==== TABLE OF CONTENTS ==== The-Selmer-converse-for-elliptic-curves-at-every-prime//figures/proof-roadmap.tex The-Selmer-converse-for-elliptic-curves-at-every-prime//main.tex The-Selmer-converse-for-elliptic-curves-at-every-prime//preamble.tex The-Selmer-converse-for-elliptic-curves-at-every-prime//sections/01-introduction.tex The-Selmer-converse-for-elliptic-curves-at-every-prime//sections/02-reduction.tex The-Selmer-converse-for-elliptic-curves-at-every-prime//sections/03-tame-deformation.tex The-Selmer-converse-for-elliptic-curves-at-every-prime//sections/04-theta-construction.tex The-Selmer-converse-for-elliptic-curves-at-every-prime//sections/05-constant-terms.tex The-Selmer-converse-for-elliptic-curves-at-every-prime//sections/06-nonconstant-coefficient.tex The-Selmer-converse-for-elliptic-curves-at-every-prime//sections/07-cusp-lifting.tex The-Selmer-converse-for-elliptic-curves-at-every-prime//sections/08-galois-parameters.tex The-Selmer-converse-for-elliptic-curves-at-every-prime//sections/09-extension-extraction.tex The-Selmer-converse-for-elliptic-curves-at-every-prime//sections/10-nonstandard-transfer.tex The-Selmer-converse-for-elliptic-curves-at-every-prime//sections/10-sylvester-cube-sums.tex The-Selmer-converse-for-elliptic-curves-at-every-prime//references.bib =========================== ==== BEGIN The-Selmer-converse-for-elliptic-curves-at-every-prime//figures/proof-roadmap.tex ==== \begin{figure}[tb] \centering \begin{tikzpicture}[ box/.style={draw=black!55,rounded corners=2pt,align=center, text width=5.4cm,minimum height=1.05cm,inner sep=6pt,font=\small}, arr/.style={-{Latex[length=2mm]},thick,draw=black!65}, node distance=0.65cm and 0.55cm] \node[box,text width=12cm] (hyp) at (0,0) {Assume the conductor-one Heegner point is torsion.\\ Choose the tame character over $R_{b_*}$.}; \node[box] (upper) at (-3.15,-1.9) {Cohomological upper bound\\ (\cref{prop:green-length})\\ $\length_{R_{b_*}}\GrH3$ with irreducible $A_0[p]$, Burungale--Castella--Skinner remove the earlier auxiliary residual-ramification conditions \citep{BCS2025}. Reducible residual representations require a different analysis: Castella--Grossi--Lee--Skinner prove the converse at odd good Eisenstein primes with a local-character restriction, and Keller--Yin remove that restriction and treat multiplicative and potentially good ordinary cases, including rational $p$-torsion and some additive reduction \citep{CGLS2022,KellerYinSemistable2024,KellerYin2024}. At good supersingular primes greater than three, Castella--Wan prove the corank-one converse for semistable curves \citep{CastellaWan2024}. Castella treats multiplicative reduction under local and residual hypotheses \citep[Theorems~1.1 and~1.3]{Castella2024}; the zeta-element constructions of Burungale--Skinner--Tian--Wan yield further supersingular rank-zero and ordinary applications \citep[Theorems~1.6 and~1.10]{BSTW2024}. Complex multiplication also permits small-prime results. Burungale--Castella--Skinner--Tian prove the corank-one converse at good ordinary primes, including $2$ and $3$, when the associated Hecke character has conductor exactly divisible by the different of the CM field \citep[Theorem~A]{BCST2022}. Burungale--Tian prove the rank-zero converse for every CM elliptic curve at every prime \citep{BurungaleTian2026}. Kriz's preprint states a low-corank converse at the ramified prime for curves with CM by the full ring of integers, together with its cube-sum application \citep[Theorems~1.1 and~1.3]{Kriz2022}. The present proof uses two distinct results of \citet{OpenAIGoldfeldDensity2026}: the unrestricted converse at $2$ settles that prime, while the analytic twist-density theorem supplies the auxiliary fields for the odd-prime argument. Their precise statements are recorded in Section~\ref{sec:reduction}. The remaining construction keeps the specified odd prime arbitrary. In particular, it must control local conditions even when the elliptic curve is not ordinary and the residual representation is reducible. \subsection{Proof strategy: two incompatible lengths} Fix an odd prime $p$ and suppose $s_p(A_0)\in\{0,1\}$. The first goal is to produce a non-torsion Heegner point. The density input selects an imaginary quadratic field $K$ and a biquadratic CM field $L$ containing $K$. The rational Selmer spaces of $A_0$ and an auxiliary quadratic twist over $K$ are both lines; the latter is generated by a known non-torsion point. Section~\ref{sec:reduction} shows that non-torsion of the conductor-one Heegner trace $y$ implies the desired conclusion by Gross--Zagier and Kolyvagin. Assume, for a contradiction, that $y$ is torsion. We will obtain more Galois extensions than the Selmer hypothesis allows. \paragraph{The cohomological upper bound.} Let $V=T_pA_0\otimes_{\Z_p}\Q_p$. At a sequence of split auxiliary primes $q_i$, cyclic $p$-quotients of growing ring class groups give a tame character $\psi$ with coefficients in \[ R_b=k[t]/(t^b),\qquad b\ge1, \] where $k/\Q_p$ is finite and $\psi\equiv1\pmod t$. We form $\psi$ by taking the $p$-adic ultralimit of the finite-stage characters on sequences of Galois elements over $K$, and restrict it to $L$ below. Its cohomology is represented by bounded finite-stage cochains whose identities hold to uniformly increasing precision; Section~\ref{sec:tame} calls these cochains \emph{admissible}. Choose a place $v$ of $K$ above $p$ and the places of $L$ above $v$. The one-sided Greenberg group $\mathcal H_b=\GrH(L,V\otimes\psi)$ allows unrestricted local classes at the chosen places and requires zero classes at their conjugates, with relaxed conditions at the remaining fixed places. If the Selmer line of $A_0/K$ has nonzero localization at $p$, then $\mathcal H_b=0$. Otherwise that line is everywhere locally zero; we call this the strict case. There a Heisenberg commutator modifies the tame direction so that no nonzero base Greenberg class lifts to first order. The exact coefficient sequence then confines every admissible class to the last $t$-layer and gives $\length_{R_b}\mathcal H_b\le2$, independently of $b$. Consequently, with \[ b_*=1\quad\text{outside the strict case},\qquad b_*=3\quad\text{in the strict case}, \] the cohomological upper bound is strictly smaller than $b_*$. \paragraph{A theta congruence retaining the tame parameter.} The weighted Heegner logarithm is a truncated $t$-series formed from unnormalized sums of local logarithms of ring-class Heegner points, weighted by the finite-stage tame characters. The torsion assumption on $y$ makes its constant coefficient tend to zero. In the strict case, conjugation and the zero localizations give the same conclusion for the next two coefficients. This logarithmic vanishing is distinct from the first-order obstruction used for the upper bound. It is the input to the automorphic construction. Sections~\ref{sec:theta}--\ref{sec:content} construct a holomorphic theta family on a unitary group of signature $(3,1)$ at both real places of the maximal real subfield of $L$. The construction uses Weil's oscillator representation and the theta and period framework of Kudla and Rallis \citep{Weil1964,Kudla1994,Rallis1984Injectivity}. An algebraic square identity relates its constant term to toric periods, one of which specializes to the Heegner logarithm. The passage from raised CM values through an inverse differential operator to that logarithm follows the method of Bertolini--Darmon--Prasanna \citep{BertoliniDarmonPrasanna2013}; the argument here includes the additional bad-reduction comparisons. Uniform integral moment estimates make the constant terms small, while a separate positive Fourier--Jacobi coefficient has bounded divisibility and prevents the entire theta series from disappearing. The geometric input to cusp lifting is the ordinary locus of the target PEL variety, not ordinary reduction of $A_0$. Using the compactification and ordinary deformation frameworks of Lan and Katz \citep{Lan2013,Lan2018Ordinary,Katz1981SerreTate}, Section~\ref{sec:cusp-lifting} lifts the theta series to genuine cusp forms modulo increasing powers of $p$. Its integral coefficient lattice and normalized Hecke traces preserve the whole truncated tame ring. Evaluating the congruence only at individual characters would discard the nilpotent parameter that measures the required length. \paragraph{The extension lower bound.} The passage from an automorphic congruence to a Galois extension goes back to Ribet and Wiles \citep{Ribet1976,Wiles1990}. Skinner--Urban use congruences on larger unitary groups \citep{SkinnerUrban2014}; the semiordinary $\mathrm{GU}(3,1)$ construction of Castella--Liu--Wan is a close antecedent allowing a nonordinary two-dimensional source \citep{CastellaLiuWan2022,CastellaLiuWanAddendum}. Here Section~\ref{sec:parameters} constructs four-dimensional characteristic-zero Galois representations from the cusp forms and proves the local identities needed to extract one-sided Greenberg classes. Two normalized parabolic operators are units, although the elliptic representation remains unrestricted at $p$. The limiting trace separates into a two-dimensional twist of $V$ and two characters; this is not a decomposition assumed for each cuspidal Galois representation. Section~\ref{sec:extensions} uses the off-diagonal modules of a finite limiting matrix algebra to record extensions between these constituents, in the spirit of the generalized matrix algebras of Bella\"iche--Chenevier \citep{BellaicheChenevier2009}. The main tasks are to eliminate an unwanted cyclotomic extension quotient between the two characters and to impose the strict local condition on extensions of one character by the two-dimensional constituent. Uniformly many algebra generators reduce each limiting functional to finitely many matrix values, so the resulting cochains remain admissible. The local identities hold over the entire Artin ring, and a Fitting ideal calculation gives \[ \length_{R_{b_*}}\mathcal H_{b_*}\ge b_*. \] This contradicts the upper bound and proves that $y$ is non-torsion. Figure~\ref{fig:proof} summarizes the comparison. \input{figures/proof-roadmap} The proof therefore separates into the arithmetic reduction and upper bound in Sections~\ref{sec:reduction}--\ref{sec:tame}, the theta and cusp congruence in Sections~\ref{sec:theta}--\ref{sec:cusp-lifting}, and the Galois construction and extraction in Sections~\ref{sec:parameters}--\ref{sec:extensions}. Section~\ref{sec:sylvester} gives the cube-sum application. The simultaneous transfer in Section~\ref{subsec:simultaneous-transfer} uses Labesse's base-change theorem, with its sign property verified for the holomorphic discrete series at both real places. Appendix~\ref{sec:nonstandard-transfer} records a separate normalized global measure comparison and a change-of-characteristic principle for nonstandard weighted orbital integrals. ==== END The-Selmer-converse-for-elliptic-curves-at-every-prime//sections/01-introduction.tex ==== ==== BEGIN The-Selmer-converse-for-elliptic-curves-at-every-prime//sections/02-reduction.tex ==== \section{Arithmetic reduction and conventions}\label{sec:reduction} The converse at $2$ settles the even prime. For an odd prime, we choose auxiliary quadratic fields so that the problem reduces to proving that one conductor-one Heegner trace is non-torsion. The resulting compact Selmer spaces will also supply the initial data for the tame deformation. \subsection{Selmer groups and established inputs} For a number field $M$, put $T_pA_0=\varprojlim_n A_0[p^n]$ and $V_pA_0=T_pA_0\otimes_{\Z_p}\Q_p$. For a finite place $v$ of $M$, the finite local condition is the Kummer image \[ \Hf(M_v,V_pA_0) =\im\bigl(A_0(M_v)^{\wedge}_p\otimes_{\Z_p}\Q_p \longrightarrow H^1(M_v,V_pA_0)\bigr). \] Here $A_0(M_v)^{\wedge}_p=\varprojlim_n A_0(M_v)/p^nA_0(M_v)$. The rational compact Selmer group is \[ H^1_f(M,V_pA_0)= \ker\left(H^1(M,V_pA_0)\longrightarrow \prod_v H^1(M_v,V_pA_0)/\Hf(M_v,V_pA_0)\right). \] Continuous cohomology and the usual archimedean local conditions are understood. The Kummer exact sequences and local/global duality give \begin{equation}\label{eq:compact-corank} \dim_{\Q_p}H^1_f(M,V_pA_0) =\corank_{\Z_p}\Sel_{p^\infty}(A_0/M). \end{equation} We use rational coefficients throughout the cohomological argument; integral lattices will be retained whenever a uniform valuation bound is required. We use the following two established results. \begin{theorem}[Established converse at $2$] \label{thm:two-converse-input} If $E/\Q$ is any elliptic curve and $\corank_{\Z_2}\Sel_{2^\infty}(E/\Q)=r\in\{0,1\}$, then \[ \ord_{s=1}L(E,s)=\rank E(\Q)=r, \qquad \Sha(E/\Q)\text{ is finite}. \] \end{theorem} \begin{theorem}[Established analytic twist-density theorem] \label{thm:twist-density-input} For any elliptic curve $E/\Q$, among nonzero squarefree integers $d$ ordered by $|d|$, the quadratic twists $E^{(d)}$ of analytic rank zero and one each have density $1/2$. In particular the exceptional set of twists of analytic rank at least two has density zero. \end{theorem} These are \citet[Theorems~1.1 and~1.2]{OpenAIGoldfeldDensity2026}. Theorem~\ref{thm:two-converse-input} treats the prime $2$. For auxiliary-field selection we use the density-zero exceptional-set consequence of Theorem~\ref{thm:twist-density-input} in fixed positive-density congruence classes. The remainder of the paper supplies the argument at odd primes. Fix $A_0$ and $r=s_p(A_0)\in\{0,1\}$. By Theorem~\ref{thm:two-converse-input}, we may assume $p>2$. Write $N$ for the conductor and $V=V_pA_0$. The Weil pairing gives $V\simeq V^*(1)$ with alternating pairing. The $p$-parity theorem \citep[Theorem~1.4]{DokchitserDokchitser2010} gives the global root number \begin{equation}\label{eq:root-sign} w(A_0)=(-1)^r. \end{equation} Modularity and the Gross--Zagier--Kolyvagin theorems will be used in their rank-zero and rank-one forms \citep{BCDT2001,GrossZagier1986,Kolyvagin1990}. \subsection{Two auxiliary discriminants} \begin{lemma}\label{lem:auxiliary-fields} There are coprime negative odd fundamental discriminants $D,D'$ such that every prime dividing $2Np$ splits in both quadratic fields, \[ \left(\frac{D}{|D'|}\right)=1, \] and each of $A_0^D,A_0^{D'},A_0^{DD'}$ has the smallest analytic rank permitted by its sign. They may also be chosen so that \[ K=\Q(\sqrt D),\qquad F=\Q(\sqrt{DD'}),\qquad L=\Q(\sqrt D,\sqrt{D'}) \] satisfy the following properties: $L/F$ is CM and unramified at every finite prime, $L$ has only the roots of unity $\{\pm1\}$, and $L$ does not contain the CM field of $A_0$ if $A_0$ has CM. \end{lemma} \begin{proof} Negative odd fundamental discriminants in any admissible fixed congruence class have positive density. The splitting conditions at $2Np$ are finitely many such conditions. We can exclude the finitely many unwanted quadratic fields and then apply Theorem~\ref{thm:twist-density-input} to choose $D$ for which $A_0^D$ has analytic rank at most one. Its rank is then minimal for its sign. Keep $D$ fixed. The conditions on $D'$ of being coprime to $D$, splitting $2Np$, and satisfying the displayed Jacobi-symbol condition again contain admissible congruence classes of positive density. For example, impose $D'\equiv-1\pmod{|D|}$ and $D'\equiv1$ at a sufficiently divisible modulus supported on $2Np$. The moduli are coprime, and $|D'|\equiv1\pmod{|D|}$ gives the required Jacobi symbol. Exclude the finitely many remaining forbidden fields. Apply the density theorem to both $A_0$ and $A_0^D$. The union of their two exceptional sets has density zero, so some $D'$ in the prescribed classes makes both $A_0^{D'}$ and $(A_0^D)^{D'}=A_0^{DD'}$ have analytic rank at most one. This use of density is unchanged if parameters are described by fundamental discriminants: for squarefree $d$ the corresponding discriminant is $d$ or $4d$, so a zero-density exceptional set remains zero-density. In the odd classes used here the parameter itself is the discriminant. The three quadratic subfields have discriminants $D,D',DD'$. Consequently the biquadratic discriminant formula gives \[ |\operatorname{disc}(L)|=D^2(D')^2 =\operatorname{disc}(F)^2. \] The discriminant formula for $L/F$ therefore has relative discriminant of norm one, proving finite unramifiedness. The two imaginary subfields give the CM involution over the real field $F$. The exclusions made above remove any extra roots of unity and the possible CM field of $A_0$. \end{proof} \begin{proposition}\label{prop:arithmetic-reduction} With these choices, \[ \dim H^1_f(K,V)=1, \qquad \dim H^1_f(K,V^{DD'})=1. \] The second line localizes nontrivially at both places of $K$ over $p$. Let $y\in A_0(K)$ be the norm of the conductor-one Heegner point under a modular parametrization taking the cusp $\infty$ to zero. If $y$ is non-torsion, Theorem~\ref{thm:main} follows. \end{proposition} \begin{proof} For a fundamental discriminant prime to $N$, the quadratic-twist root-number formula is $w(A_0^d)=w(A_0)\chi_d(-N)$. The splitting conditions and the signs of $D,D'$ therefore give \[ w(A_0^D)=w(A_0^{D'})=-(-1)^r, \qquad w(A_0^{DD'})=(-1)^r. \] By Lemma~\ref{lem:auxiliary-fields}, their analytic ranks are $1-r,1-r,r$, respectively. Gross--Zagier--Kolyvagin gives the same Mordell--Weil and $p$-power Selmer ranks and finite Sha for these three twists. Restriction and corestriction for the quadratic extension $K/\Q$ give \[ H^1_f(K,V) =H^1_f(\Q,V)\oplus H^1_f(\Q,V^D), \] and the analogous decomposition for $V^{DD'}$. Thus the two dimensions are $r+(1-r)=1$ and $r+(1-r)=1$. For the second representation both summands come from twists with known analytic rank at most one. Its Selmer line is therefore spanned by a non-torsion global point on the corresponding twist over $K$. Such a point has nonzero image under the local logarithm at either place above $p$. Indeed, on a fixed $p$-adic completion, a finite-index open subgroup is a formal group on which the logarithm is injective after shrinking; a point with zero logarithm consequently has a torsion multiple in that subgroup, and is torsion. This argument uses no reduction hypothesis. Every prime dividing $N$ splits in $K$, so the classical Heegner hypothesis holds. The Gross--Zagier formula identifies the nonzero height of $y$ with simple vanishing of \[ L(A_0/K,s)=L(A_0,s)L(A_0^D,s) \] at $s=1$. Since the second factor has order $1-r$, a non-torsion $y$ implies $a(A_0)=r$. The analytic rank-zero/one theorems then give $\rank A_0(\Q)=r$ and finiteness of the whole $\Sha(A_0/\Q)$. \end{proof} For the rest of the proof we assume \begin{equation}\label{eq:torsion-assumption} y=0\quad\text{in }A_0(K)\otimes\Q. \end{equation} We will derive a contradiction. In particular, every subsequent choice is made after $A_0,p,D,D'$ have been fixed. The first Selmer line in Proposition~\ref{prop:arithmetic-reduction} may have zero localization, whereas the second cannot. Section~\ref{sec:tame} distinguishes these two possibilities before deforming the representations. \subsection{Coefficient, character, and measure conventions} \label{sec:conventions} Fix embeddings into $\C$ and $\overline{\Q}_p$. Let $\Sigma$ be the CM type on $L$ induced from one embedding of $K$, and let $\Sigma_p$ be the corresponding half of the places of $L$ above $p$. Complex conjugation is denoted by $c$. All the fields introduced above are split at $p$, and the places of $K$ above $p$ will be distinguished according to this choice. Smooth characters and normalized Satake parameters are evaluated on uniformizers using the reciprocity convention corresponding to geometric Frobenius. Write $\epsilon$ for the cyclotomic character. A de Rham weight of $\epsilon^{-1}$ is called $1$. When a local cohomology computation instead uses arithmetic Frobenius, this will be stated; inversion changes neither the vanishing assertions nor the invertibility requirements used below. Characters of $\U(1)_{L/F}=\ker\Norm_{L/F}$ are pulled back to characters of $L^\times\backslash\A_L^\times$ by $z\mapsto z/z^c$; the same convention applies to $K/\Q$. Choose a character $\mathcal C$ of $\U(1)_{K/\Q}$ with infinity type $s\mapsto s$ in the chosen embedding and with finite conductor supported on split primes away from $pDD'N$. These primes may be required to split completely in $L$. The ray-class construction gives such a character: a sufficiently deep split conductor removes the finite unit obstruction, and the prescribed algebraic character on principal ideles then extends across the finite ray-class quotient. We use the same symbol for its norm pullback to $\U(1)_{L/F}$ and for the corresponding algebraic $p$-adic character. At the distinguished split component above $p$, its finite smooth part has