mathbf Q_2)$; the determinant expressing this spanning
property is a fixed nonzero $2$-adic number. Every one of these
Tate matrices is realized by choices of Galois elements in each
component. Their accompanying character scalars are units and
do not change the spanning assertion. Its determinant remains
nonzero in $L_\infty$. This proves absolute irreducibility in
that field. Also $1+t$ has infinite order already in every
$\Omega_j$: for a nonzero integer $n$, the first nonzero term of
$(1+t)^n-1$ occurs in finite degree. It remains a unit modulo
$2$ in every component, so no such power becomes one in the
outer field.
If $\Psi^2=1$, extension across conjugation gives the two quadratic
twists. They are inequivalent, since a quadratic self-twist of the
Tate representation here would make its restriction to $G_K$
reducible over an algebraic closure. In the CM case this would
force $K$ to be the CM field, which was excluded.
\end{proof}
\begin{lemma}[Simultaneous finite Chebotarev realization]
\label{rl:chebotarev-evaluation}
For any element $u$ of an exact kernel in Lemma~\ref{rl:kernel},
derivative primes can be chosen, at increasing finite precision,
whose rational Frobenius is $\tau u$ on all retained data. Their
finite localization is evaluation on $(\tau u)^2$. The assertion
is compatible with the outer construction of
Proposition~\ref{rl:outer-evaluation} and with any fixed finite
list of previous derivative operations.
\end{lemma}
\begin{proof}
At one Artin stage, a coordinate of a cochain in the image of a
finite model is a locally constant function. The finitely many
coordinates being used factor through one finite quotient of the
absolute Galois group. Enlarge it to contain the Tate action at
the required precision, the finitely many ring-class fields and
quadratic characters currently in use, and all old finite data.
Chebotarev realizes the conjugacy class of $\tau u$ on this finite
quotient, with a chosen place realizing the desired representative.
It gives a rational prime inert in $K$, outside the old finite set.
Because $u$ is exactly Tate-trivial, its Frobenius tends to
$\rho_T(\tau)$; in particular $a_\ell\to0$ and $\ell\to-1$.
Because $u$ is exactly ring-class trivial, the square Frobenius on
the $K$-prime is the identity on every retained ring-class field.
The unramified local coordinate is therefore the stated
Frobenius-square evaluation.
Repeat at each finite stage with increasing precision. All
conditions remain finite, and no density bound or prime-size
bound is required. For an outer sequence do this separately
for each inner sequence, retaining the relevant finite evaluation
matrices and degree-two identities before taking the outer
quotient. Approximate inner cycles cause no problem, by the
integrality calculation in Proposition~\ref{rl:outer-evaluation}.
Previously imposed finite diagrams are unchanged.
\end{proof}
In induced-module notation, restriction to the exact kernel is additive.
For a crossed cocycle $b$ and an exact-kernel element $u$,
\[
b((\tau u)^2)=(1+\tau)b(u).
\]
Indeed $\tau^2=1$, $u$ acts trivially on the coefficients, and the
cocycle identity cancels the two occurrences of $b(\tau)$. Projection
to either induced summand identifies $\operatorname{im}(1+\tau)$
with a two-dimensional coefficient module, also in characteristic two.
\begin{theorem}[The limit and evaluation interface]
\label{rl:limit-evaluation-package}
For the ring data fixed at the start of this section, retain the finite
target-side arithmetic diagrams with their maps to original cochains, homotopies,
and dualities as in Lemma~\ref{rl:finite-limits}. The moving characters,
local conditions, and evaluation constructions then have the following
simultaneous properties:
\begin{enumerate}
\item Every specified finite target-side arithmetic diagram admits a bounded free
model together with maps to its original cochains; the specified
maps, homotopies and dualities survive the inner limit.
\item At the stated height-one tests the relaxed active local
complexes are acyclic, and the Selmer duality pairs degrees one
and two. Integral Kummer conditions at $2N$ include their full
derived degree-zero invariants.
\item The residual sizes needed for the outer construction are
bounded independently of the active conductor support, with
dependence only on $E$, the bound on $\omega(\operatorname{disc}K)$,
and the number of retained derivative primes.
\item Inner and outer $H^1$ are detected by actual crossed
cocycles, including on the exact common kernel. The evaluations
and their degree-two identities can be realized by finite
Chebotarev conditions without disturbing earlier finite data.
\end{enumerate}
These assertions do not assert an Euler-system reciprocity law or
an integral derivative class that has not first been constructed
at finite level. Once those finite arithmetic diagrams are
constructed, this Theorem supplies their limit compatibility.
\end{theorem}
\begin{proof}
Combine Lemmas~\ref{rl:finite-model} and~\ref{rl:finite-limits},
Lemma~\ref{rl:kummer}, Proposition~\ref{rl:characters},
Lemmas~\ref{rl:local-acyclic} and~\ref{rl:residual-bound},
Lemmas~\ref{rl:evaluation} and~\ref{rl:kernel},
Proposition~\ref{rl:outer-evaluation}, and
Lemma~\ref{rl:chebotarev-evaluation}. Every passage uses actual
integral matrices of bounded size; arbitrary evaluations are
retained by their original contraction maps, not by an inference
from a finite list of identities.
\end{proof}
==== END A-pointwise-2-converse-for-elliptic-curves-with-rational-two-torsion//ring-limits.tex ====
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}
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@misc{BurungaleSkinner2024Appendix,
author = {Burungale, Ashay and Skinner, Christopher},
title = {A {$p$}-converse theorem for {CM} elliptic curves},
year = {2024},
howpublished = {Appendix A to Levent Alp{\"o}ge, Manjul Bhargava and Ari Shnidman, \emph{Integers expressible as the sum of two rational cubes}},
eprint = {2210.10730v3},
archivePrefix = {arXiv},
primaryClass = {math.NT},
url = {https://arxiv.org/abs/2210.10730v3},
note = {Version 3, 19 October 2024; Theorems A.1, A.3 and Remark A.5; \href{https://arxiv.org/abs/2210.10730v3}{arXiv:2210.10730v3}}
}
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author = {Burungale, Ashay and Castella, Francesc and Skinner, Christopher and Tian, Ye},
title = {{$p^{\infty}$}-{Selmer} groups and rational points on {CM} elliptic curves},
journal = {Annales math{\'e}matiques du Qu{\'e}bec},
volume = {46},
pages = {325--346},
year = {2022},
doi = {10.1007/s40316-022-00203-y},
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archivePrefix = {arXiv},
primaryClass = {math.NT},
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note = {Preprint, version 5, 19 October 2022; \href{https://arxiv.org/abs/2002.04767v5}{arXiv:2002.04767v5}}
}
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@misc{OpenAIGoldfeldTwoConverse2026,
author={{OpenAI}},
title={{Goldfeld's analytic density conjecture and the $2$-converse for elliptic curves}},
howpublished={OpenAI Math Release preprint
\href{https://github.com/openai/math/blob/main/preprints/Goldfelds-analytic-density-conjecture-and-the-2-converse-for-elliptic-curves-September-23-2026/paper.pdf}{OAI:Goldfelds-analytic-density-conjecture-and-the-2-converse-for-elliptic-curves-September-23-2026}},
year={2026}
}
==== END A-pointwise-2-converse-for-elliptic-curves-with-rational-two-torsion//references.bib ====