uniformizer value of valuation $-1$ with these conventions. This records the algebraic infinity contribution; it is not a claim that the smooth character is unitary in the $p$-adic norm. All archimedean weights $m\to+\infty$ below also satisfy $m\to0$ $p$-adically on a fixed residue branch, including the finite torsion orders of the characters raised to the $m$th power. Let $k$ be a sufficiently large fixed finite extension of $\Q_p$, with ring of integers $\cO$. We enlarge it finitely when fixed algebraic normalization factors require this. Valuation bounds refer to fixed bases and integral lattices. A \emph{uniform} bound may depend on the fixed data and on a fixed truncation degree, but not on the moving prime, its growing $p$-power quotient, or the growing weights. When coefficient fields for classical eigensystems vary, we retain their embeddings in $\overline{\Q}_p$ and use valuation bounds; no compactness of their union is assumed. For an algebraic group $H$ over a number field $M$, write $[H]=H(M)\backslash H(\A_M)$. Measures on compact source groups and archimedean integration tori have algebraic normalization, and maximal compact subgroups have volume one at almost every finite place. Passing between these measures and invariant norm formulas costs a fixed global scalar. All toric sums called \emph{raw} below are sums without division by the varying ring-class order. This distinction is essential for the estimates at the moving primes. ==== END The-Selmer-converse-for-elliptic-curves-at-every-prime//sections/02-reduction.tex ==== ==== BEGIN The-Selmer-converse-for-elliptic-curves-at-every-prime//sections/03-tame-deformation.tex ==== \section{A tame deformation and its Selmer length}\label{sec:tame} We retain the fields and the contradiction assumption of \cref{sec:reduction}. In particular, the conductor-one Heegner trace $y\in A_0(K)$ is torsion. Put $T=T_pA_0$ and $V=T\otimes_{\Z_p}\Q_p$. Initially all cohomology in this section has $\Q_p$ coefficients; extension to the fixed finite coefficient field $k$ is made by tensor product. Enlarge the finite, conjugation-stable set $S$ to contain the places over $p$, the bad places of $A_0$, and all the fixed conductors used below. It may be enlarged by a further \emph{fixed} finite set at any stage. In particular, the auxiliary finite characters and tests of Section~\ref{sec:content} are chosen independently of the moving primes, and their conductors are included before the final prime sequence is realized. All the constructions below allow this fixed enlargement. Write $G_{K,S}$ for the Galois group unramified outside $S$. We need two conclusions for the same tame deformation. The one-sided local condition, full at one $p$-adic place and zero at its conjugate, will give a cohomology module of length zero or at most two. Under the torsion assumption on $y$, the first one or three coefficients of the corresponding weighted Heegner logarithms will tend to zero, respectively. These are \cref{prop:green-length,prop:heegner-log}. The logarithm estimate supplies the constant-term estimate of \cref{prop:constant-term-small}; the length bound will contradict the extensions extracted in Section~\ref{sec:extensions}. \subsection{The undeformed Selmer spaces at the two places above \texorpdfstring{$p$}{p}} The two places $v,v^c$ of $K$ above $p$ have completion $\Q_p$. The alternating Weil pairing \[ e:V\times V\longrightarrow\Q_p(1) \] and the local invariant give a symmetric perfect pairing on $H^1(K_w,V)$: the sign from interchanging two degree-one cochains cancels the sign of $e$. The local Kummer line $H^1_f(K_w,V)$ is its own orthogonal complement. Thus, for $w\mid p$, this is a hyperbolic plane. These statements include arbitrary reduction at $p$: local torsion is finite, the rational Kummer space has dimension one, and the local Euler characteristic gives $\dim H^1(K_w,V)=2$. At a finite place $w\nmid p$ we instead have \begin{equation}\label{eq:away-p-vanishing} H^0(K_w,V)=H^2(K_w,V)=H^1(K_w,V)=0. \end{equation} Indeed $H^0=0$ by finiteness of local torsion, $H^2=0$ by local duality, and the Euler characteristic away from $p$ is zero. We use the local and global duality sequences in \cite[Chapter~I, Sections~2--4]{Milne1986} and \cite[Section~8.6]{NSW2008}. For the moment let $W$ denote either $V$ or its twist $V^{DD'}$ over $K$. Its compact Selmer space has dimension one by \cref{prop:arithmetic-reduction}. Let $H^1_{\mathrm{rel}}(K,W)$ have full conditions at $v,v^c$, and the usual unramified conditions outside $S$; conditions at the other places of $S$ make no difference by \cref{eq:away-p-vanishing}. Let $H^1_{\Gr}(K,W)$ have full condition at $v$ and zero condition at $v^c$, and let $H^1_{\overline{\Gr}}(K,W)$ have the opposite conditions. These are the two one-sided Greenberg conditions. \begin{lemma}\label{lem:base-green} If the compact Selmer line of $W/K$ has nonzero localization, then $H^1_{\Gr}(K,W)=H^1_{\overline{\Gr}}(K,W)=0$. The only remaining possibility is the following \emph{strict case} in the $V$ summand: $r=1$, and the compact Selmer space is generated by a class $a$ such that $a^c=a$ and every localization of $a$ is zero. In that case \begin{equation}\label{eq:base-green-bases} H^1_{\Gr}(K,V)=\langle a,h\rangle, \qquad H^1_{\overline{\Gr}}(K,V)=\langle a,h^c\rangle, \end{equation} where $h$ has nonzero localization at $v$ and zero localization at $v^c$. The $V^{DD'}$ summand always has zero Greenberg space. \end{lemma} \begin{proof} Let $F_w=H^1_f(K_w,W)$ and let $Q_w=H^1(K_w,W)/F_w$. Poitou--Tate duality identifies the image of \[ H^1_{\mathrm{rel}}(K,W)\longrightarrow Q_v\oplus Q_{v^c} \] with the annihilator of the localization of the compact Selmer line in $F_v\oplus F_{v^c}$. If that localization is nonzero, both of its coordinates are nonzero: the one-dimensional global space is stable under $c$, which interchanges the two places. Its annihilator is therefore a line whose two coordinates are both nonzero for every nonzero vector. A class satisfying a zero condition at one place has zero singular image, hence belongs to the compact Selmer line; the zero condition then kills that line as well. Suppose the compact Selmer line is everywhere locally zero. The same duality sequence makes the singular image all of $Q_v\oplus Q_{v^c}$. The kernel of localization is precisely the compact Selmer line. Consequently the relaxed local image has dimension two and maps isomorphically to the singular plane. In each local hyperbolic plane choose the isotropic complement $E_w$ of $F_w$, choosing the two complements compatibly with $c$. The relaxed local image is the graph of a map \[ E_v\oplus E_{v^c}\longrightarrow F_v\oplus F_{v^c}. \] Global reciprocity makes the graph isotropic. In mutually dual bases its matrix is therefore skew-symmetric, hence of the form $\left(\begin{smallmatrix}0&u\\-u&0\end{smallmatrix}\right)$. Conjugation interchanges both pairs of basis vectors and preserves the local pairings. Invariance of the graph therefore changes $u$ to $-u$, so $u=0$. Lifting the first coordinate line gives $h$; conjugating gives $h^c$ and proves \cref{eq:base-green-bases}. The auxiliary summand has a nontorsion global point and nonzero localization, by \cref{prop:arithmetic-reduction}. If $r=0$, the unique Selmer line of $V/K$ likewise comes from a nontorsion point on the quadratic twist $A_0^D/\Q$, so it cannot be strict. When $r=1$, the twist by $D$ has zero Selmer space and the line comes from $H^1_f(\Q,V)$, giving $a^c=a$. \end{proof} \subsection{Primes with a controlled tame logarithm} In the prime construction and cohomological calculations of this section, Galois Frobenius elements are arithmetic Frobenius elements. The following construction gives a tame direction that is trivial at the fixed places and has uniform local bounds. In the strict case we will refine its limiting Frobenius element after determining the obstruction to first-order lifting. \begin{proposition}\label{prop:tame-primes} There are rational primes $q_i\notin S$, split completely in $L$, distinguished places $\mathfrak q_i$ of $K$ above them, cyclic $p$-quotients $\Delta_i$ of the ring class group of conductor $q_i$, and surjections \[ \ell_i:G_K\longrightarrow\Delta_i\simeq\Z/p^{n_i}\Z, \qquad n_i\longrightarrow\infty, \] with the following properties. \begin{enumerate}[label=\textup{(\roman*)}] \item $\ell_i$ is unramified outside the pair above $q_i$, its restriction to inertia at $\mathfrak q_i$ is surjective, $\ell_i^c=-\ell_i$, and $\ell_i|_{G_{K_s}}=0$ for every $s\in S$. \item $v_p(q_i-1)-n_i$ is bounded. At either place above $p$, the full ring class field of conductor $q_i$ is unramified and its local degree has bounded $p$-part. \item Frobenius at $\mathfrak q_i$, on all fixed data unramified outside $S$, tends to an element $\gamma\in G_L$ such that \[ \eps(\gamma)=1,\qquad \det(V(\gamma)-1)\ne0,\qquad \cC(\gamma)\text{ is not a root of unity}. \] Moreover $\gamma$ fixes all $p$-power radicals of a fixed, conjugation-stable finite list in $K^\times$ spanning $\cO_{K,S}^{\times}\otimes\Q_p$ under the Kummer map. \end{enumerate} \end{proposition} \begin{proof} Choose representatives $\mathfrak a_j$ for generators of the ideal class group, prime to $S$. For their orders, their multiplication relations, and the expressions of the primes in $S$ in these generators, choose principal generators. Adjoin generators of the finite-rank $S$-unit group and their conjugates to this finite list, denoted $\alpha_1,\ldots,\alpha_d$. Fixed class-group orders and roots of unity will cost only a fixed index. For a split prime $q$, the kernel of the ring class group map to the ordinary ideal class group is the quotient of $(\cO_K/q\cO_K)^\times$ by diagonal residue units and global units. Its $p$-part is cyclic, of order $p^{v_p(q-1)}$ up to a fixed factor. Require the $\alpha_j$ to have zero residue logarithm modulo $p^{n_i}$ at the pair over $q_i$. This is ensured by splitting in their $p^{n_i}$-radical extensions, together with the corresponding roots of unity. The ring class presentation now allows us to set the logarithms of all $\mathfrak a_j$ equal to zero and to take the surjective logarithm on the residue quotient. Each relation is respected, and the expressions chosen for the primes of $S$ give $\ell_i|_{G_{K_s}}=0$. Conjugation inverts the residue quotient and the ring class action, giving $\ell_i^c=-\ell_i$. We verify that imposing the full radical condition leaves room for $\gamma$. Over the cyclotomic tower the Kummer Galois group is a closed subgroup of $\Z_p(1)^d$. The joint image of $V$ and $\cC$ is a $p$-adic Lie group. A common Lie quotient with this translation group is abelian, so it kills the derived Lie algebra of the image of $V$. On the remaining torus Lie algebra, conjugation by an open cyclotomic subgroup is trivial; on the Kummer translations it is multiplication by the cyclotomic character. Any equivariant common quotient is consequently zero. This argument can be made after a fixed finite extension, which does not change Lie dimensions. Thus killing the radicals removes no Lie dimension from the joint image over the cyclotomic tower. For a non-CM elliptic curve the determinant-one image has Lie algebra $\mathfrak{sl}_2$, by the open image theorem \cite{Serre1972}; for a CM curve its identity component contains the norm-one torus, by the CM description of the Tate-module character \cite{Serre1968}. In either case the locus $\det(V(g)-1)\ne0$ is nonempty and open in every sufficiently small identity neighborhood. The infinity type of $\cC$ makes its restriction over the cyclotomic tower nonconstant; its torsion locus is a proper closed subset in a sufficiently small identity neighborhood. The two complements intersect. We may also impose $g\in G_L$ and any fixed open-kernel conditions. This gives $\gamma$ fixing the full radicals and the cyclotomic tower with the required properties. It remains to control $v_p(q_i-1)$, rather than merely make it large. The cyclotomic--radical image is a closed subgroup of \[ \Z_p(1)^d\rtimes\Z_p^\times \] with open cyclotomic projection. Choose an element in a pro-$p$ subgroup whose cyclotomic projection is nontorsion. Its $p^m$-th powers tend to the identity; their translation coordinates are divisible by $p^{m-C_0}$, and the valuation of their cyclotomic coordinate minus one is $m+C_1$, for fixed $C_0,C_1$. Multiplying $\gamma$ by such a power with $m=n_i+C_2$ therefore leaves all radicals trivial modulo the requested precision and makes the cyclotomic valuation exactly $n_i+C$, for a fixed $C$. Chebotarev applied to a growing sequence of finite quotients realizes these conditions and increasingly accurate approximations on all fixed data. Choosing a place above each prime selects the indicated Frobenius representative. The primes can be taken outside any fixed finite set. Finally the $p$-part of the order of the full ring class group is $p^{v_p(q_i-1)+O(1)}$. The kernel of its map to $\Delta_i$ therefore has bounded $p$-part. The Frobenius of every $s\in S$ belongs to this kernel. At $p$ the extension is unramified, so the $p$-part of the local degree, which is the $p$-part of the Frobenius order, is bounded. This proves the proposition. \end{proof} \subsection{Admissible cochains and division by the parameter} Fix a nonprincipal ultrafilter $\cU$ on the prime indices. A statement ``eventually'' in the construction below means on a set in $\cU$. For every fixed $b\ge1$, put \[ R_b=k[t]/(t^b). \] At finite stages the substitution of $(1+t)^{\ell_i}$ is defined only to finite $p$-adic precision. There is a constant $c_b$ such that \begin{equation}\label{eq:binomial-loss} u\equiv u'\pmod {p^n}\quad\Longrightarrow\quad \binom{u}{j}\equiv\binom{u'}{j}\pmod {p^{n-c_b}} \quad(0\le j2$, these identities give both diagonal zeros in \cref{eq:obstruction-matrix} and the opposite signs of its off-diagonal entries. It remains to make $\kappa=H_\gamma(a,h^c)$ nonzero. Set $d^-=h^c-h$. It is anti-invariant, and its cup product with $a$ is everywhere locally zero. Average a primitive to be anti-invariant as well. Then \[ H_\gamma(a,d^-)=2z_\gamma(a,d^-;u),\qquad H_\gamma(a,d^-)=2H_\gamma(a,h^c). \] It is enough to make $z_\gamma(a,d^-;u)$ nonzero. Let $G=G_{K,S}$ and $N_V=\ker(G\to\GL(V))$. The classes $a,d^-$ are linearly independent: the former is nonzero and invariant under $c$, whereas the latter is anti-invariant with nonzero local image. The image of $G$ on $V$ contains a central scalar different from one. The usual central-scalar argument gives $H^1(\im(G\to\GL(V)),V)=0$: for a cocycle $f$ and a central scalar $z\ne1$, the cocycle identity is $(z-1)f(g)=(g-1)f(z)$, so $f$ is a coboundary. Inflation--restriction thus makes restriction to $N_V$ injective on $H^1(G,V)$. On $N_V$, both cocycles are additive homomorphisms. Their joint image spans $V\oplus V$: its span is a $G$-submodule, and absolute irreducibility with $\End_G(V)=\Q_p$ says that a proper submodule projecting onto each factor would impose a scalar linear relation between the two restrictions. That contradicts the preceding injectivity. The absolute irreducibility here follows from open image in the non-CM case; in the CM case it follows because neither $K$ nor $L$ contains the CM field. Passing to the finite-index subgroup over $L$ leaves the span unchanged. The Weil pairing implies $\eps=1$ on $N_V$. On this group, \cref{eq:cup-primitive} reads \[ u(gg')=u(g)+u(g')-e(a(g),d^-(g')). \] Consequently \begin{equation}\label{eq:heisenberg-commutator} u([g,g'])=-e(a(g),d^-(g'))+e(a(g'),d^-(g)). \end{equation} The alternating form on $V\oplus V$ on the right is nonzero and nondegenerate. Since the joint translations span $V\oplus V$, some $g,g'\in N_V\cap G_L$ have a nonzero right-hand side. Write $\sigma=[g,g']$. Both translations vanish on $\sigma$, but $u(\sigma)\ne0$. Every Kummer character in our radical list is an additive homomorphism on $N_V$, because $\eps=1$ there. It therefore vanishes on $\sigma$, so $\sigma$ fixes all the radicals. A one-dimensional character such as $\cC$ also kills $\sigma$. Replacing $\gamma$ by $\gamma\sigma^m$, for an integer $m$, thus preserves its $V$, $\cC$, cyclotomic, and radical values. The two translations entering the correction in \cref{eq:corrected-primitive} do not change, whereas its primitive value changes by $m u(\sigma)$. Some $m$ makes the result nonzero. If further fixed open conditions are required, first replace $\sigma$ by a suitable nonzero power; its nonzero primitive value survives. This constructs the required $\gamma$. In particular the inverse matrix $V(\gamma)-1$, and hence its denominator bound, is unchanged by the modification. Applying the prime construction to this element proves \cref{eq:obstruction-matrix} for both signs of the tame twist. \end{proof} In the strict case we henceforth use this refined $\gamma$ in \cref{prop:tame-primes}, and $q_i,\ell_i$ denote the sequence realized from it. With the same ultrafilter $\cU$, we henceforth form $\ell$, $\psi$, $M_b^\eta$, and the admissible global and local cochain complexes from this reselected sequence. The preserved properties of the prime construction ensure that the specialization, division, and obstruction arguments above still apply. As stipulated at the start of the section, the final realization is made after all fixed conductors and tests have been chosen. Any additional fixed open-kernel conditions can be imposed in the construction and preserved by the commutator modification. \subsection{The exact Artin length bound} \begin{proposition}\label{prop:green-length} For the primes of \cref{prop:tame-primes}, chosen in the strict case as in \cref{lem:nonzero-obstruction}, every fixed $b\ge1$, and either $\eta=1$ or $\eta=-1$, one has \begin{equation}\label{eq:green-length} \length_{R_b}\GrH(L,V\otimes\psi^{\eta}) \begin{cases} =0,&\text{outside the strict case},\\ \le2,&\text{in the strict case}. \end{cases} \end{equation} In the strict case this module is killed by $t$ and is the image of its two-dimensional base space under multiplication by $t^{b-1}$. Put \begin{equation}\label{eq:b-star} b_*=1\quad\text{outside the strict case},\qquad b_*=3\quad\text{in the strict case}. \end{equation} Thus in both cases the length in \cref{eq:green-length} is strictly smaller than $b_*$. \end{proposition} \begin{proof} It suffices to work in each of the two summands over $K$. Let $\mathcal S_b^\bullet$ be the cone, shifted by $-1$, of the map from the admissible global cochain complex to the local complex at the strict $p$-place. Its degree-one cohomology is exactly the indicated Greenberg group. Indeed global and local $H^0$ vanish for $V$ and, by the $t$-filtration, for every $M_b$. The coefficient sequence in \cref{eq:coefficient-sequence} gives a short exact sequence of these cone complexes. The resulting cohomology sequence therefore identifies \begin{equation}\label{eq:selmer-kernel} \ker\bigl(H^1(\mathcal S_b^\bullet) \longrightarrow H^1(\mathcal S_1^\bullet)\bigr) =\im\bigl(t:H^1(\mathcal S_{b-1}^\bullet) \longrightarrow H^1(\mathcal S_b^\bullet)\bigr). \end{equation} There is no flatness assumption on these cohomology groups. If the base group is zero, induction in \cref{eq:selmer-kernel} gives zero at every order. This deals with the auxiliary summand and with both summands outside the strict case. In the strict summand, a class at any order $b\ge2$ reduces to an order-two class, whose base specialization is zero by \cref{lem:obstruction-necessary,lem:nonzero-obstruction}. Equation~\eqref{eq:selmer-kernel}, iterated, gives \[ H^1(\mathcal S_b^\bullet) =\im\bigl(t^{b-1}:H^1(\mathcal S_1^\bullet) \longrightarrow H^1(\mathcal S_b^\bullet)\bigr). \] This image is killed by $t$ and has $k$-dimension at most two. Its $R_b$-length is therefore at most two, as asserted. \end{proof} \subsection{A uniform local logarithm estimate} The cohomological length bound is now proved. Under the torsion assumption on $y$, we next establish the weighted Heegner-logarithm vanishing needed for the theta congruence. The argument first gives vanishing of local Kummer classes. The following estimate converts this into a valuation statement, uniformly in the unramified fields of definition that occur at $p$. \begin{lemma}\label{lem:uniform-local-log} Fix an integer $d_0\ge0$. There is a constant $c$ such that, for every finite unramified extension $E/\Q_p$ with $v_p([E:\Q_p])\le d_0$, \begin{equation}\label{eq:uniform-local-log} \log_{A_0,p}(A_0(E))\subseteq p^{-c}\cO_E. \end{equation} If $P\in A_0(E)$ and the Kummer class of $p^C P$ vanishes in $H^1(E,A_0[p^n])$, where $C,n\ge0$ are integers, then \[ v_p(\log_{A_0,p}P)\ge n-C-c. \] \end{lemma} \begin{proof} On a fixed sufficiently small formal subgroup, the formal logarithm and its inverse are integral, uniformly under unramified base extension. The intervening formal-group quotient has bounded $p$-exponent. In the N\'eron special fiber the geometric component group has fixed order, the additive part has bounded $p$-exponent, and a torus in characteristic $p$ has no $p$-torsion. For the \'etale $p$-power torsion in the abelian part, let $\alpha$ be a unit Frobenius eigenvalue. It is not a root of unity, by its complex absolute value. If $[E:\Q_p]=d$, the exponent is controlled by $v_p(\alpha^d-1)$. Writing $\alpha$ as its finite torsion part times a principal unit shows \[ v_p(\alpha^d-1)\le C_\alpha+v_p(d) \] whenever the left side is positive. It is therefore bounded for these extensions. A fixed $p$-power, followed if necessary by an integer prime to $p$, moves any point into the formal subgroup. Dividing its logarithm by that multiplier proves \cref{eq:uniform-local-log}, including at bad reduction. For the precision assertion, the Kummer injection \[ A_0(E)/p^n A_0(E)\hookrightarrow H^1(E,A_0[p^n]) \] gives $p^C P=p^n Q$ for some $Q\in A_0(E)$. Consequently \[ v_p(\log_{A_0,p}P)\ge n-C-c \] by \cref{eq:uniform-local-log}. Thus the comparison loses only a fixed precision, without division by the local extension degree. \end{proof} \subsection{Weighted Heegner traces and their local logarithms} Let $H_i/K$ be the full ring class field of conductor $q_i$, and let $y_i\in A_0(H_i)$ be the corresponding Heegner point. At the split prime $q_i$ it is the conductor-$q_i$ translate obtained by $n(1/q_i)$ in torus-adapted coordinates, with the opposite sign if the embedding convention requires it. Write $\mathcal G_i=\Gal(H_i/K)$, and view $\ell_i$ also as its quotient character. The sums in the following statement are raw sums over $\mathcal G_i$. \begin{proposition}\label{prop:heegner-log} At the distinguished place over $p$, and for either sign, \begin{equation}\label{eq:heegner-log} \sum_{g\in\mathcal G_i}(1+t)^{\pm\ell_i(g)} \log_{A_0,p}(y_i^g) \longrightarrow0\pmod {t^{b_*}}. \end{equation} More precisely, every coefficient of degree less than $b_*$ has valuation tending to $+\infty$ along $\cU$. The coefficients through any other fixed degree have a common lower valuation bound. These assertions hold in a common $p$-adic algebraic closure, with its fixed extension of the valuation. \end{proposition} \begin{proof} Choose integral Kummer cocycles for $y_i$ and corestrict with the tautological group-ring coefficients. Corestriction is a sum: its terms have integral coefficients regardless of the size of $\mathcal G_i$. Truncate by the substitution in \cref{eq:binomial-loss}. We obtain bounded admissible classes with coefficients $M_b^{\pm}$, for every fixed $b$, satisfying Kummer conditions at both $p$-places. The split Heegner norm relation gives augmentation equal to the Kummer class of \[ (a_{q_i}-2)y. \] Indeed the $q_i+1$ cyclic subgroups in the Hecke correspondence comprise two horizontal CM isogenies, corresponding to the two prime ideals over $q_i$, and the $q_i-1$ descending isogenies whose target order has conductor $q_i$. The latter are the orbit over the conductor-one ring class field. Thus the relative trace is the Hecke action minus the two horizontal translates. Tracing further to $K$ turns each horizontal translate into $y$, and the Hecke eigenvalue on $A_0$ is $a_{q_i}$. This proves the displayed relation; see also the correspondence and CM action conventions in \cite[Chapter~I, Sections~2--3]{GrossZagier1986}. Since $y$ is a fixed torsion point, one fixed scalar kills all these augmentations in integral Kummer cohomology. Their rational augmentation is zero. For $b_*=1$ this proves the cohomological assertion. In the strict case divide the limiting Kummer trace once by $t$, as in \cref{lem:admissible-specialization}; its first specialization, say $z_1$, is an ordinary compact Selmer class of $V/K$. The cohomology sequence of \cref{eq:coefficient-sequence} and $H^0(K,V\otimes k)=0$ make multiplication by $t$ injective on $H^1(M_{b-1}^{\pm})\to H^1(M_b^{\pm})$, so this divided class is unique. All local Kummer conditions survive this division because the tame character is trivial at $p$. The operation merely shifts bounded coordinates after subtraction of a fixed bounded coboundary; no denominator depending on $[H_i:K]$ is introduced. We verify the conjugation identity with its effect on this coefficient, using the CM coordinates of \cite[Chapter~II, Section~1]{GrossZagier1986}. Write a CM point in fractional-ideal coordinates as \[ x(\mathfrak a,\mathfrak n)= (\C/\mathfrak a\longrightarrow\C/\mathfrak n^{-1}\mathfrak a), \qquad \mathfrak n\overline{\mathfrak n}=(N). \] The dual cyclic isogeny gives $w_Nx(\mathfrak a,\mathfrak n) =x(\mathfrak n^{-1}\mathfrak a,\overline{\mathfrak n})$. Consequently \[ x(\mathfrak a,\mathfrak n)^c =w_Nx(\mathfrak n\overline{\mathfrak a},\mathfrak n). \] In the ring class group, $[\overline{\mathfrak a}]=[\mathfrak a]^{-1}$, so this is inversion followed by a fixed translation. The level ideal is invertible in every order in question because $(q_i,N)=1$. If $\varphi$ is the modular parametrization, the differentials of $\varphi\circ w_N$ and $-\operatorname{sign}(A_0)\varphi$ agree. Their difference is therefore a constant $Q$, the image of a cusp. The Manin--Drinfeld theorem \cite{Manin1972,Drinfeld1973} makes $Q$ of fixed torsion order. In the strict case $r=1$, so the Fricke scalar $-\operatorname{sign}(A_0)$ is $+1$. Conjugation also inverts the anticyclotomic weight, namely \begin{equation}\label{eq:weight-involution} t\longmapsto(1+t)^{-1}-1=-t+t^2-\cdots. \end{equation} The translation is multiplication by a group-ring unit of augmentation one. Because the constant Kummer coefficient has already vanished, that unit has no effect on the leading linear coefficient. Equation~\eqref{eq:weight-involution} thus gives $z_1^c=-z_1$. But the compact Selmer line is generated by the invariant class $a$. Therefore $z_1=0$. Divide once more by $t$. Its specialization $z_2$ again belongs to the compact Selmer line, by the same local Kummer and bounded division argument. That line is everywhere locally zero in the strict case. Thus the original trace has zero local Kummer class through degree two, which is the assertion required for $b_*=3$. It remains to convert the local Kummer vanishing into \cref{eq:heegner-log}. At $p$, Proposition~\ref{prop:tame-primes} makes $H_i/K$ unramified with local degree of bounded $p$-part, so \cref{lem:uniform-local-log} applies with a common constant $c$. Restricting the local twisted Kummer trace to the unramified field of definition makes the coefficient action trivial. It becomes exactly the Kummer class of the raw weighted sum, modulo the precision allowed by \cref{eq:binomial-loss}. The vanishing through degree $b_*-1$ holds to precision tending to infinity as in \cref{eq:precision-choice}. After one fixed factor $p^C$ clears the cochain denominators, the local comparison gives the lower bound $n-C-c$ at Kummer precision $n$. Thus the corresponding logarithm coefficients have valuations tending to infinity. For any fixed higher truncation, the Kummer cocycles, logarithms, and binomial coefficients all have a fixed denominator, so those coefficients remain bounded as well. This proves the proposition. \end{proof} When a subsequent construction squares a series, we form the preceding traces first at any fixed order $b\ge2b_*$. Boundedness of all the retained coefficients and \cref{eq:heegner-log} then permit the required product estimates without an interchange of an unbounded truncation and a limit. ==== END The-Selmer-converse-for-elliptic-curves-at-every-prime//sections/03-tame-deformation.tex ==== ==== BEGIN The-Selmer-converse-for-elliptic-curves-at-every-prime//sections/04-theta-construction.tex ==== \section{The holomorphic theta family}\label{sec:theta} We construct forms on a unitary group of signature $(3,1)$ at each real place of $F$, with unit eigenvalues for two normalized positive parabolic operators at $p$ and coefficients of uniformly bounded denominator. These are the forms whose constant and positive Fourier--Jacobi coefficients will be compared in the next two sections. The operator calculation allows arbitrary reduction of $A_0$ at $p$; the denominator bound must hold as both the weight and the tame level grow. Throughout this section a finite-order character at the moving prime means the universal character with values in a group algebra. All integrality assertions are made before specializing that algebra at its characters. \subsection{Hermitian spaces and the compact automorphic input} Let $\delta=\sqrt D$, with $\operatorname{Im}\sigma(\delta)>0$ for $\sigma\in\Sigma$. We use the Hermitian space and the skew-Hermitian space \begin{equation}\label{eq:theta-spaces} A=(L^2,1_2),\qquad W=B\oplus\mathbb H,\qquad B=(L^2,b_0\delta\,1_2), \end{equation} where $\mathbb H=Le\oplus Lf$ is a hyperbolic plane and $b_0\in F^{\times,+}$ is a unit at the primes above $p$. Choose $b_0$ so that \begin{equation}\label{eq:lattice-parities} v(b_0\delta\mathfrak d_F)\equiv0\pmod2 \quad\text{at every nonsplit finite place of }L/F. \end{equation} Here $\mathfrak d_F$ is the different and we use the trace additive character. To check the existence of this choice, prescribe the local norm classes of $b_0$ needed to cancel the parity of the additive conductor on each paired line. The exact norm-invariant sequence for the quadratic extension $L/F$ says that these prescriptions come from a global element if and only if their product is one. The possible odd contributions are at the primes of $D'$ that are inert in the relevant quadratic extension; the condition $(D/|D'|)=1$ in Lemma~\ref{lem:auxiliary-fields} makes their number even. The discriminants are odd and coprime, and the places over $2$ are split. The real prescriptions are positive. The product condition therefore holds. Multiplication by a global norm, using approximation at the split $p$-places, makes $b_0$ a $p$-unit without changing the prescribed classes. At a nonsplit finite place the extension is unramified. Scaling a line coefficient by a norm changes its valuation by an even integer, so \eqref{eq:lattice-parities} gives a self-dual Schr\"odinger lattice for each paired line, compatible with the trace character. Choose compatible hyperbolic bases. These choices give the unramified models and spherical tests used below. At $p$ all coordinates are unit self-dual coordinates. We pair the sign of the alternating form with the additive character so that a positive unary theta series has exponential $\exp(2\pi i a x^2z)$ with $a>0$. Write $\pi$ for the representation on the compact group $U(A)$ obtained from the weight-two form of $A_0$ by real quadratic base change and Jacquet--Langlands transfer, followed by pullback to $U(A)$; see \cite{ArthurClozel1989,JacquetLanglands1970}. More explicitly, $PU(A)$ is the projective multiplicative group of the quaternion algebra ramified at the two real places of $F$ and at no finite place. The real component of $\pi$ is trivial, as is its adelic central character. The successive base changes remain cuspidal: the only possible quadratic inducing field in the dihedral case was excluded in the choice of $L$. Its standard base change to $L$, denoted $\pi_L$, is thus the cuspidal base change of the representation of $A_0$. At a nonsplit finite place take hyperspecials with the $B$-lattice homothetic to the $A$-lattice. The hyperspecial maps onto the hyperspecial of the adjoint quotient because the unramified kernel is connected. On dual groups the quotient map is the inclusion $\mathrm{SL}_2\hookrightarrow\mathrm{GL}_2$ with trivial pinned Galois action on the former. Consequently the spherical standard parameters are the restrictions of those of the quaternionic transfer. We use algebraic-valued vectors in the compact automorphic model of $\pi$; a rational tensor vector can be taken real algebraic. They are spherical at nonsplit finite places. No local condition is imposed on $\pi_p$. The precise multiplicity and pullback assertion needed for the binary lift is proved in Section~\ref{sec:constant-terms}. For the moving prime $q_i$, let $\nu_i$ denote the universal character of $\Delta_i$, pulled back to the norm-one torus of $L/F$ via the norm $N_{L/K}$. Put \begin{equation}\label{eq:theta-character} \eta=\cC^m\nu_i,\qquad k_0=m-1. \end{equation} The integer $m>0$ will grow in the ordinary sense and tend to zero $p$-adically on the fixed torsion branch. The finite smooth conductor of the factor $\cC^m$ is fixed, or is killed by that branch. The tame character $\nu_i$ is trivial at every fixed place where this is required by Proposition~\ref{prop:tame-primes}. We often suppress $i$ in $\eta$. \subsection{The Fock vector and its algebraic weight} Take the oscillator representation on $A\otimes_L W$ with both even unitary splitting characters trivial, in the conventions of \cite{Kudla1994}. For a bare compact vector $z\in\pi^\vee$, the theta integral has the form \begin{equation}\label{eq:upper-theta-integral} \Theta_{i,m}(g;\Phi,z) =\int_{[U(A)]}\theta_\Phi(g,h)z(h)\eta^{-1}(\det h)\,dh. \end{equation} The quotient of integration is compact, so this is an ordinary convergent theta integral. At each real place use Fock polynomials of degree $k_0$ in the $2\times2$ minors of the $2\times3$ positive-coordinate matrix. Their space is $\operatorname{Sym}^{k_0}(\bigwedge^2\operatorname{Std}_3)$. The vacuum contributes $\det$ on the source $U(2)$, while each minor contributes another determinant. Thus the source type is $\det^m$, as required by \eqref{eq:upper-theta-integral}. On the target the vacuum has weight $(1,1,1;-1)$, and the polynomial contributes $(m-1,m-1,0;0)$. The resulting weight is \begin{equation}\label{eq:theta-weight} \lambda=(m,m,1;-1). \end{equation} The first three entries refer to the distinguished holomorphic Lie quotient; the last refers to the remaining distinguished direction, whose conjugate is holomorphic. The antiholomorphic lowering operators are contractions involving the negative coordinates, which do not occur in these polynomials. They therefore annihilate the vector. The mixed expansion below proves holomorphy also at the cusps. We specify the integral coefficient lattice, including its frame convention. Let $\mathcal H=\omega^+$ be the rank-three positive Hodge differential bundle and put $d=m-1$. The positive creation directions belong to the holomorphic Lie quotient, dual to $\mathcal H$. Write frames as row tuples. If a differential frame changes from $e$ to $eg$, its dual creation frame changes by $g^{-t}$, and the minor frame changes by $\bigwedge^2g^{-t}$. This passive change of differential frame is distinct from the active compact-group action on the Fock polynomials used to compute \eqref{eq:theta-weight}: the Lie quotient has active type $\operatorname{Std}_3$, while its differential dual has type $\operatorname{Std}_3^\vee$. Theta evaluation is linear in its Fock test polynomial. Equivalently, the covariance identity $\Theta(gk;P)=\Theta(g;\omega(k)P)$ says that its coefficient array transforms dually to the input polynomial lattice. In the preceding differential frames this gives \begin{equation}\label{eq:dual-fock-lattice} \mathcal P_d(\mathcal H) =\operatorname{Sym}^d((\textstyle\bigwedge^2\mathcal H)^\vee), \qquad \mathcal D_d(\mathcal H) =\mathcal P_d(\mathcal H)^\vee =\Gamma^d(\textstyle\bigwedge^2\mathcal H). \end{equation} Here $\Gamma^d$ denotes the divided-power module, defined by the displayed duality for finite locally free modules. We use the literal dual basis to the Fock monomials throughout. In particular, extracting a displayed monomial coefficient does not multiply it by a binomial coefficient or a factorial. Including the vacuum, the positive coefficient bundle is $\det(\mathcal H)\otimes\mathcal D_d(\mathcal H)$, and the opposite differential line occurs to exponent $1$. Section~\ref{sec:cusp-lifting} uses this same integral bundle. All constructions are tensored over the two embeddings of $F$. We use the PEL model with common rational similitude to give these coefficient bundles their algebraic meaning, and restrict to the isometry components. Analytically these are the Hermitian symmetric domain times finite isometry-group cosets modulo the rational isometry group. Fixing the multiplier class of the finite frame and absorbing positive principal multipliers into the polarization realizes them as components or finite covers of the PEL model, with compatible cyclotomic constants. The positive rational units preserving a fixed compact multiplier level are trivial. Thus the isometry Hecke correspondences and the expansion principles below are compatible with this model. No extension of the theta kernel to a chosen similitude theta representation is required. \subsection{Split finite tests and the two parabolic operators} At a split finite place identify the isometry groups with their actions on the distinguished factors. In the linear-pair polarization the Schr\"odinger variables are dual covectors $Y\in M_{2,4}$, and the action is \begin{equation}\label{eq:split-weil-action} \omega(h,g)\Phi(Y) =|\det h|^2|\det g|\Phi({}^thYg). \end{equation} For the binary target the exponent on $|\det h|$ is $1$ instead of $2$. There is no further determinant character: both even splitting characters are trivial. At nonsplit unramified places the matched self-dual lattice test is spherical. At every other fixed finite place we use integral tests up to a fixed denominator and at fixed level. Finite additive Fourier transforms away from $p$ introduce only powers of the residue characteristic, roots of unity, and square roots of moduli. These are $p$-units; in particular no inverse order of a multiplicative residue group is introduced. Order the distinguished columns at $p$ as $(e,B_1,B_2,f)$. The $e$ direction is multiplicative. Our tests have the form \begin{equation}\label{eq:p-pivot-test} \Phi_p(Y)=\phi_p(Y_{[1,2]}) \1_{\Z_p^2}(Y_3)\1_{\Z_p^2}(Y_4), \qquad \supp\phi_p\subset\GL_2(\Z_p), \end{equation} where $\phi_p$ is locally constant at a fixed depth $s$. They are fixed by a subgroup $J\subset\GL_4(\Z_p)$ with upper block pattern $(2,1,1)$: its upper block unipotent is full, its first Levi is a principal congruence subgroup of depth $s$, its two remaining unit factors are full, and its lower block entries have depth $s$. Increasing the fixed $s$ if needed makes the same description work for all the tests. On the canonical component the rank-three distinguished flag member modulo $p^s$ is the multiplicative subgroup. This level records only flags and frames in that subgroup; it does not choose a splitting of the \'etale quotient. Its integral geometric description will be given in Section~\ref{sec:cusp-lifting}. At either distinguished $p$-place put \begin{equation}\label{eq:theta-u-operators} t_j=\diag(1_j,p^{-1}1_{4-j}),\qquad U_j=[Jt_jJ],\qquad j=2,3, \end{equation} where $U_j$ is the unnormalized right-coset sum. \begin{lemma}\label{lem:split-u-operators} Every test \eqref{eq:p-pivot-test} satisfies $U_2\Phi_p=p^2\Phi_p$ and $U_3\Phi_p=p^2\Phi_p$. Thus both operators $p^{-2}U_j$ have value $1$ on the theta form of weight \eqref{eq:theta-weight}. \end{lemma} \begin{proof} The upper shifts in the double coset are indexed by $M_{j,4-j}(\F_p)$, of cardinality $p^{j(4-j)}$. The expanding columns must become divisible by $p$ before the multiplication by $p^{-1}$. Since the first two columns have rank two modulo $p$, these equations have $p^{(j-2)(4-j)}$ solutions whenever the original test is supported, and none otherwise. The equations do not change the chosen head function. The determinant factor in \eqref{eq:split-weil-action} is $p^{4-j}$. Hence the surviving factor is \[ p^{4-j}p^{(j-2)(4-j)}=p^{(j-1)(4-j)}=p^2 \] for both $j=2$ and $j=3$. The calculation is independent of the source representation and of its reduction at $p$. \end{proof} At each of the two places of $F$ over $q=q_i$ take the corresponding pivot test \begin{equation}\label{eq:q-pivot-test} \Phi_q(Y)= \1_{\GL_2(\cO_q)}(Y_{[1,2]}) \eta_q(\det Y_{[1,2]}) \1_{M_{2,2}(\cO_q)}(Y_{[3,4]}). \end{equation} Only the tame determinant factor is relevant on the displayed compact support. Its transformation under $\GL_2(\cO_q)$ cancels the $\eta_q^{-1}$ in source integration. Consequently this integration uses a compact subgroup of fixed volume, rather than the kernel of a character of growing order. The coefficients in \eqref{eq:q-pivot-test} are group-like units in the tame group algebra. The corresponding two-column test is the reference test in the binary kernel. Section~\ref{sec:content} will impose the other finite local controls without changing either \eqref{eq:p-pivot-test} or \eqref{eq:q-pivot-test}. \begin{lemma}\label{lem:theta-eigenparameters} At a split spherical place the standard parameter of the theta form, in unitary normalization, is \begin{equation}\label{eq:theta-eigenparameters} (\pi\otimes\eta)\ \oplus\ |\cdot|^{-1/2} \ \oplus\ |\cdot|^{1/2}. \end{equation} Here $\pi\otimes\eta$ means tensoring the standard two-dimensional parameter by the determinant-twist character. \end{lemma} \begin{proof} This is the unramified theta correspondence for the matched lattices. At split places it can also be read directly from the open rank-two matrix orbit: its coinvariants are the normalized parabolic induction of the determinant-twisted source with the trivial binary representation. The latter has standard parameters $|\cdot|^{-1/2},|\cdot|^{1/2}$. The open-orbit calculation, including the exclusion of the lower ranks near the comparison point, is given in Lemma~\ref{lem:fixed-place-spanning}. \end{proof} The spaces, coefficient lattice, and local tests are now fixed. We record the resulting family before proving the uniform denominator bound, which occupies the rest of this section. \begin{proposition}\label{prop:theta-family} For the characters \eqref{eq:theta-character} and every sufficiently large positive $m$ on the prescribed branch, the theta construction gives holomorphic algebraic forms $\Theta_{i,m}$ of weight \eqref{eq:theta-weight}. In the canonical ordinary monomial-dual lattice their coefficients have a common bounded denominator, coefficientwise in the tame group algebra and in every fixed truncated substitution $R_b$. They have spherical parameters \eqref{eq:theta-eigenparameters}, and at both distinguished $p$-places they satisfy \[ p^{-2}U_2\Theta_{i,m}=\Theta_{i,m},\qquad p^{-2}U_3\Theta_{i,m}=\Theta_{i,m}. \] These assertions allow arbitrary fixed split tests of the stipulated bounded levels. They remain valid for the particular tests with all the finite local controls in Proposition~\ref{prop:bounded-content}. \end{proposition} \subsection{Frames, periods, and unary moments} The remaining bound reduces, through the mixed Fourier--Jacobi expansion, to moments of unary theta series. We first fix their periods and differential frames. The essential estimate will bound every polynomial combination of moments by the supremum norm of that polynomial on the position lattice; bounds for individual degrees alone would not give the required interpolation. Use the PEL Hodge bundles, with logarithmic differentials in toric directions. In complex linear coordinates analytic-to-de Rham comparison uses $2\pi i\,du$ for each holomorphic differential, including those of the conjugate Lie quotient. Normalize each Fock creation generator so that its leading Schr\"odinger position coordinate is linear, rather than multiplied by $\pi^{1/2}$. Diagonal CM and hyperbolic frames can be chosen with algebraic $p$-unit scales, allowing square roots. The base heights, quadratic coefficients, and comparison bases can likewise be chosen to be units at $p$. On one positive paired line, restriction to the symplectic $\mathrm{SL}_2/F$ gives a Hilbert unary theta function at the CM point of type $\Sigma$. Write $d=e_0+2j$ with $e_0\in\{0,1\}$, independently at each embedding when necessary. The $d$-th Fock vector is the $j$-fold normalized Maass--Shimura raise of the base theta of weight $\frac12+e_0$. Indeed the compact-circle raising operator multiplies by the square of the Fock variable, while the leading term of the normalized derivative of the Gaussian exponential is its quadratic form. Comparison of leading terms identifies the two vectors up to a fixed algebraic unit raised to the degree. There is no factor $j!$. If $\Omega_\sigma$ compares the linear differential with the chosen integral algebraic CM differential, normalization multiplies by \begin{equation}\label{eq:unary-period-normalization} \left(\frac{2\pi i}{\Omega_\sigma}\right)^{1/2+d}. \end{equation} Use compatible square roots in products. Since the type is induced from $K$, rational tensor decompositions of homology over $K$, with bases localized at $p$, compare these periods with the elliptic CM period by algebraic $p$-units. Equivalently use the Serre tensor construction and prime-to-$p$ isogenies; $p$ is completely split. Passing to canonical Igusa differentials gives the analogous comparison with $p$-adic CM periods, again with unit factors. \begin{convention}\label{conv:scaled-support} The position-moment functional is defined first in fixed Igusa differential frames. Fixed powers of algebraic unit scales arising from conversion to another CM frame are restored afterwards. An equivalent formulation puts the polynomial norm on the actual scaled compact support. We do not identify that support with $\Z_p$ merely because the scale is an algebraic $p$-unit: a unit in a coefficient extension need not belong to $\Z_p$. The restored unit powers are analytic on a sufficiently narrow weight branch and have valuation zero. \end{convention} \begin{lemma}[Uniform unary moments]\label{lem:unary-moments} Fix the integral position supports, the locally constant tests at $p$, and their levels. At a fixed ordinary CM test object, the period-normalized unary jets, in the frames of Convention~\ref{conv:scaled-support}, are the moments of a bounded measure on the product of their integral position lattices. Its norm is bounded in terms of the coefficients of the fixed tests and their $p$-levels. The bound is independent of the moment degrees and of the moving tame level $q_i$. Cutting a position measure to units gives at classical degrees the theta jet with the corresponding unit cut on the finite position test. The corresponding moments converge to these unit moments when every degree tends to infinity and converges on its fixed $p$-adic branch. On a sufficiently narrow branch the unit moments, including the fixed CM unit-scale powers, are analytic functions of the degrees. \end{lemma} \begin{proof} For a scalar position test $\phi$, write $J_{\mathbf d}$ for the normalized jet of multi-degree $\mathbf d$ at the fixed CM object. For a polynomial $P(X)=\sum_{\mathbf d}a_{\mathbf d}X^{\mathbf d}$ on the product position lattice $\Lambda$, put \[ L_\phi(P)=\sum_{\mathbf d}a_{\mathbf d}J_{\mathbf d}, \qquad \|P\|_\Lambda=\sup_{x\in\Lambda}|P(x)|_p. \] We will prove $|L_\phi(P)|_p\leq C\|P\|_\Lambda$, where $C$ depends only on the stated test bounds and $p$-levels. The proof constructs the square of this combination as an algebraic Igusa function, bounds its expansions, and removes its initial denominator. The square argument in this proof is scalar. For a test with values in a tame group algebra, first expand it in the group basis and apply the argument separately to each scalar coefficient test. These tests have the same coefficient bound and the same position supports and levels at $p$. Only after obtaining the scalar estimates do we reassemble the group-algebra-valued moments. In particular, no coefficientwise bound is inferred by taking a square root inside a growing group algebra. \paragraph{Algebraic squares in a common Igusa ring.} For a fixed tame Schwartz level $N_1$, which may contain $q_i$, let $\mathcal I_{N_1}$ be the complete ordinary Hilbert Igusa tower with its canonical multiplicative trivialization. In its canonical logarithmic differential frames all integral Hodge weights are functions. Let $\mathcal A_{N_1}$ be the $p$-adically complete ring of these regular functions. Finite sums of different weights therefore lie in one ring; a finite Igusa level may be chosen separately at each reduction precision. We use Katz's expansion principle \citep[Section~5.2]{Katz1976} and its Hilbert formulation \citep[Theorem~6.10 and Sections~11.1--11.7]{AndreattaGoren2005}. The coefficient-multiplier formula for differentiation is \citet[Sections~15.21--15.24]{AndreattaGoren2005}, which identifies these operators with those of \citet[Corollary~2.6.25]{Katz1978}. For the CM comparison we use the equality of the ordinary unit-root and CM Hodge projections in \citet[Section~1.4, Theorem~1.2]{HidaTilouine1993}, attributed there to Katz~(2.6.7). The prime $p$ splits completely in our CM field, and the chosen multiplicative direction defines the required ordinary CM type. After the fixed tame-level enlargement, the level and polarization hypotheses in that comparison hold. We apply it to the algebraic integral-weight products constructed below; their algebraization and descent are part of this proof. We first show that the square of a finite combination of jets belongs to $\mathcal A_{N_1}[1/p]$. Enlarge the fixed prime-to-$p$ level, including its level at $2$, to fix the metaplectic multipliers. Every product of two base half-weight thetas is then an ordinary integral-weight algebraic form, by its Fourier expansion. Products of their normalized nearly holomorphic derivatives are rational expressions in the jets of these base products: apply Leibniz and Gauss--Manin, initially allowing powers of base squares in the denominators. There are no horizontal poles. On a smooth characteristic-zero Hodge-frame chart the square of a base theta has even horizontal divisor, as is seen in its holomorphic half-weight realization over $\C$. Locally write this square as $h^2u$, with $u$ invertible. Adjoining $\sqrt u$ is \'etale, and the half-weight coefficient $h\sqrt u$ is regular. On the torsor of complements to the Hodge filtration, projected Gauss--Manin differentiation has a regular connection matrix; its half-weight action multiplies the scalar weight by $1/2$. Differentiating $h\sqrt u$ therefore introduces powers of $2$ and $u$ in denominators, but no inverse of $h$. All fixed iterates are regular. Products descend because the base cross-products fix the signs of the local square roots. In logarithmic cusp charts, choose fixed widths clearing the half-integral exponents; the operators $q_\sigma\partial/\partial q_\sigma$ preserve nonnegative orders. The same argument applies there, and specializing the splitting variable gives the required jets. Omit an identically zero base theta. Projection by the ordinary unit-root splitting consequently produces a function $F_P\in\mathcal A_{N_1}[1/p]$ from the square of the combination defining $L_\phi(P)$. Its initial vertical denominator may depend on the orders and on $N_1$. At a CM object the unit-root splitting is the CM splitting, so these products agree with the normalized analytic jets, including products of different orders and weights. In particular, $F_P(\mathrm{CM})=L_\phi(P)^2$. Compatible square roots also give algebraicity of the individual normalized values. The common Igusa ring is what makes this construction possible when $P$ combines different weights. \paragraph{Compatible cusps and the position norm.} We next choose cusp frames in which the bound by $\|P\|_\Lambda$ is preserved. On each paired unary line, choose the position vector in the distinguished $L$-factor selected by $\Sigma_p$, the same factor used for the linear-pair coordinates $Y$. This is the CM multiplicative eigenspace. The unit local pairing makes this an integral symplectic frame. Specify the finite position tests in this frame from the outset. At a place over $p$, choose $s$ such that all these tests are supported on $\Z_p$ and constant modulo $p^s\Z_p$. Their additive Fourier transforms are supported on $p^{-s}\Z_p$. With the unramified additive character and unit quadratic coefficient, the lower unipotent with parameter $c\in p^{2s}\Z_p$ fixes each test: conjugating by Fourier transform makes its action multiplication by the trivial phase $\psi(cQ(x))$ on that support. Upper unipotents with integral parameter preserve integral position support, and diagonal units rescale it by units. Enlarge a fixed depth $M$ to cover all tests and fixed character conductors. Their congruence stabilizer then contains the subgroup with $a,d\equiv1$, $c\equiv0\pmod {p^M}$ and $b\in\Z_p$. Use the canonical $\Gamma_0(p^M)$ section over the tame ordinary moduli space, together with the multiplicative Igusa marking. At a Tate cusp its connected subgroup is $\mu_{p^M}$, the same position direction used at the CM object. Choose cusp representatives integral at $p$ with unit diagonal entries, by choosing their cusp ideals prime to $p$ and applying approximation to the bases. Their effective frame changes lie in $B(\Z_p)K(p^M)$, where $B$ is the upper triangular subgroup and $K(p^M)$ the principal congruence subgroup. The congruence part fixes the finite test; the triangular part preserves its support and coefficient norm. Thus this choice involves no initial Fourier interchange of the position and momentum directions. These cusps test every component. Indeed the finite Igusa cover over each connected tame ordinary component is a Galois torsor under the units of the multiplicative character group \citep[Sections~7.2 and~11.1]{AndreattaGoren2005}. Its deck group acts transitively on the connected components of the cover. The canonical unramified Tate-cusp chart of \citet[Sections~6.5 and~11.7]{AndreattaGoren2005} gives one such cusp; translating its marking by the deck units gives a cusp on each component. These changes are diagonal units and preserve the position norm. We use these canonical charts and their unit-deck translates as the compatible cusps. The expansion principle applies componentwise, so one such cusp per component will suffice to detect vertical divisibility. All of this concerns the position frames before restoring the separate unit scales of Convention~\ref{conv:scaled-support}. \paragraph{The uniform bound and removal of vertical denominators.} By scaling $P$, it suffices to prove the estimate when $\|P\|_\Lambda\leq1$. At a compatible cusp the base expansion is $\sum_x\phi(x)q^{Q(x)}$. The normalized jet of degree $e_0+2j$ has expansion \[ \sum_x\phi(x)x^{e_0}Q(x)^j q^{Q(x)}. \] Katz differentiation multiplies by the quadratic index. The quadratic coefficient is a $p$-unit; normalize the position coordinate first, or record its power as a separate fixed unit-scale factor. A polynomial $P$ of supremum norm at most $1$ on the integral position lattice therefore gives coefficients \begin{equation}\label{eq:moment-cusp-coefficient} \sum_{Q(x)=\beta}\phi(x)P(x), \end{equation} each bounded by the fixed bound for $\phi$. Every sum is finite. The square has bounded convolution coefficients, with twice the valuation loss. In several variables the same argument applies to multiorders. Parity pieces are separated by the sign projectors $2^{-r}\sum_{\varepsilon\in\{\pm1\}^r}\varepsilon^e[\varepsilon]$, which are integral because $p>2$; the base cross-products give their common descent data. At primes other than $p$, changes of polarization preserve this bound, since their normalizations are residue-characteristic powers and roots of unity. At $q_i$ the scalar coefficient tests obtained from the determinant character have integral coefficients. Additive Fourier transforms there likewise have $p$-unit normalization. Thus $F_P$ has the same expansion bound at every compatible cusp, independently of the degree and $q_i$. Clear that fixed bound and call the resulting function $G\in\mathcal A_{N_1}[1/p]$. Its expansions are integral. Choose the least $r\geq0$ for which $\varpi^rG\in\mathcal A_{N_1}$. If $r>0$, all expansions of $\varpi^rG$ vanish modulo $\varpi$. At a finite level containing this reduction, the expansion principle on each ordinary component implies that its reduction is zero: every component meets one of the compatible cusps just constructed, and the finite \'etale Igusa cover introduces no denominator. Thus $\varpi^rG$ is divisible by $\varpi$, contradicting minimality. This saturation argument removes the entire initial vertical denominator, regardless of its dependence on $N_1$ and the degrees. Evaluation at the integral ordinary CM object now bounds $F_P(\mathrm{CM})=L_\phi(P)^2$. Taking a scalar square root proves $|L_\phi(P)|_p\leq C$ and hence the required polynomial-norm estimate. \paragraph{The measure and its unit restriction.} We have obtained a bounded linear functional on polynomials on the compact integral position lattice. Polynomial density extends it to continuous functions, hence to a bounded measure. Approximate the characteristic function of the units by polynomials. Applying the same square and cross-product argument to the difference proves that cutting this measure corresponds exactly to cutting the finite test. On the complement of the product unit locus, the monomials tend uniformly to zero when every coordinate degree tends to infinity. On units, after fixing the torsion component, the formula $x^s=\omega(x)^s\exp(s\log\langle x\rangle)$ proves analyticity on a sufficiently narrow branch. The finitely many fixed coefficient-field unit scales have the same property after narrowing the branch once. \end{proof} \subsection{Mixed Fourier--Jacobi expansions and their integrality} At the rank-one boundary use $x\in A$ for the hyperbolic variables and the Schr\"odinger realization of $A\otimes B$ for the remaining variables. The Fourier--Jacobi indices are the positive norms $(x,x)$, together with zero. At zero Jacobi argument a power of the positive hyperbolic Fock column inserts the corresponding power of the coordinates of $x$. There are no lower Hermite terms: the used creations are the isotropic half of the variables for the contractions, namely holomorphic linear coordinates on the positive trace form of $A_\infty$. The positive and conjugate multiplicative frames in this direction are toric logarithmic frames, so these insertions incur no CM periods. The $B$-columns have precisely the unary normalization of Lemma~\ref{lem:unary-moments}. At a rational Jacobi torsion point, use Heisenberg translation of the finite tests and its translated Fourier trivialization. A rational unipotent moves this point to zero; for prime-to-$p$ torsions this is a prime-to-$p$ isogeny move and supplies a splitting of the semiabelian differentials. Consequently the same assertion holds monomial by monomial according to column degree. No estimate in an unrelated analytic trivialization at a nonzero Jacobi argument is required. Restricting to $B\oplus\mathbb H$ gives another description: after removing $\exp(2\pi i(x,x)\tau)$, the hyperbolic coefficient is the harmonic position polynomial. In particular, at index zero only the minor using the two $B$-columns remains; it is the binary theta lift of parallel determinant weight $m$. \begin{lemma}\label{lem:mixed-expansions} The mixed Fourier--Jacobi expansions of \eqref{eq:upper-theta-integral}, in algebraic ordinary differential frames and in the monomial-dual lattice, have a common bounded denominator. The bound is independent of $m$, $i$, and the tame character, and is coefficientwise in the tame group ring. The same assertion holds after substitution into any fixed $R_b$. \end{lemma} \begin{proof} Choose source integration representatives integral at $p$, by weak approximation in the rational unitary group, or by changing the rational Hermitian basis. The source quotient then gives finite sums at a level of fixed volume denominator. This remains true at $q_i$ because the determinant equivariance of \eqref{eq:q-pivot-test} cancels the varying source character. The archimedean transformation factors cancel in the integral, and character evaluations on the remaining representatives are algebraic $p$-units. A boundary representative may be chosen integral at $p$ by Iwasawa decomposition, sending the specified multiplicative member to $e+B$ modulo the fixed level. The translated $B$-tests remain integrally supported. The test in the column Fourier-dualized to produce $x$ is invariant under integral translations, so $x$ too is integral at $p$. Rational Jacobi torsions can use unipotent parameters divisible by the fixed $p$-level; these torsions are still dense and preserve this structure. Representatives and unipotents can simultaneously be chosen integral at the moving and fixed exceptional places, using approximation in the cusp parabolic and the rational Cayley chart on its binary unitary factor. After the source-equivariance cancellation just made, expand each remaining finite test in the tame group basis and decompose each scalar coefficient test into simple unary tensors, retaining integral support and the original coefficient bound. Apply the scalar unary estimate before reassembling the group-algebra coefficients. Additive Fourier transforms away from $p$ cost only $p$-units; at $p$ the levels are fixed. Each index coefficient at a prime-to-$p$ Jacobi torsion is consequently a finite sum of the bounded moments in Lemma~\ref{lem:unary-moments}, multiplied by integral powers of the coordinates of $x$ and by the unit scales already separated in Convention~\ref{conv:scaled-support}. The nonarchimedean bound is unaffected by the number of summands. At each fixed weight and Fourier--Jacobi index, these algebraic evaluations first give an algebraic coefficient section. Indeed the coefficient bundle on the abelian chart is algebraic. By proper algebraic--analytic comparison, the holomorphic coefficient lies in the complexification of its finite-dimensional algebraic space of sections. Choose finitely many algebraic prime-to-$p$ torsion evaluations that separate this space, in the algebraic split differential frames just used. In an algebraic basis their evaluation matrix has full rank. Algebraicity of the values therefore gives algebraicity of the section, with an initial denominator that may depend on the weight, index, and tame level. The full family of these torsion evaluations detects vertical integrality. The prime-to-$p$ torsions, and those satisfying the fixed congruences just imposed, are dense on the special fiber. Their split differential frames are obtained by prime-to-$p$ isogenies. After clearing an individual denominator, a section whose normalized torsion evaluations vanish modulo the uniformizer is therefore zero modulo it. The same least-denominator argument as above proves the uniform bound for the coefficient section. These algebraic coefficients first give algebraicity of the form by the characteristic-zero expansion principle. The ordinary Fourier--Jacobi expansion principle then transfers their uniform bound to the form itself. The canonical cusp charts with their finite \'etale level are those of Lemma~\ref{lem:canonical-ordinary-level}. Their construction uses only ordinary PEL geometry and the flag and frame data of $J$, so it is independent of the theta estimates and the cusp-lifting conclusion. Finally the substitution defining $R_b$ uses only the finitely many binomial coefficients through degree $b-1$. Its factorial loss is a fixed constant depending on $b$. This proves the last assertion. \end{proof} \begin{proof}[Proof of Proposition~\ref{prop:theta-family}] The Fock calculation gives the weight and holomorphy, the mixed expansion gives its extension at cusps and its algebraic integral interpretation, Lemma~\ref{lem:split-u-operators} gives the exact operator values, and Lemma~\ref{lem:theta-eigenparameters} gives the spherical parameters. Lemma~\ref{lem:mixed-expansions} gives the uniform bounds. \end{proof} ==== END The-Selmer-converse-for-elliptic-curves-at-every-prime//sections/04-theta-construction.tex ==== ==== BEGIN The-Selmer-converse-for-elliptic-curves-at-every-prime//sections/05-constant-terms.tex ==== \section{The constant terms and a square-root period comparison} \label{sec:constant-terms} We retain the theta data of Proposition~\ref{prop:theta-family} and the integer $b_*$ of Section~\ref{sec:tame}. At each of the two places of $F$ above $q=q_i$, the upper theta test, in the column order $(e,B_1,B_2,f)$, is supported on \[ Y_{[e,B_1]}\in\GL_2(\cO_{F_v}),\qquad Y_{B_2},Y_f\in\cO_{F_v}^{2}, \] and has factor $\eta_v(\det Y_{[e,B_1]})$ on the invertible block. Its transformation under the full source maximal compact cancels the determinant twist in the source integral. The corresponding two-column test is the reference test for the binary theta lift. All unspecified exceptional tests have fixed support and level and uniformly bounded coefficients. \begin{proposition}\label{prop:constant-term-small} Assume that the conductor-one Heegner point $y$ is torsion. The positive weights $m_i$ in Proposition~\ref{prop:theta-family} can be chosen with $m_i\to+\infty$ and $m_i\to0$ on the prescribed $p$-adic branch so that every constant Fourier--Jacobi coefficient of the upper theta form tends coefficientwise to zero modulo $t^{b_*}$. The assertion holds in bounded algebraic ordinary differential bases, uniformly over all boundary representatives and all polynomial components. The weights may be required to satisfy any additional lower bounds and any further $p$-adic approximation conditions tending to zero. \end{proposition} Here and below, a coefficientwise limit uses the full tame Artin deformation of Section~\ref{sec:tame}. In particular, it is stronger than a statement about separate specializations at finite-order characters. At index zero, the hyperbolic variable in the mixed expansion is zero. Every minor involving its column vanishes, and the remaining Fock polynomial is the determinant power on $B$. A constant coefficient is therefore a value of the binary theta lift. We first identify the single global scalar in that lift. Its square will be expressed by two raw toric periods, one of which approaches the Heegner logarithm from Proposition~\ref{prop:heegner-log}. \subsection{The binary scalar and its multiplicity} Since multiplying a Hermitian form by the scalar $b_0\delta$ does not change its isometry group, $U(B)=U(A)$. The binary theta lift of the source data belongs to the compact automorphic representation $\pi\eta(\det)$. We record the multiplicity assertion needed here. At split places equal-rank local theta correspondence gives the prescribed representation; at nonsplit finite places we use only its spherical line, and at infinity the binary polynomial gives the determinant character. Removing $\eta(\det)$ gives trivial adelic central action. There is a direct description of the resulting automorphic space that avoids a multiplicity formula for general inner-form packets. Let $D_{\mathrm{quat}}/F$ be the quaternion algebra ramified at the two real places and split at all finite places, so that $PU(A)=P(D_{\mathrm{quat}}^\times)$. Write $\omega_{L/F}$ for the quadratic idele-class character. The image of the central quotient of $U(A)$ in the projective quaternion automorphic quotient is the part on which \[ \omega_{L/F}(\operatorname{Nrd}(g))=1. \] Locally the obstruction to lifting is the reduced norm modulo norms from $L$. The Hasse norm theorem supplies rational lifts when the local obstructions vanish, and rational quaternion norms are precisely the totally positive elements. Quadratic norm reciprocity then identifies the adelic image modulo rational translation with the displayed part. Extend a function on this part by zero to the projective quaternion quotient. This respects all split-place actions and the trivial archimedean weight. After Jacquet--Langlands transfer, the split good Hecke data determine the prescribed cuspidal Hilbert representation up to the twist $\omega_{L/F}$. To see the last assertion, use the Galois representations of the parallel-weight-two Hilbert transfers \cite[Theorem~1.1 and Remark~1.2]{Newton2015} and Chebotarev at the places splitting in $L$. Their restrictions to $G_L$ agree and are absolutely irreducible, so their extensions to $G_F$ differ by at most the quadratic character of $L/F$. Strong multiplicity one gives the assertion for the automorphic representations. The one-dimensional reduced-norm representations do not have these cuspidal Hecke data. The two possible quadratic twists restrict identically to the indicated part. Moreover restriction of the prescribed representation is injective: a nonzero function supported only on the complementary part would belong also to its quadratic twist, contradicting their distinctness. Finally each nonsplit hyperspecial factor contributes a single spherical line. This proves the required multiplicity one with exactly the local levels being used. The base-change and Jacquet--Langlands inputs are the standard cuspidal ones \cite{ArthurClozel1989,JacquetLanglands1970}; the avoidance of the inducing CM field ensures the required cuspidality. At a split finite place the binary local intertwiner is \begin{equation}\label{eq:binary-intertwiner} J_v(\Phi,z)=\int_{\GL_2(F_v)} \Phi({}^th)|\det h|_v\, \pi_v^\vee(h)\eta_v^{-1}(\det h)z\,d^\times h, \qquad\vol(\GL_2(\cO_{F_v}))=1. \end{equation} Here $z$ belongs to the bare $\pi_v^\vee$ space; the determinant twist has been written in the integrand. The output is read by inverse transpose, which is field conjugation in the chosen split bases. Equation~\ref{eq:binary-intertwiner} is the Godement--Jacquet integral at the unitary center, interpreted by rational continuation where necessary. Equivariance, local theta uniqueness, and a nonzero open test identify it with the theta intertwiner. We normalize by this map at the exceptional split places, by spherical-to-spherical maps elsewhere, and by the determinant highest vector at infinity. The global binary map is consequently a single scalar $S_i(m)$ times these fixed intertwiners. Include in $S_i(m)$ the conversion to CM evaluation at a unit-framed object, namely \[ \prod_{\sigma:F\hookrightarrow\R} (2\pi i/\Omega_\sigma)^{2m}, \] with the fixed algebraic unit powers prescribed in Section~\ref{sec:theta}. This scalar has bounded tame group-ring coefficients. Indeed, choose integral open tests at $q$ and fixed-level open tests at $p$ that map a fixed compact vector to itself under Equation~\ref{eq:binary-intertwiner}; the volume loss at $p$ is fixed, whereas the full source maximal compact at $q$ has volume one. The binary moment estimate bounds the reference lift. Its values are $S_i(m)$ times values of one fixed nonzero algebraic compact automorphic function, times determinant-character units. Evaluate at a fixed nonzero value, choosing finite representatives integral at $p$. Division by that value costs a fixed denominator. The cancellation of determinant twists at finite compact levels prevents the source summation level from growing with $q$. Applying these intertwiners to the projected test at any boundary component expresses its constant coefficient as $S_i(m)$ times a compact automorphic evaluation and local operators. Thus two estimates remain: $S_i(m)$ must tend to zero through tame degree $b_*-1$, and the projected local operators must have uniform bounds. The latter will use the same local factors as the period comparison. \subsection{Raw toric periods and the comparison identity} Let $f_0$ and $f_1$ be the weight-two newforms attached to $A_0$ and $A_0^{D'}$, respectively, and let $\pi_0,\pi_1$ denote their unitary automorphic representations. For $m\ge1$, form toric periods over $K$ of the $(m-1)$st normalized Maass--Shimura raises of these forms, against $\eta_K^{-1}=(\cC^m\nu)^{-1}$. Normalize CM evaluations by integral differentials. We denote the resulting periods by $P_{0,i}(m)$ and $P_{1,i}(m)$, suppressing the tautological tame character from the notation. The following local choices fix their normalization. \begin{enumerate} \item At $q$, translate the spherical Whittaker vector by $n(1/q)$ in torus coordinates whose first direction is the distinguished multiplicative direction. Multiply the fixed-measure torus integral by $q-1$. Thus $P_{j,i}$ is a raw sum at conductor $q$, with no averaging denominator depending on $q$. \item At $p$, use the Kirillov function $W(\diag(x,1))=\1_{\Z_p^\times}(x)$. This is the canonical orientation of the $p$-depletion of the newvector. The function belongs to the Kirillov model of every generic local representation of $\GL_2$, including ramified principal series, special representations, and supercuspidal representations. \item At all other places use fixed tests with nonzero local toric pairing at weight zero. For $P_{0,i}$ these can be the oriented Heegner newvectors. At a split level prime their diagonal Mellin integrals are the nonzero local central $L$-factors. For $P_{1,i}$, a prime dividing $D'$ that is nonsplit in $K$ presents no obstruction: the local representation is a tempered principal series, and the field torus acts transitively on the projective line. In the induced model this gives the toric functional directly, since the central character is trivial. A fixed deeper level is allowed at these finitely many places. \end{enumerate} The coefficients of both $P_{j,i}$ have a common lower valuation bound. Indeed, all CM objects are ordinary with canonical multiplicative structure, the $p$-tests are integral of fixed level, and Katz differentiation gives bounded moments in integral differential frames. The expansion and coefficient-multiplier results are \citet[Section~5.2]{Katz1976} and \citet[Theorem~6.10, Sections~11.1--11.7 and~15.21--15.24]{AndreattaGoren2005}. The ordinary CM comparison is \citet[Section~1.4, Theorem~1.2]{HidaTilouine1993}; its application to the algebraized theta moments is Lemma~\ref{lem:unary-moments}. Fixed CM-conductor and ideal-class denominators can be cleared once. Changes to Igusa differential frames contribute fixed unit powers, and the raw conductor-$q$ sum introduces no further denominator. All these assertions hold before projecting the tame group algebra to characters. The relation between these periods and the binary scalar is the following cleared identity. Its augmentation condition is exactly what will allow division after a fixed Artin truncation. \begin{lemma}[Cleared period comparison] \label{lem:cleared-period-comparison} There are group-ring polynomial factors $A_i(m),B_i(m)$ with uniformly bounded coefficients such that \begin{equation}\label{eq:cleared-period-comparison} A_i(m)S_i(m)^2 =B_i(m)P_{0,i}(m)^2P_{1,i}(m)^2. \end{equation} For each $i$, after taking $m$ sufficiently close to zero on its branch, $A_i(m)$ has nonzero augmentation whose valuation is bounded above by a constant independent of $i$. The required neighborhood of zero may depend on $i$. The identity holds in the full characteristic-zero tame group algebra, including at zeros of the global central $L$-values. \end{lemma} We prove this comparison below by matching the two bilinear period formulas and their local factors. First we show why $P_{0,i}$ is small. Its square will then be small through twice the required tame order; the bounded coefficients of $S_i$ will recover the required order for $S_i$ itself. \subsection{The Heegner logarithm limit} We claim that \begin{equation}\label{eq:toric-period-small} P_{0,i}(m)\longrightarrow0\pmod {t^{b_*}}, \end{equation} where for each fixed $i$ one first takes $m$ sufficiently close to zero on its branch, and then lets $i$ increase. The passage from raised CM values to a Heegner logarithm follows the $p$-adic period method of \citet[Introduction and Proposition~3.24]{BertoliniDarmonPrasanna2013}. Their construction is made under $p\nmid N$. We give the primitive and logarithm comparison for the reduction types occurring here. Write $u=q_{\mathrm{Tate}}$ for a modular cusp parameter, to distinguish it from the auxiliary prime $q_i$, and put \[ f_0^{[p]}(u)=\sum_{p\nmid n}a_nu^n,\qquad F_0(u)=\theta^{-1}f_0^{[p]}(u) =\sum_{p\nmid n}\frac{a_n}{n}u^n. \] On the ordinary Igusa tower, the normalized derivatives $\theta^{m-1}f_0^{[p]}$ converge as $m\to0$ on the chosen branch, because $n^{m-1}\to n^{-1}$ uniformly on the unit indices. Their limiting Igusa weight is zero. The ordinary expansion principle therefore constructs a global analytic function $F_0$ on the completed ordinary locus of the good tame modular curve; weight-zero invariance descends it from the Igusa tower. Its differential and its cusp constants are \begin{equation}\label{eq:inverse-katz-primitive} dF_0=f_0^{[p]}(u)\,d\log u, \qquad F_0|_{\text{each compatible cusp}}=0. \end{equation} Each component in use has such a compatible cusp. The canonical $\Gamma_0(p^{\ord_pN})$ section and the canonical-subgroup Frobenius maps allow the $p$-depleted form to be read on this tame ordinary locus, even when $p\mid N$. To make the logarithm comparison explicit, choose a modular parametrization $\varphi$ with $\varphi^*\omega_0=\kappa f_0(u)d\log u$, where $\omega_0$ is a fixed invariant differential of $A_0$ and $\kappa\ne0$ is fixed. If the depletion polynomial is $E_p(X)=\sum_r e_rX^r$, and $\mathrm{Fr}$ is canonical-subgroup Frobenius, then \begin{equation}\label{eq:depletion-differential} dF_0=\kappa^{-1}\sum_r e_rp^{-r} (\varphi\circ\mathrm{Fr}^{r})^*\omega_0. \end{equation} The factors $p^{-r}$ arise because substitution $u\mapsto u^{p^r}$ multiplies $d\log u$ by $p^r$. There are only finitely many such maps, all determined by the fixed newform. They extend over a wide-open thickening of the ordinary affinoid. The primitive itself extends to a slightly smaller such thickening. For completeness, clear one denominator and write its expansion near a missing special-fiber point, with local parameter $z$, in the completed localization of $\cO[[z]][1/z]$: \[ F_0=\sum_{n\in\Z}a_nz^n. \] Overconvergence of $dF_0$ implies exponential decay of the negative coefficients $na_n$. Division by $n$ loses only the subexponential factor $|1/n|_p\le |n|$, so $a_n$ still decays exponentially on any slightly smaller collar. Exactness rules out a logarithmic term. The positive part is bounded and converges for $|z|<1$. These expansions retain the coefficients of the original $F_0$, hence introduce no independent constants. One can glue them on a model with collar $zw=\varpi'$ and $|\varpi'|<1$ sufficiently close to one: modulo each uniformizer power the usual localization--completion sequence \[ A\longrightarrow A[1/z]\oplus\widehat A \longrightarrow\widehat A[1/z] \] is injective at the first term and exact at the middle term. This follows from flatness of completion and invariance of local cohomology with support. Thus the continued primitive is an analytic function on a connected wide open in every component under consideration. We use Berkovich--Coleman integration on this neighborhood, with the functoriality and fundamental theorem of \cite[Definition~3.5]{KatzRabinoffZureickBrown2016}. After a finite extension, $A_0$ has either good or split multiplicative reduction. In the good reduction case integration of its invariant differential equals its abelian logarithm \cite[Corollary~3.18]{KatzRabinoffZureickBrown2016}. In the multiplicative case write $A_0^{\mathrm{an}}=\Gm^{\mathrm{an}}/Q_0^{\Z}$ and normalize the pullback differential to $dz/z$. Starting with any branch $\operatorname{Log}$, choose \[ \operatorname{Log}_{Q_0}(z) =\operatorname{Log}(z) -\frac{v_p(z)}{v_p(Q_0)}\operatorname{Log}(Q_0). \] This branch kills $Q_0$, and integration along every lifted path is the abelian logarithm on the quotient; see \cite[Example~3.11]{KatzRabinoffZureickBrown2016}. All maps in Equation~\ref{eq:depletion-differential} have the same elliptic target, so this single branch suffices. For a path from a compatible cusp $c$ to $x$, functoriality and Equations~\ref{eq:inverse-katz-primitive}--\ref{eq:depletion-differential} now give \begin{equation}\label{eq:katz-abelian-log} F_0(x)=\kappa^{-1}\sum_r e_rp^{-r} \log_{A_0,p}\bigl(\varphi(\mathrm{Fr}^{r}x)\bigr). \end{equation} Indeed the cusp constant is zero, and every image of a cusp is torsion by the Manin--Drinfeld theorem \citep{Manin1972,Drinfeld1973}. This is an equality obtained by path integration on the connected wide open; equality of derivatives alone would not suffice to remove constants on different residue discs. At the CM points entering $P_{0,i}$, the canonical Frobenius quotients are the horizontal translations at $p$ of the conductor-$q_i$ Heegner points, including when a nontrivial canonical $p$-level is present. Taking their raw tame weighted sums in Equation~\ref{eq:katz-abelian-log} therefore gives a fixed finite linear combination of the logarithm sums in Proposition~\ref{prop:heegner-log}. Both signs of the tame character are covered by Equation~\ref{eq:heegner-log}. The CM differential and character unit powers tend to one as $m\to0$. This proves Equation~\ref{eq:toric-period-small}, with bounded higher coefficients in every fixed truncation. No ordinary hypothesis on $A_0$ at $p$ has entered this argument. \subsection{Period identities and the two local functional equations} \begin{proof}[Proof of Lemma~\ref{lem:cleared-period-comparison}] The equal-rank Rallis inner product formula and the Waldspurger toric period formula give \begin{equation}\label{eq:binary-period-comparison} S_i(m)^2=C_i(m)\,P_{0,i}(m)^2P_{1,i}(m)^2. \end{equation} We use these formulas with bilinear dual data, so the squares are algebraic squares, rather than absolute squares. For the theta identity, the precise Siegel--Weil input is the anisotropic boundary case of \cite[Theorems~2.1(i) and~2.2(i)]{Yamana2011}, also stated in \cite[Theorem~7.1]{GanQiuTakeda2014}. In the notation of those results, $m=n=2$, $r=0$, and $s_0=(m-n)/2=0$. On the doubled group $U(2,2)$ the Siegel Eisenstein series is holomorphic at $s_0$ and $E(0,\Phi)=2I(\Phi)$, where $I$ is the ordinary convergent theta integral over the compact binary group with quotient volume one. The conversion to our global measure convention contributes only the fixed scalar permitted in \cref{sec:conventions}. Restrict this identity to the two copies of $U(A)$ and integrate against the source cusp forms. The theta seesaw gives their bilinear theta pairing on one side; unfolding the Eisenstein series gives the doubling zeta integral on the other. Its Euler factorization is the central standard $L$-value times the normalized local integrals used below. This proves the equal-rank Rallis identity needed here directly from that boundary Siegel--Weil identity. The doubling and local factorization conventions are those of \cite[Sections~11.3, 11.5--11.6]{GanQiuTakeda2014}. The source groups are compact, the standard base-change parameter is cuspidal and tempered, and its zero-space theta lift vanishes, since that lift could only be one-dimensional. The Waldspurger identities \cite[Section~II.7 and Section~III.3, Proposition~7]{Waldspurger1985} apply to cuspidal transfers with compatible central character and opposite torus characters in the two slots. Thus the identities involve central values, not residues or derivatives. For the theta pairing, pull back the same data by $(h,g)\mapsto(h^c,g^c)$. Field conjugation is anti-symplectic on the tensor product and identifies the oscillator with its dual by Poisson summation, without a global conductor factor. Pair fixed compact vectors nontrivially; if necessary two different vectors can be used, since the scalar is the same. On the toric side, translate the vector by a rational torus normalizer inducing conjugation. This turns the second period into the first, up to a fixed character value from a base-point translation. At $p$ and $q$ the normalizer exchanges the distinguished coordinates with unit scales. The global central $L$-values in the formulas match by quadratic base change over $K$: \begin{equation}\label{eq:rankin-factorization} L\bigl(\tfrac12,\pi_L\otimes\eta_L\bigr) =L\bigl(\tfrac12,\pi_{0,K}\otimes\eta_K\bigr) L\bigl(\tfrac12,\pi_{1,K}\otimes\eta_K\bigr). \end{equation} Here $\eta_L$ is induced by norm from $\eta_K$, and simultaneous inversion of the character gives the same comparison. One may retain raw local integrals at every removed place and use partial $L$-values. Fixed adjoint, norm, and global zeta factors then contribute a single fixed scalar. We compare the remaining local factors explicitly. \paragraph{The matrix functional equation.} Let $E$ be a split local field, write $\chi=\eta_v^{-1}$ in the distinguished multiplicative coordinate, and let $\psi_{\mathrm{add}}$ be the trace additive character. Write $\pi$ for the bare elliptic local representation, so $\pi\simeq\pi^\vee$ and its central character is trivial. For a matrix coefficient $c$ and $F(X)=\Phi({}^tX)$, put \[ Z_G(F,c,\chi,s)=\int_{\GL_2(E)} F(g)c(g)\chi(\det g)|\det g|^{s+1/2}\,d^\times g. \] At $s=1/2$ this is a scalar coefficient of Equation~\ref{eq:binary-intertwiner}. The Fourier transform uses the pairing $\psi_{\mathrm{add}}(\tr(XY))$. Field conjugation exchanges the two dual Lagrangians and acts on the source by inverse transpose, so the second integral uses the full matrix Fourier transform and $c^\vee(g)=c(g^{-1})$. With these identifications the functional equation is \begin{equation}\label{eq:matrix-functional-equation} Z_G(\widehat F,c^\vee,\chi^{-1},1-s) =\gamma(s,\pi\otimes\chi,\psi_{\mathrm{add}}) Z_G(F,c,\chi,s). \end{equation} Apply the matrix zeta identity \cite[Theorem~13.1(iii),(v)]{JacquetLanglands1970} to $\sigma=\pi\otimes(\chi\circ\det)$: its determinant exponent is $s+1/2$ and its trace Fourier transform uses the self-dual additive measure. The $L$- and epsilon factors are the same as in the Whittaker functional equation of Section~2.18 of that reference. In particular, normalizing the first map to be one on the chosen vector gives the factor \begin{equation}\label{eq:local-gamma-factor} \Gamma_v(\pi\otimes\chi) =\varepsilon\bigl(\tfrac12,\pi\otimes\chi,\psi_{\mathrm{add}}\bigr) \frac{L(\tfrac12,\pi^\vee\otimes\chi^{-1})} {L(\tfrac12,\pi\otimes\chi)}. \end{equation} There is only one application of the functional equation in this bilinear pairing. To relate this calculation to the Rallis local integral, integrate first over invertible matrices in $M_2(E)$. Additive measure becomes a fixed multiple of $|\det h|^2d^\times h$, so the integral factors as the pairing of the two maps in Equation~\ref{eq:binary-intertwiner}. For self-dual additive measure the spherical factor is $(1-Q^{-1})(1-Q^{-2})$, where $Q$ is the residue cardinality. It cancels exactly when the local integral is normalized by its spherical value: the spherical intertwiner itself gives $L(1/2,\pi^\vee\otimes\chi)$ with no such factor. In particular this normalization never leaves a denominator $1-Q^{-1}$ at the moving prime. \paragraph{The toric functional equation.} Put $a(x)=\diag(x,1)$, let $w=\left(\begin{smallmatrix}0&1\\-1&0\end{smallmatrix}\right)$, and write \[ M(W,\chi,s)=\int_{E^\times}W(a(x))\chi(x)|x|^{s-1/2}\,d^\times x, \qquad W^w(g)=W(gw). \] The corresponding Whittaker functional equation is \begin{equation}\label{eq:toric-functional-equation} M(W^w,\chi^{-1},1-s) =\gamma(s,\pi\otimes\chi,\psi_{\mathrm{add}}) M(W,\chi,s). \end{equation} For the twist $\sigma$, use $W_\sigma(g)=\chi(\det g)W(g)$ and its central character $\chi^2$ in \citet[Theorem~2.18(ii)--(iv)]{JacquetLanglands1970}. Since $\det w=1$, the dual integral has character $\chi(x)\chi(x)^{-2}=\chi(x)^{-1}$. The identity $w a(x^{-1})=x^{-1}a(x)w$ then gives the displayed Weyl translate, with no extra $\chi(-1)$ factor. Both first integrals, in Equations~\ref{eq:matrix-functional-equation} and~\ref{eq:toric-functional-equation}, use the same $\chi$ in the same distinguished coordinate. The invariant Kirillov pairing factors the toric local matrix integral into these Mellin integrals; its proportionality constant also cancels under spherical normalization. Consequently its factor is \[ \Gamma_v(\pi\otimes\chi)\,M(W,\chi,1/2)^2, \] with exactly the same gamma orientation as the matrix factor. At $p$, the character $\chi$ is unramified and the chosen Kirillov test gives $M(W,\chi,1/2)=1$. Thus all gamma, Euler, and epsilon factors cancel between the two period formulas. No bound for $\Gamma_v(\pi\otimes\chi)$ itself is asserted or needed; indeed $v_p(\chi(p))=m$ varies with the weight. The functional equations are identities of rational zeta functions and apply by continuation at these parameters. Their validity includes special representations \cite[Proposition~3.6 and Corollary~3.7]{JacquetLanglands1970}; a fixed compact open test normalizes the matrix map even when $\pi_p$ is ramified. Additive-character signs and changes of the chosen unit frames introduce only character values on units and fixed unit powers. Since $p$ splits completely in $L$, there are two matching local comparisons; the quadratic twist by $D'$ does not change either local elliptic representation. \paragraph{The raw factor at $q$.} Let $L_{\mathrm{fake}}$ be the degree-two unramified central Euler factor formed using $\chi(q)$ while ignoring its tame inertia. The spherical Whittaker expansion after translation by $n(1/q)$ gives \begin{equation}\label{eq:raw-q-factor} F_q(\chi):=(q-1)M(W,\chi,1/2) =\sum_{a\in\F_q^\times}\chi(a) \bigl(\psi_{\mathrm{add}}(a/q)+L_{\mathrm{fake}}-1\bigr). \end{equation} To obtain the formula, separate valuation zero from positive valuations. On units the translated Whittaker function has the additive factor $\psi_{\mathrm{add}}(a/q)$; on positive valuations its sum is $L_{\mathrm{fake}}-1$. Multiplication by $q-1$ converts the unit integrals to the displayed raw sum. For nontrivial tame $\chi$ the second sum vanishes, leaving the Gauss sum. At augmentation one instead has \[ F_q(1)=-1+(q-1)(L_{\mathrm{fake}}-1). \] In the group ring put $D_q=L_{\mathrm{fake}}^{-1}$. Clearing this Euler denominator gives the polynomial factor \begin{equation}\label{eq:cleared-q-factor} H_q=D_qF_q =D_q\sum_{a\in\F_q^\times}\chi(a)\psi_{\mathrm{add}}(a/q) +(1-D_q)\sum_{a\in\F_q^\times}\chi(a), \qquad H_q(1)=(q-1)-qD_q(1). \end{equation} Both $D_q$ and $H_q$ have bounded group-ring coefficients. The no-eigenvalue-one condition on $V(\gamma)$ implies that the limit of $D_{q_i}(1)$ at weight zero is nonzero; the conventions of Section~\ref{sec:conventions} identify this with the corresponding Frobenius Euler determinant, up to factors tending to one. Thus its valuation is bounded above. Since $q_i\to1$, the same is true of $H_{q_i}(1)$, once $m$ is sufficiently close to zero for each $i$. The two raw toric periods contribute $F_q^2$ in their respective comparisons. Therefore the ratios in Equation~\ref{eq:binary-period-comparison} divide by these full factors, and never by an isolated $(q-1)^2$, by a conductor projector, or by individual Gauss sums. The identical gamma factors cancel just as at $p$. \paragraph{The archimedean factors.} Write $k_0=m-1$. In the $2\times2$ Fock model the Gaussian norm ratio of $\det(Z)^{k_0}$ to the vacuum is \begin{equation}\label{eq:factorial-cancellation} k_0!(k_0+1)!. \end{equation} For example, expand the determinant by the binomial theorem. The different monomials are orthogonal, and each of the $k_0+1$ summands contributes $(k_0!)^2$ after including its squared binomial coefficient. Their sum is Equation~\ref{eq:factorial-cancellation}. The definite quaternion real polarization has positive quadratic norm equal to this determinant. Its scalar symplectic $\SL_2$ generates from the vacuum the holomorphic discrete series of lowest weight two, and raising is multiplication by the determinant in Fock coordinates. The norm ratio of its $k_0$th raised vector is the same $k_0!(k_0+1)!$. This is also the ratio in the archimedean toric pairing with the Weyl-dual raised weight-two vector. The factorials therefore cancel. The remaining comparison with normalized Maass--Shimura raising is a fixed algebraic $p$-unit to the $k_0$th power: this follows by comparing the leading linear Schr\"odinger coordinates, as in Lemma~\ref{lem:unary-moments}, with quaternion frames chosen as unit bases at $p$ and CM heights chosen as algebraic $p$-units. The anti-symplectic duality contributes only phases of the same kind. There is one comparison at each real embedding of $F$, and the compatible CM normalization on both sides has weight $2m$ per embedding, with the factors $(2\pi i/\Omega_\sigma)^{2m}$ already included in $S_i(m)$. No factorial denominator remains. \paragraph{Fixed factors and zeros.} At fixed exceptional finite places away from $p$, the local factors are rational functions of unramified smooth-character parameters, with fixed inertia. On our branch those parameters tend to their weight-zero values. At the tempered center the zeta denominators have no pole, and the toric functionals were chosen nonzero there. After shrinking the branch, the denominators have nonzero specialization of bounded valuation, so these ratios cost a fixed denominator. The same statement applies to the fixed rational local intertwiners. The remaining global measure and norm constant is independent of $i,m,\nu$. It is algebraic by evaluating the comparison at any parameter for which both sides are nonzero, since all normalized periods there are algebraic. If there is no such parameter, the vanishing theta comparison already gives the required zero scalar. The factors just computed, notably $D_q$ and $H_q$, supply the polynomial clearing factors in Equation~\ref{eq:cleared-period-comparison}, with bounded coefficients and with nonzero augmentation of $A_i(m)$ of bounded valuation. Fixed algebraic constants and unit powers are absorbed into $A_i$ and $B_i$. The period formulas prove this identity first at every finite-order character. Clearing denominators gives an identity in the full characteristic-zero tame group algebra, because those characters separate that algebra. There is no division by a global central $L$-value: use Equation~\ref{eq:rankin-factorization} in the two identities and cross-multiply the local factors. Hence the cleared identity also holds at their global zeros. We do not assert that the inverse of $A_i$ is a bounded measure on the entire finite group. \end{proof} \subsection{Projected tests at every cusp} The period comparison now controls the scalar shared by all binary theta values. To apply it to every constant coefficient, we must bound the projected local operators in the factorization at the start of this section, uniformly in the boundary representative. The moving prime will use the factor $D_q$ just computed. At $p$, canonical projection leaves the binary test supported on integral matrices. Expanding Equation~\ref{eq:binary-intertwiner} by the determinant valuation gives the factor \[ p^{(m-1)v_p(\det h)}. \] The right-coset volumes have fixed level, and translates of the fixed compact automorphic function have bounded sup norm. For large $m$ the integral-matrix expansion therefore converges $p$-adically with a uniform bound. Its determinant truncations are the usual Godement--Jacquet series, so by continuation it computes the same rational zeta operator. This argument applies at both places over $p$, without imposing any condition on the local elliptic representation. At $q$, the projected test is supported integrally, depends only on reduction modulo $q$, and retains determinant equivariance under the source $\GL_2(\cO_{F_v})$. On the full-rank stratum this gives a bound directly with maximal-compact volume one. On a deficient-rank stratum the determinant of the stabilizer surjects onto $\Delta_i$. Therefore its group-ring coefficient is invariant under multiplication by every element of $\Delta_i$ and is a bounded multiple of the norm element \[ N_{\Delta_i}=\sum_{\delta\in\Delta_i}[\delta]. \] This is an integral invariance statement: all its group coefficients are equal. It involves no division by $|\Delta_i|$. The remaining integral, now untwisted on units, is a sum over integral image lattices $h\cO_{F_v}^{2}$ with weight $|\det h|\chi(\det h)$, and a coefficient constant on each prescribed reduced image subspace. For a line or the zero subspace, express the condition of having that exact reduced image by subtraction of the conditions that the image be contained in it. Each containment sum is a translate and a determinant power times the spherical full-lattice series $L_{\mathrm{fake}}$. All translation determinants and volume factors are powers of $q$, hence $p$-units. Thus the already controlled factor $D_q$ clears the operator denominator. This rank-stratum argument is valid coefficientwise in the full group ring. At all other places the operators are spherical or among fixed rational operators with the no-pole property proved above. Boundary Iwasawa representatives can be chosen integral at $p$, compatible with the canonical multiplicative flag. Evaluation of the compact model at the conjugate representatives then introduces only unit factors. Source sums have a fixed volume denominator, and all nonsplit spherical lines and fixed-level coefficients have common bounds. Although the number of boundary representatives may grow at $q_i$, their projected tests have precisely the uniformly bounded forms just described. This proves uniformity over every cusp, rather than only over a chosen finite list of evaluations. \subsection{Truncation, square roots, and choice of weights} We finish by spelling out the algebra behind the coefficientwise estimate. Fix $h\ge2b_*$. In a coefficient extension containing the finite test values, substitute \[ [g]\longmapsto(1+t)^{\ell_i(g)}\pmod {t^h} \] in the tame group algebra. At finite precision this is a ring map modulo $p^{n_i-c_h}$ for a constant $c_h$ depending only on $h$: the relation $[g]^{p^{n_i}}=1$ maps to $(1+t)^{p^{n_i}}-1$, whose coefficients below degree $h$ have valuations at least $n_i-c_h$. Consequently products and identities may be truncated with an error tending to zero. Bounded group-ring coefficients give bounded coefficients in this fixed Artin truncation. All estimates below are unaffected by these errors. If $D(t)=d_0+\cdots+d_{h-1}t^{h-1}$ has bounded coefficients and $v_p(d_0)$ is bounded above, its inverse in the truncated rational ring is bounded by a constant depending only on $h$ and those bounds: \begin{equation}\label{eq:bounded-artin-inverse} D(t)^{-1}=d_0^{-1} \sum_{r=0}^{h-1}\bigl(-d_0^{-1}(D(t)-d_0)\bigr)^r \pmod {t^h}. \end{equation} Thus the clearing factors must be cleared in the group algebra \emph{before} truncation, after which their inverses cost a fixed denominator. No assertion about bounded character idempotents or bounded inverses on the full group ring is required. By Equation~\ref{eq:toric-period-small}, the coefficients of $P_{0,i}$ below degree $b_*$ tend to zero, and all coefficients below degree $h$ of both toric periods are bounded. Every coefficient of $P_{0,i}^2$ below degree $2b_*$ contains a factor of degree below $b_*$. Hence \[ P_{0,i}^2P_{1,i}^2\longrightarrow0\pmod {t^{2b_*}}. \] Apply Equation~\ref{eq:bounded-artin-inverse} to Equation~\ref{eq:cleared-period-comparison} to obtain $S_i(m)^2\to0\pmod {t^{2b_*}}$. The passage to the square root uses only valuations, not compactness of the fields of coefficients. Write $S_i=\sum_{j0$, the coefficient is a section on the abelian extension-data chart $C$ of \[ \mathcal L_\alpha\otimes\mathcal E_\lambda|_C, \] where the Jacobi line bundle $\mathcal L_\alpha$ is ample. This description persists on the finite etale abelian covers in \cref{lem:canonical-ordinary-level}. The semiabelian differential sequence has constant locally free graded pieces over the fixed boundary base. Apply \cref{lem:dual-fock-functor}, and tensor with the determinant and opposite differential line factors. This gives $\mathcal E_\lambda|_C$ a finite filtration with constant graded pieces, in precisely the integral lattice used by the theta section. This construction needs no rational splitting of the semiabelian differential sequence. We recall why the required vanishing is valid in every residue characteristic. If $L$ is ample on an abelian variety, choose multiplication by an integer $n$ prime to the characteristic. Its trace splits $\mathcal O$ inside $[n]_*\mathcal O$, so $H^r(L)$ is a summand of $H^r([n]^*L)$. For sufficiently large such $n$, the latter vanishes for $r>0$: $[n]^*L$ has arbitrarily large ample degree, and Serre vanishing is uniform under the algebraically trivial twists occurring in the theorem of the square, by properness of the dual abelian variety. The filtration just described gives the same vanishing after tensoring with $\mathcal E_\lambda|_C$. Cohomology and base change, or induction through the exact sequences for successive uniformizer powers, then lifts every positive coefficient modulo $p^M$ integrally. It remains to lift the arithmetic identifications between coefficients. Modulo translations, the stabilizer acts on positive indices by unit norms. Fixing a positive index forces this norm to be one; the resulting CM norm-one unit group is finite. The remaining binary arithmetic stabilizer is finite by definiteness. Neatness removes these finite stabilizers. We may therefore choose one lift in each free index orbit and transport it around the orbit, without averaging or dividing by an orbit order. The resulting tuples converge on every formal cone. In fact, an interior rational ray gives a linear functional strictly positive on the nonnegative index cone. A bound on its value bounds both real coordinates of a positive index, and hence leaves only finitely many points of the index lattice. This is exactly the formal convergence condition on the completed monoid algebra. The formal-chart description and formal functions now identify these tuples with the completed pushforward sections; see \citet[Section~7.1.2]{Lan2013} and \citet[Chapter~III, Theorem~11.1]{Hartshorne1977}. Thus the required reduction map is surjective at a cusp. Its injectivity follows from applying the left-exact pushforward to multiplication by a uniformizer on the torsion-free source sheaf. Off the boundary the assertion is immediate from the finite level normalization. Faithfulness of completion for coherent modules gives the assertion on the ordinary open. The branch decomposition allows the zero lift on every other branch. Finally, $\cO[t]/(t^b)$ is a finite free $\cO$-module, so the entire construction applies coefficientwise and gives the stated Artin-ring assertion. This proves the required special case directly; the general ordinary subcanonical vanishing theorem is \citet[Theorem~8.2.1.3, with the setup of Section~8.1]{Lan2018Ordinary}. \end{proof} \subsection{Hasse extension and the shifted weight} The base-change lemma turns vanishing constant terms into a cusp section modulo the working precision on the ordinary open. We now extend that section over the compactification and lift it to characteristic zero, retaining its coefficients and Hecke relations modulo the same precision. Choose positive integers $m_i$ tending to infinity and tending $p$-adically to zero on the fixed torsion branch, as in the preceding sections. Choose integers $M_i\to\infty$ sufficiently slowly along the filter that, after the fixed clearing of denominators, all constant coefficients vanish modulo $p^{M_i}$ through degree $b-1$. The theta section is then a cusp section on the ordinary open over $\cO[t]/(p^{M_i},t^b)$ by \cref{lem:cusp-base-change}. We also require $M_i\leq n_i-c_b$, with $c_b$ a fixed constant large enough for the truncated tame substitution of \cref{sec:tame}. Indeed, the coefficients of $(1+t)^{p^{n_i}}-1$ below degree $b$ have valuations at least $n_i-c_b$. Thus the finite group relation is respected at this precision. All subsequent congruences use this one common precision for the full ring. The ordinary locus is the nonvanishing locus of the total Hasse invariant. A section of a coherent sheaf on this locus extends over $X^{\min}$ after multiplication by a sufficiently high power of Hasse. The assertion follows from localization on a finite affine cover. It applies to the sheaf $\mathcal M_\lambda$ and to each of the finitely many $t$-coefficients simultaneously. For precision $M$, a sufficiently large $p$-power of Hasse lifts modulo $p^M$. To see this directly, choose local lifts of Hasse. On overlaps their differences are divisible by $p$; raising to $p^r$ makes those differences divisible by $p^{r+1}$, with the bundle transition functions included. They therefore glue for $r$ sufficiently large. In multiplicative differential frames, the resulting lift reads $1$ modulo $p^M$. Its exponent can be increased arbitrarily while retaining this property. Twisting farther by the ample Hodge line gives, by Serre vanishing \citep[Chapter~III, Theorem~5.2 and Proposition~5.3]{Hartshorne1977}, surjectivity of reduction on global sections. We may consequently lift the extended section to characteristic zero, coefficientwise in $\cO[t]/(t^b)$. Choose the Hasse weight shift $H_i$ arbitrarily large and divisible by $(p-1)p^{M_i+c}$, with $c$ any needed fixed constant. The resulting weight, at each real place, is \begin{equation} \lambda'_i =(m_i+H_i,\ m_i+H_i,\ 1+H_i;\ -1-H_i). \label{eq:shifted-weight} \end{equation} These generic sections are genuine cusp forms: the sheaf before pushforward was twisted by minus the pulled-back boundary. The size of $H_i$, the point at which Serre vanishing applies, and the dimensions of the resulting spaces may depend on $i$, $m_i$ and $M_i$. None of these choices consumes congruence precision. If a finer neat cover was used, descent by its fixed averaging operator costs only the valuation of its fixed index. We absorb this constant by reducing $M_i$ once. There is no averaging at the varying prime $q_i$. \subsection{The two integral normalized operators} Write $U_j$ for the actual double-coset operator, with the measures and expanding-lattice convention used in the theta construction. For either the original or shifted weight put \begin{equation} \widetilde U_j=p^{-N_j(\lambda')}U_j, \qquad N_2(\lambda')=2+\lambda'_3+\lambda'_4, \qquad N_3(\lambda')=3+\lambda'_4. \label{eq:ordinary-normalization} \end{equation} \begin{lemma}[Integral trace calculation] \label{lem:ordinary-trace} The operators in \cref{eq:ordinary-normalization} preserve the dual-Fock differential lattice on the canonical ordinary branch and its reductions modulo every power of $p$. After the Hasse shift, both comparison eigenvalues are $1$ modulo the prescribed precision. \end{lemma} \begin{proof} It suffices to work at one distinguished $p$-adic factor. The split PEL idempotents select the rank-three multiplicative character lattice and the rank-one etale lattice; polarization determines the conjugate factor. Choose Serre--Tate coordinates $q_1,q_2,q_3$ for their bilinear pairing \citep[Theorem~2.1(1),(2)]{Katz1981SerreTate}. Locally their deformation ring is \[ R=\cO[[q_1-1,q_2-1,q_3-1]]. \] For $U_j$, expand the lattice transverse to a multiplicative $j$-summand by $p^{-1}$. The expanded multiplicative directions are selected by $p^{j(3-j)}$ integral graphs. Fix one graph. Functoriality of the Serre--Tate pairing \citep[Theorem~2.1(4)]{Katz1981SerreTate} for the lattice inclusion and $p$ times the reverse inclusion shows that the remaining extension choices replace the $j$ retained coordinates by their $p$th roots; the other $3-j$ coordinates are unchanged. Indeed, expanding the etale generator requires a root when paired with a retained multiplicative character, whereas the simultaneous expansion of a transverse multiplicative direction cancels this change for the other coordinates. The corresponding finite free extension is the appropriate completion of \[ R'=R[u_1,\ldots,u_j]/(u_a^p-q_a:1\leq a\leq j). \] Its basis consists of $\prod_a u_a^{e_a}$ with $0\leq e_a1$, the same rank-one constituent occurs in all its consecutive shifts. Its weight $3-h_4$ at $\bar w$ would then have an adjacent integer weight in the full representation. This is impossible: it is the lowest weight there, separated from the next by $h_4-h_3$, which was chosen arbitrarily large. Hence $a=1$. At every unramified place, the roots of a unitary generic cuspidal local representation have absolute values strictly between $Q^{-1/2}$ and $Q^{1/2}$ \citep[Theorem~7.5]{Tadic1986} and \citep[Theorem~9.7]{Zelevinsky1980}. After the cohomological shift in \cref{eq:four-dimensional-compatibility}, the value of $\xi$ has absolute size strictly between $Q$ and $Q^2$. On the other hand, for an algebraic Hecke character with parallel conjugate weight sum $w(\xi)$ its absolute value is $Q^{w(\xi)/2}$: dividing out this norm power leaves a unitary character. Consequently \begin{equation} w(\xi)=3. \label{eq:rank-one-weight} \end{equation} There is a uniform open subgroup of the local units above $p$ on which the smooth factors of all these characters are trivial. Indeed the compact open level $J$ at $p$ is fixed. Bernstein's decomposition theorem gives finitely many inertial cuspidal-support classes among representations with $J$-invariants \citep[Section~2.3]{BernsteinDeligneKazhdan1986}; local Langlands therefore gives finitely many semisimple inertia types. The smooth factor of a rank-one constituent belongs to this finite list. This bounds its conductor independently of the growing algebraic weights. Choose in this fixed open subgroup an element $g$ with $\eps(g)\ne1$, deep enough that $\eps(g)$ is not torsion, and use its conjugate $g^c$ at the opposite place. \Cref{eq:rank-one-weight} implies at every actual constituent under consideration \begin{equation} \xi(g)\xi(g^c)=\eps(g)^{-3}. \label{eq:weight-three-unit} \end{equation} Thus it implies the same identity in the evaluated ambient field. But the scalar corner element \[ (e_y\rho_*(g)e_y)(e_y\rho_*(g^c)e_y) -\eps(g)^{-3}e_y \] is a unit in $T e_y$. Its reduction in the residual diagonal representation is $(\eps(g)^{-4}-\eps(g)^{-3})e_y\ne0$, since the residual $y$ is $\eps^{-2}$ and cyclotomic is invariant under conjugation. A unit cannot vanish in a field evaluation, contradicting \cref{eq:weight-three-unit}. The assumed vanishing of $D_{fy}$ is impossible. \end{proof} \begin{corollary} \label{cor:fitting-vanishing} One has \begin{equation} \Fitt_{0,T}(D_{fy})=0. \label{eq:fitting-vanishing} \end{equation} \end{corollary} \begin{proof} If $a\in T$ annihilates $D_{fy}$, then its image at every ambient field is zero by \cref{lem:no-stable-y-line}. Since $T$ embeds in the reduced ring $\mathcal R$, this gives $a=0$. For a finite module over an arbitrary commutative ring its zeroth Fitting ideal is contained in its annihilator: choose finitely many generators and apply the adjugate identity to each square matrix of relations. This does not require a finite presentation. It proves \cref{eq:fitting-vanishing}. \end{proof} \subsection{The Greenberg extension module and the contradiction} \begin{proposition}[Extension extraction] \label{prop:extension-extraction} There is an injective $R_b$-linear map \begin{equation} \Hom_{R_b}(D_{fy}\otimes_{T,\lambda}R_b,R_b) \hookrightarrow \GrH(L,V\otimes\psi^{\pm1}). \label{eq:green-injection} \end{equation} Its source has $R_b$-length at least $b$. \end{proposition} \begin{proof} Put $M=D_{fy}\otimes_{T,\lambda}R_b$. A functional $M\to R_b$ gives a triangular extension of $y$ by $f$ while retaining the separate $x$ block. We check the only additional product relation. The path from $y$ through $x$ to $f$ has image in \[ D_{fx}D_{xy}=D_{fx}D_{xf}D_{fy} \] by \cref{eq:xy-factorization}. Its first two factors form a crossing diagonal cycle, killed by \cref{eq:crossing-cycles} after applying $\lambda$. All other discarded products are killed by the same cycle relation or by the diagonal representation. Hence the triangular assignment is an algebra homomorphism out of $B$. As in \cref{eq:xy-cohomology}, a split extension must have its splitting preserved by $B$ and then by its idempotents; it forces the functional to be zero. This proves injectivity into extension cohomology. Moreover \[ \Hom(y,f)=V\otimes\psi^{\pm1}, \] and \cref{lem:finite-value-admissibility} proves admissibility with coefficients in the full ring $R_b$. At a conjugate $p$-place, use the same common-kernel stack $S_g=\rho_*(g)-Y_{\bar w}(g)$. Choose four rows of types $f,f,x,f$, with either of the two $f$ positions as the final row. By \cref{lem:corner-minors} the vanishing four-row determinants are identities in the corresponding entry of $D_{fy}$, and they survive in the triangular extension. The first two $f$ rows and one $x$ row give a $3\times3$ minor on the competing columns whose residue is invertible by $H^0(L_{\bar w},V)=0$ and $x\ne y$. It is therefore a unit over $R_b$. Solve these equations with $y$ coordinate one. Varying the final $f$ row proves that every other $f$ equation is satisfied. The $x$ coordinate is zero and the $y$ rows vanish. This produces a common $y$-eigenline and splits the local extension over the full Artin ring, including its nilpotent coefficients. The Greenberg condition is full at $\Sigma_p$, so no splitting is required there; it is relaxed at the remaining fixed bad places. Outside those places and the moving places the representations are unramified, as required for admissible classes. Thus the preceding local splitting is precisely the one-sided condition of \cref{prop:green-length}, and gives \cref{eq:green-injection}. Fitting ideals commute with base change for finite modules, so \cref{eq:fitting-vanishing} gives $\Fitt_{0,R_b}(M)=0$. The finite module $M$ over the principal Artin ring $R_b$ has an elementary-divisor decomposition \[ M\simeq\bigoplus_j R_b/(t^{a_j}), \qquad 1\leq a_j\leq b. \] Its zeroth Fitting ideal is $(t^{\sum_j a_j})$, with the evident interpretation for free summands $a_j=b$. Vanishing is equivalent to $\sum_j a_j\geq b$. Each summand and its $R_b$-dual have length $a_j$, since $\Hom_{R_b}(R_b/(t^{a_j}),R_b)=(t^{b-a_j})$. Thus the source in \cref{eq:green-injection} has length at least $b$. \end{proof} We can now finish the arithmetic argument. Outside the strict case, $b=b_*=1$, whereas \cref{prop:green-length} gives a Greenberg module of length zero. In the strict case, $b=b_*=3$, whereas the same proposition bounds its length by two. Both contradict \cref{prop:extension-extraction}. Consequently the torsion alternative for the conductor-one Heegner point in \cref{eq:torsion-assumption} cannot occur. The arithmetic reduction of \cref{prop:arithmetic-reduction}, Gross--Zagier and Kolyvagin then give the analytic order, the Mordell--Weil rank and the finiteness of the whole Tate--Shafarevich group asserted in \cref{thm:main}. ==== END The-Selmer-converse-for-elliptic-curves-at-every-prime//sections/09-extension-extraction.tex ==== ==== BEGIN The-Selmer-converse-for-elliptic-curves-at-every-prime//sections/10-nonstandard-transfer.tex ==== \section{Measure comparison and change of characteristic} \label{sec:nonstandard-transfer} We give two comparisons for nonstandard weighted orbital integrals, for fixed root data and outside a finite set of residue characteristics. First, we compare the normalized global centralizer factors in the equal-characteristic argument of \citet[Theorem~17.7]{HalleckDubeWFLDraft}. The factors agree even though the centralizer tori need only be isogenous. Second, we express the paired local orbital-integral identity as the vanishing of a constructible function, with common parameters and compatible measures. The transfer principle then gives the required mixed-characteristic identity. \subsection{The global centralizer factor under an isogeny} \label{subsec:global-torus-factor} A rational identification of character spaces need not identify the corresponding tori. In the global nonstandard comparison, the factor that is invariant is the quotient of the centralizer constant by the component-group order. The following lemma includes the degree-lattice term needed over a function field. \begin{lemma}[Comparison of global torus factors] \label{lem:global-torus-factor} Let $K=\mathbb F_q(X)$, where $X$ is a smooth projective geometrically connected curve of genus $g$, and let $\Gamma_K$ be its absolute Galois group. Let $T_1,T_2$ be $K$-tori with isomorphic rational character modules as Galois representations, and let $\ell:\operatorname{Lie}T_1\simeq\operatorname{Lie}T_2$ be a $K$-linear isomorphism. Choose local Haar measures $\eta_{i,v}$ compatible under $\ell$ on the Lie algebras, and give the smooth integral compact subgroups volume one at almost all places. Write $\eta_i$ for the resulting adelic measures, using counting measure on $T_i(K)$. For a torus $T$, put $Y=X^*(T)$, $r=\operatorname{rank}Y^{\Gamma_K}$, and \[ \deg_T(t)(\chi)=\sum_v [k(v):\mathbb F_q]\, v\bigl(\chi(t_v)\bigr),\qquad I(T)=\bigl[\operatorname{Hom}(Y^{\Gamma_K},\mathbb Z): \deg_T T(\mathbb A_K)\bigr]. \] Let $T(\mathbb A_K)^1=\ker\deg_T$, and set \[ \Sha^1(K,T)=\ker\!\left(H^1(K,T)\longrightarrow \prod_v H^1(K_v,T)\right), \qquad b(T)=\bigl|\pi_0(\widehat T^{\Gamma_K})\bigr|. \] If \[ c_{\eta}(T)= \operatorname{vol}_{\eta}\bigl(T(K)\backslash T(\mathbb A_K)^1\bigr) \,|\Sha^1(K,T)|, \] then \begin{equation} \label{eq:torus-factor-degree-index} \frac{c_{\eta_1}(T_1)/b(T_1)}{c_{\eta_2}(T_2)/b(T_2)} =\frac{I(T_1)}{I(T_2)}. \end{equation} In particular the two normalized factors agree if each torus splits at a place of degree one. The places may be different. \end{lemma} \begin{proof} We will express each normalized factor as the same scalar times $I(T_i)$. First, the exponential sequence for the complex dual torus is \[ 0\longrightarrow Y\longrightarrow Y\otimes_{\mathbb Z}\mathbb C \longrightarrow\widehat T(\mathbb C)\longrightarrow 1. \] The Galois action factors through a finite group. Taking its invariants, and averaging on the complex vector space, identifies the component group of $\widehat T^{\Gamma_K}$ with $H^1(K,Y)$. Indeed, the image of the invariant vector space is precisely the identity component of the fixed torus. Thus \begin{equation} \label{eq:torus-component-cohomology} b(T)=|H^1(K,Y)|. \end{equation} We specify the measure comparison. Choose invariant top differential forms $\omega_i$ over $K$ with $\ell^*\omega_2=\omega_1$. Compatibility of the local Haar measures means \[ \eta_{i,v}=a_v|\omega_i|_v \] with the same positive constant $a_v$ for $i=1,2$. Put $d=\dim T_i$. Let $P_v(T,t)$ be the local Artin polynomial and $\rho(T)=\lim_{s\to1}(s-1)^rL(T,s)$, in the conventions of \citet[Section~6]{GeisserSuzuki2022}. The rational Galois-module isomorphism gives the same $d,r,P_v$, and $\rho$ for both tori. In these conventions the Tamagawa measure is \begin{equation} \label{eq:torus-tamagawa-measure} \mu_i=\frac{1}{\rho\,q^{(g-1)d}} \prod_v P_v(q_v^{-1})^{-1}|\omega_i|_v, \qquad q_v=|k(v)|. \end{equation} Consequently $\eta_i=\alpha\mu_i$ with the same scalar \[ \alpha=\rho\,q^{(g-1)d}\prod_v a_vP_v(q_v^{-1}). \] At almost all places the forms are integral generators and the tori have good reduction; there the volume of the integral compact under $|\omega_i|_v$ is $P_v(q_v^{-1})$. The displayed product therefore has only finitely many factors different from $1$. Replacing a global form by a $K^\times$-multiple does not change the global measure, by the product formula. This also explains why clearing denominators in a rational torus correspondence adds no unrecorded global scalar. For clarity, the function-field Tamagawa number includes a degree correction. The definition in \citet[Section~6]{GeisserSuzuki2022} and their Proposition~6.3 give \[ \operatorname{vol}_{\mu_i} \bigl(T_i(K)\backslash T_i(\mathbb A_K)^1\bigr) =(\log q)^r I(T_i)\tau(T_i), \qquad \tau(T_i)=\frac{|H^1(K,X^*(T_i))|}{|\Sha^1(K,T_i)|}. \] Here their discriminant $\operatorname{Disc}(h_{T_i})$ is exactly $I(T_i)$: the degree pairing identifies the torsion-free adelic class group with the image lattice of $\deg_{T_i}$ in $\operatorname{Hom}(X^*(T_i)^{\Gamma_K},\mathbb Z)$. Together with \eqref{eq:torus-component-cohomology}, this proves the more explicit formula \begin{equation} \label{eq:torus-factor-explicit} \frac{c_{\eta_i}(T_i)}{b(T_i)} =\alpha(\log q)^r I(T_i), \end{equation} and hence \eqref{eq:torus-factor-degree-index}. Finally suppose $T$ splits at a degree-one place $v_0$. The invariant sublattice $Y^{\Gamma_K}$ is saturated in $Y$: if $ny$ is invariant for $n\ne0$, torsion-freeness of $Y$ makes $y$ invariant. Therefore restriction gives a surjection \[ X_*(T_{K_{v_0}})=\operatorname{Hom}(Y,\mathbb Z) \longrightarrow\operatorname{Hom}(Y^{\Gamma_K},\mathbb Z). \] For a local cocharacter $\lambda$ and uniformizer $\varpi_{v_0}$, the adele equal to $\lambda(\varpi_{v_0})$ at $v_0$ and to $1$ elsewhere has degree $\chi\mapsto\langle\chi,\lambda\rangle$. Thus the degree map is onto and $I(T)=1$. \end{proof} \paragraph{Application to the global nonstandard comparison.} In \citet[Corollary~17.11, equation~(17.2.2)]{HalleckDubeWFLDraft}, take $K=F_0$ and let $T_i$ be the generic fiber of the regular centralizer $J_{i,a_M}$. The rational root-ray correspondence is equivariant for the common cameral monodromy. It gives isomorphic rational character modules; after clearing the fixed denominators it gives a torus isogeny. At the good characteristics under consideration its differential is invertible and induces the Lie-algebra identification used for the local measures, as in \citet[Section~1.12, before Theorem~1.12.7]{Ngo2010FundamentalLemma}. There is no requirement that the integral character lattices be isomorphic. The marked point in the global construction is rational, and the rigidification makes each $T_i$ split there \citep[Section~14.1 and the proof of Lemma~15.11]{HalleckDubeWFLDraft}. Hence $I(T_1)=I(T_2)=1$. Moreover, \citet[Lemma~12.2.5, pp.~1735--1736]{ChaudouardLaumon2012II} identify the Frobenius-coinvariant kernel used in that draft with $\Sha^1(F_0,T_i)$: \[ \ker\bigl(T_i(F)_\tau\longrightarrow T_i(\mathbb A)_\tau\bigr) \simeq\Sha^1(F_0,T_i). \] Here $F=\overline{\mathbb F}_q(X)$, $\mathbb A$ is its geometric adele ring in that construction, and $\tau$ is arithmetic Frobenius. This statement concerns coinvariants for the infinite cyclic Weil group; it does not replace them silently by continuous Galois cohomology. Also $\widehat T_i^{\Gamma_{F_0}}$ is the fixed dual torus denoted $\widehat T_i^{W_a}$ in the trace formula. To see this even when $W_a$ is defined geometrically, use the section of the arithmetic fundamental group supplied by the rational marked point. Its image acts trivially on the character lattice because the torus is split there. The arithmetic and geometric monodromy images on that lattice are consequently equal. The lemma therefore gives the exact cancellation needed in equation~(17.2.2): \begin{equation} \label{eq:nonstandard-centralizer-cancellation} \frac{c(J_{1,a_M})}{|\pi_0(\widehat T_1^{W_a})|} =\frac{c(J_{2,a_M})}{|\pi_0(\widehat T_2^{W_a})|}. \end{equation} The two numerators and the two denominators need not agree separately. Equation~\eqref{eq:nonstandard-centralizer-cancellation} gives their combined cancellation in place of the generic-torus identifications and separate cancellations on page~169 of that draft. \paragraph{The remaining coroot covolumes.} It remains to identify the factors in its equation~(17.2.3) with Waldspurger's coefficient. This calculation uses the coroot lattices within each group, rather than an identification of integral torus lattices across the pair. Let $S_i$ be the pinned maximal torus of $G_i$, and let $\Gamma$ be the based-root-datum monodromy group. Denote the rational root-ray correspondence by $\psi$, and write \[ \psi_*:X_*(S_1)\otimes_{\mathbb Z}\mathbb Q \xrightarrow{\ \sim\ } X_*(S_2)\otimes_{\mathbb Z}\mathbb Q \] for its $\Gamma$-equivariant map on geometric cocharacters. For $H_i=M_i,G_i$, put $\Lambda_{H_i}=\mathbb Z\langle\Phi_{H_i}^{\vee}\rangle$ inside the geometric cocharacter lattice $X_*(S_i)$. The map $\psi_*$ carries $\Lambda_{H_1}\otimes\mathbb Q$ onto $\Lambda_{H_2}\otimes\mathbb Q$ for $H=M,G$, and therefore induces isomorphisms on the corresponding rational coinvariant quotients. The lattice and real space in \citet[Lemma~17.10]{HalleckDubeWFLDraft} are \[ \mathcal X_*(H_i)= \bigl((X_*(S_i)/\Lambda_{H_i})_\Gamma\bigr)_{\mathrm{tf}}, \qquad \mathfrak a_{H_i}=\mathcal X_*(H_i)\otimes_{\mathbb Z}\mathbb R, \] where the subscript $\mathrm{tf}$ means quotient by torsion. For $S_i$ the same definition uses the zero coroot lattice. Write $\mathfrak a_{S_i}^{H_i}$ for the kernel of $\mathfrak a_{S_i}\to\mathfrak a_{H_i}$, with its induced measure. We also write $\psi_*$ for the induced real isomorphisms $\mathfrak a_{H_1}\simeq\mathfrak a_{H_2}$ and $\mathfrak a_{S_1}^{H_1}\simeq\mathfrak a_{S_2}^{H_2}$. Set \[ V_{H_i}=\operatorname{vol}\bigl( \mathfrak a_{S_i}^{H_i}/(\Lambda_{H_i})_\Gamma \bigr). \] First suppose $\mathfrak a_{G_i}=0$. Dividing the formula of that lemma for $H_i=G_i$ by the formula for $H_i=M_i$ gives \[ \frac{V_{G_i}}{V_{M_i}} =\frac{|\pi_0(Z_{\widehat G_i}^{\Gamma})|} {|\pi_0(Z_{\widehat M_i}^{\Gamma})|} \operatorname{vol}\bigl(\mathfrak a_{M_i}/\mathcal X_*(M_i)\bigr). \] The torus covolume and the torus component factor cancel within each fixed $i$. Transporting the real measures through $\psi_*$ gives \[ \frac{V_{H_1}}{V_{H_2}}= [ (\Lambda_{H_2})_\Gamma: \psi_*((\Lambda_{H_1})_\Gamma)],\qquad H=M,G. \] Here $[A:B]=[A:A\cap B]/[B:A\cap B]$ is the relative index of commensurable lattices: it is the covolume of $B$ divided by that of $A$ in their common real space. Thus the remaining ratio is Waldspurger's coefficient \[ c_\psi=\frac{V_{G_1}V_{M_2}}{V_{G_2}V_{M_1}}= \frac{[(\Lambda_{G_2})^\Gamma: \psi_*((\Lambda_{G_1})^\Gamma)]} {[(\Lambda_{M_2})^\Gamma: \psi_*((\Lambda_{M_1})^\Gamma)]}. \] The simple coroots form permutation bases for $\Gamma$, and $\psi_*$ scales their rays by scalars constant on each $\Gamma$-orbit. It acts with the same scalars on orbit sums in the invariants and orbit classes in the coinvariants. The relative indices therefore agree, as in the proof of \citet[Corollary~17.11]{HalleckDubeWFLDraft}. For general $G_i$, that proof reduces to this case by quotienting by the maximal split central tori. Its reduction, using Theorem~15.16, preserves the weighted orbital integrals, the stable recursions, and the coroot-lattice coefficient. This proves the required global centralizer normalization. The geometric inputs remain the isogeny statement for neutral Picard components and the cohomological comparison on the anisotropic locus \citep[Proposition~4.18.1 and Theorem~8.8.2]{Ngo2010FundamentalLemma}, together with the support statement used in the draft. The split rational marked point identifies arithmetic and geometric monodromy \citep[Proposition~2.21 and Corollary~2.22]{HalleckDubeWFLDraft}; thus, after the split-center reduction, the center satisfies the geometric anisotropy hypothesis of that comparison. The calculation does not assert a generic isomorphism $J_1^0\simeq J_2^0$ or constitute a new proof of those geometric results. \subsection{Change of characteristic} We now transfer the local nonstandard identity, with the compatible normalizations just described. The uniformity is over a fixed finite collection of root data and all its integral generically regular matching parameters. \begin{lemma}[Change of characteristic for fixed nonstandard data] \label{lem:nonstandard-change-characteristic} Fix a finite collection of paired based root data with rational root-ray maps $\psi$, matching Levi data, and compatible finite cyclic outer actions. Suppose the nonstandard weighted fundamental lemma holds over local fields of sufficiently large positive characteristic for their unramified forms, including all cyclic-generator variants. Then it holds over characteristic-zero local fields of sufficiently large residue characteristic, uniformly for this collection and for all integral generically regular matching parameters. Here the ambient hyperspecial subgroups have volume one; the Arthur weight-space measures are transported through $\psi$; and the paired regular-centralizer measures are transported through the induced Lie-algebra isomorphism. \end{lemma} \begin{proof} We apply the general transfer principle for parameterized integrals \citep[Theorems~2.8.1--2.8.3]{CluckersHalesLoeser2011}. To do so, we construct a common parameter space for the paired integrals, express their measures by differential forms, and remove the volume and discriminant normalizations. The resulting identity will be the vanishing of one constructible function. \paragraph{Common parameters and the identity.} Choose an integer $N$ excluding the bad primes for the finitely many root data, Chevalley and Kostant constructions, and outer actions, and making $\psi$ and its inverse integral over $R=\mathbb Z[1/N]$. Include the ordinary endoscopic descendants in this finite list. Write $G_i$ and $M_i$ for the groups and Levi subgroups attached to the two paired data, and $\mathfrak t_i$ for their chosen Cartan Lie algebras. The Weyl-equivariant Cartan identifications of \citet[Sections~1.12.4--1.12.6]{Ngo2010FundamentalLemma} give \[ C_M=\mathfrak t_1/W_{M_1}\simeq\mathfrak t_2/W_{M_2}, \qquad C_G=\mathfrak t_1/W_{G_1}\simeq\mathfrak t_2/W_{G_2} \] over $R$, compatibly with $C_M\to C_G$ and with the common outer action. They identify the integral models and the generically $G$-regular loci. Consequently the same statements hold after a common unramified outer twist. For a local field $F$ and an integral generically regular parameter $b$ in this common base, let $Y_{i,b}$ be its Kostant lift to $\operatorname{Lie}(M_i)$. Write $S_{M_i}^{G_i}(b)$ for the stable weighted orbital integral at $Y_{i,b}$ with test function $1_{\operatorname{Lie}(G_i)(\mathcal O_F)}$. With the measures in the statement, the identity to be transferred is \begin{equation} \label{eq:nonstandard-local-identity} S_{M_1}^{G_1}(b)=c_\psi S_{M_2}^{G_2}(b), \end{equation} in the convention of \citet[Definition~16.4 and Theorem~17.7]{HalleckDubeWFLDraft}. The coroot-lattice coefficient $c_\psi$ is fixed on each cyclic-action stratum. We next parameterize the unramified splitting extensions and outer twists so that both sides of \eqref{eq:nonstandard-local-identity} are defined on the same space. Encode an unramified splitting extension of fixed degree $d$ by a monic polynomial with integral coefficients and irreducible separable reduction, together with the matrix of an algebra automorphism $\tau$ of exact order $d$. All these conditions are first-order conditions of fixed degree, and every unramified extension has such a presentation. We use the general coefficient encoding of \citet[Section~3.2]{CluckersHalesLoeser2011}; restriction to binomials is unnecessary. Use the same extension and generator on both sides. The resulting common definable space is \[ \Lambda_F= \{(\mathbf a,\tau,b): b\in C_{M,\mathbf a,\tau}(\mathcal O_F),\quad b\text{ is generically }G\text{-regular}\}. \] The subscript denotes the descended integral model: in extension coordinates, its points satisfy the common semilinear fixed-point equations. The map $\psi$ intertwines those equations. Thus every required matching pair occurs over $\Lambda_F$. Varying $\tau$ only adds the specified cyclic-generator variants. \paragraph{Compatible measures.} Over this regular locus, the centralizers $J_i$ in $M_i$ are the twists of the pinned tori along the same cameral Weyl torsor. Multiplying $\psi$ by an integer produces an equivariant torus isogeny. Descending it and dividing its differential by that integer gives an algebraic isomorphism \[ \lambda:\operatorname{Lie}(J_1)\simeq \operatorname{Lie}(J_2). \] Enlarge $\Lambda$ by a nonzero top covector $\omega_1$ on $\operatorname{Lie}(J_1)$, and set $\omega_2=(\lambda^{-1})^*\omega_1$. Their invariant extensions give compatible Haar measures, as in \citet[Section~1.12, before Theorem~1.12.7]{Ngo2010FundamentalLemma}. Every field-valued parameter has such a frame. Any other compatible pair differs by a common positive scalar, which rescales both sides of the desired identity by its inverse. For every ambient group $H$ in either finite stable recursion, also include a nonzero invariant top-form frame $\eta_H$. On a regular stable orbit $X_b$ in $\operatorname{Lie}(H)$, the exact sequence \[ 0\longrightarrow\operatorname{Lie}(J_b) \longrightarrow\operatorname{Lie}(H) \xrightarrow{Z\mapsto[Z,Y]}T_YX_b\longrightarrow0 \] gives the quotient orbital form from $\eta_H$ and $\omega_J$. On finite smooth charts its comparison with the motivic measure is the absolute value of a rational determinant, hence a constructible factor $\mathbb L^{-\operatorname{ord}(r)}$. This is the differential-form realization of quotient measures used in \citet[Sections~7 and~8.6]{CluckersHalesLoeser2011}. If $K_H$ is the hyperspecial subgroup, put \[ V_H=\int_{K_H}|\eta_H|>0. \] This is a constructible integral. The normalized ambient measure is $|\eta_H|/V_H$; below we clear the finitely many factors $V_H$. \paragraph{Discriminants and stable recursion.} We must also account for the discriminant normalization $|D_H(Y)|^{1/2}$ in the weighted orbital integrals. In either stable recursion, write $M$ for its fixed Levi. Every ordinary endoscopic descendant has this same actual $F$-Levi. Choose an $F$-parabolic $P_H=MU_H$ and set \[ R_{H,M}(Y)=\det\bigl(\operatorname{ad}(Y) \mid\operatorname{Lie}(U_H)\bigr). \] Opposite roots give \[ D_H(Y)=(-1)^{\dim U_H}D_M(Y)R_{H,M}(Y)^2. \] Thus division of the specialized stable recursion by $|D_M(Y)|^{1/2}$ replaces every discriminant factor by $|R_{H,M}(Y)|$, an integral valuation power of the residue cardinality. Under the pairing, the two Levi discriminants differ by a fixed rational scalar $u_\psi$. Enlarge $N$ by its numerator and denominator. Then $|u_\psi|=1$, so the two removed Levi-discriminant factors agree at every matched parameter. The ordinary recursion is \[ S_M^H=J_{M,M}^H- \sum_{H'\ne H}\iota_M(H,H')S_M^{H'}. \] Its index sets are finite and its rational coefficients depend only on the fixed root data and cyclic action; each $H'$ has fewer roots and the same Levi $M$ \citep[Section~9.1]{CluckersHalesLoeser2011}. The weighted integrands and hyperspecial-lattice indicators are the definable, relatively integrable families constructed in \citet[Sections~6 and~8.6]{CluckersHalesLoeser2011}. Use their integer-valuation weights with the fixed compatible weight-space volumes; real constants are permitted by Section~2.10 of that reference. The form comparisons and resultant factors above preserve constructibility and relative integrability. Indeed, the intrinsic quotient form and the Lie-algebra/Chevalley quotient form are invariant on each geometrically transitive regular stable orbit. Their ratio comes from the base, including the frames, and can be evaluated at the Kostant lift. Its absolute value is a constructible valuation power, so the projection formula preserves the relative integrability of the CHL families. Define the modified functions by this recursion using $|R_{H,M}|$ times the quotient orbital integrals. Induction shows that their specializations equal $S_M^H/|D_M|^{1/2}$. No division by a square root in the motivic ring is being made. With this common discriminant factor removed, expand the finite recursions and multiply \eqref{eq:nonstandard-local-identity} by the product of the ambient volumes $V_H$, with multiplicities if necessary. The result is the vanishing of one constructible function $\Phi$ on the enlarged common parameter space. Each specialized multiplier is positive. \paragraph{Transfer and recovery of the normalized identity.} By hypothesis $\Phi_{k((t))}$ vanishes identically for every finite residue field $k$ of sufficiently large characteristic, including all permitted generator variants. The abstract transfer principle therefore makes $\Phi_F$ identically zero for every sufficiently large-residue-characteristic local field $F$ with residue field $k$, in particular every characteristic-zero such $F$. Undo the positive ambient-volume multiplier and the common nonzero Levi-discriminant factor. The surjectivity of the extension and frame parameterizations yields the required identity for every matching parameter. Taking the maximum of the finitely many bounds proves the assertion. \end{proof} ==== END The-Selmer-converse-for-elliptic-curves-at-every-prime//sections/10-nonstandard-transfer.tex ==== ==== BEGIN The-Selmer-converse-for-elliptic-curves-at-every-prime//sections/10-sylvester-cube-sums.tex ==== \section{Sylvester's positive prime cube-sum cases} \label{sec:sylvester} Sylvester's positive prime cases ask whether every prime $\ell\equiv4,7,8\pmod9$ is a sum of two rational cubes. Dasgupta--Voight proved the $4,7$ cases when $3$ is not a cube modulo $\ell$ \citep[Theorem~1.2.1]{DasguptaVoight2018}. Yin's recent construction removes this restriction, with a companion Gross--Zagier formula giving analytic rank one \citep{Yin2026Sylvester47,Yin2026SylvesterGZ}. Burungale--Tian prove the remaining $8$ case and state the resulting full prime cube-sum theorem \citep[Theorems~1.1 and~1.2]{BurungaleTian2026Sylvester}. The ramified-prime CM converse and its Sylvester consequence are also stated in the earlier preprint of \citet[Theorems~1.1 and~1.3]{Kriz2022}. The proof below instead deduces these cases uniformly from Theorem~\ref{thm:main} at $p=3$. This is a prime of additive reduction for the cube-sum curves. The role of a rank-one converse at this prime is also explained by \citet[Section~1.2.2]{BurungaleTian2026Sylvester}. Our input is Satg\'e's classical paired isogeny descent: retaining its Tate--Shafarevich terms bounds the full Selmer corank, rather than only the Mordell--Weil rank. \begin{corollary}[Sylvester's positive prime cases] \label{cor:sylvester} Let $\ell$ be a rational prime with $\ell\equiv4,7,8\pmod 9$, and equip the projective cubic \[ E_\ell:\quad X^3+Y^3=\ell Z^3 \] with origin $(1:-1:0)$. Then \[ \ord_{s=1}L(E_\ell,s)=\rank_\Z E_\ell(\Q)=1, \qquad \#\Sha(E_\ell/\Q)<\infty. \] In particular, there are $x,y\in\Q$ such that $x^3+y^3=\ell$. \end{corollary} \begin{proof} Use the $\Q$-isomorphic Weierstrass model $E_\ell:y^2=x^3-27(4\ell)^2$ and its isogenous companion $E'_\ell:y^2=x^3+(4\ell)^2$. Satg\'e's degree-three isogeny and its dual are \[ \lambda:E_\ell\longrightarrow E'_\ell,\qquad \lambda':E'_\ell\longrightarrow E_\ell,\qquad \lambda'\circ\lambda=[3]. \] These are the models and directions of Section~I, page~336, of \citet{Satge1987CubeSums}. Let $S$ and $S'$ be the Selmer groups for $\lambda$ and $\lambda'$, respectively, with their local Kummer conditions. Write $\Sha(E_\ell/\Q)[\lambda]$ for the kernel of the map on $\Sha$ induced by $\lambda$, and similarly for $\lambda'$, and put \[ a=\dim_{\F_3}\Sha(E_\ell/\Q)[\lambda],\qquad b=\dim_{\F_3}\Sha(E'_\ell/\Q)[\lambda']. \] These dimensions are finite: Satg\'e's descent sequences (1) and (1') exhibit the two kernels as quotients of $S$ and $S'$. For $D=\ell$, Proposition~1.2 of \citet{Satge1987CubeSums} gives \[ \dim_{\F_3}S=\dim_{\F_3}S'=1. \] Since $\ell>3$, his equation~(2), obtained from Lemma~1.1 and the two descent sequences, applies and yields the exact equality \[ \rank_\Z E_\ell(\Q) =\dim_{\F_3}S+\dim_{\F_3}S'-1-a-b =1-a-b. \] The isogeny maps on Weil--Ch\^atelet groups commute with localization, so they induce functorial maps on $\Sha$ with $\lambda'_*\circ\lambda_*=[3]$. Thus $\lambda_*$ maps $\Sha(E_\ell/\Q)[3]$ into $\Sha(E'_\ell/\Q)[\lambda']$, and the kernel of this restriction is $\Sha(E_\ell/\Q)[\lambda]$. Consequently \[ \dim_{\F_3}\Sha(E_\ell/\Q)[3]\leq a+b. \] The full Kummer sequence is \[ 0\longrightarrow E_\ell(\Q)\otimes\Q_3/\Z_3 \longrightarrow \Sel_{3^\infty}(E_\ell/\Q) \longrightarrow \Sha(E_\ell/\Q)[3^\infty] \longrightarrow 0. \] The last group is a cofinitely generated $\Z_3$-module, as a quotient of the full Selmer group. The structure theorem for its Pontryagin dual gives \[ \corank_{\Z_3}\Sha(E_\ell/\Q)[3^\infty] \leq \dim_{\F_3}\Sha(E_\ell/\Q)[3]. \] Combining these facts gives the required full-corank bound \[ s_3(E_\ell) =\rank_\Z E_\ell(\Q) +\corank_{\Z_3}\Sha(E_\ell/\Q)[3^\infty] \leq (1-a-b)+(a+b)=1. \] No finiteness of $\Sha$ has been assumed in obtaining this bound. Now $s_3(E_\ell)\in\{0,1\}$, so Theorem~\ref{thm:main} at $p=3$ gives equality of analytic rank, Mordell--Weil rank, and this corank, as well as finiteness of the entire group $\Sha(E_\ell/\Q)$. For the displayed prime classes, the functional-equation sign is $-1$ by \citet[Section~1.1, equation~(1.1.3)]{DasguptaVoight2018}. The functional equation therefore forces $L(E_\ell,1)=0$, excluding analytic rank zero. The common rank is consequently one. This step uses no separate parity hypothesis. A non-torsion rational point on the projective cubic is not the origin. The origin is its only rational point with $Z=0$, so such a point has $Z\ne0$. Dividing the cubic equation by $Z^3$ gives the asserted rational cube-sum representation. \end{proof} ==== END The-Selmer-converse-for-elliptic-curves-at-every-prime//sections/10-sylvester-cube-sums.tex ==== ==== BEGIN The-Selmer-converse-for-elliptic-curves-at-every-prime//references.bib ==== @article{BertoliniDarmonPrasanna2013, author = {Bertolini, Massimo and Darmon, Henri and Prasanna, Kartik}, title = {Generalized {Heegner} cycles and {$p$}-adic {Rankin} {$L$}-series}, journal = {Duke Mathematical Journal}, volume = {162}, number = {6}, pages = {1033--1148}, year = {2013}, doi = {10.1215/00127094-2142056}, url = {https://www.math.mcgill.ca/darmon/pub/Articles/Research/51.BDP1/duke-publishedversion.pdf} } @misc{Yin2026Sylvester47, author = {Yin, Hongbo}, title = {A proof of the {$4,7$} cases of {Sylvester}'s conjecture on cube sums}, year = {2026}, eprint = {2605.25917}, archivePrefix = {arXiv}, primaryClass = {math.NT}, note = {Preprint, version 3, 4 June 2026; Theorem 1.1}, url = {https://arxiv.org/abs/2605.25917v3} } @misc{Yin2026SylvesterGZ, author = {Yin, Hongbo}, title = {{Gross--Zagier} formula for the {$4,7$} cases of {Sylvester}'s conjecture}, year = {2026}, eprint = {2607.01744}, archivePrefix = {arXiv}, primaryClass = {math.NT}, note = {Preprint, version 1, 2 July 2026; 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F.}, title={Notes on elliptic curves. {II}}, journal={Journal f{\"u}r die reine und angewandte Mathematik},volume={218},pages={79--108},year={1965}, note={\href{https://doi.org/10.1515/crll.1965.218.79}{doi:10.1515/crll.1965.218.79}} } @article{Skinner2020, author={Skinner, Christopher}, title={A converse to a theorem of {Gross}, {Zagier}, and {Kolyvagin}}, journal={Annals of Mathematics},volume={191},number={2},pages={329--354},year={2020}, note={\href{https://annals.math.princeton.edu/wp-content/uploads/annals-v191-n2-p01-s.pdf}{primary text}} } @misc{Casselman1995, author = {Casselman, W.}, title = {Introduction to the theory of admissible representations of {$p$}-adic reductive groups}, year = {1995}, note = {Unpublished notes, draft of 1 May 1995, 80 pp.}, url = {https://www.math.ubc.ca/~cass/research/pdf/p-adic-book.pdf} } @article{Tadic1986, author = {Tadi{\'c}, Marko}, title = {Classification of unitary representations in irreducible representations of general linear group (non-{Archimedean} case)}, journal = {Annales scientifiques de l'{\'E}cole Normale Sup{\'e}rieure}, series = {4}, volume = {19}, number = {3}, year = {1986}, pages = {335--382}, doi = {10.24033/asens.1510}, url = {https://numdam.org/articles/10.24033/asens.1510/} } @book{Mok2015, author = {Mok, Chung Pang}, title = {Endoscopic Classification of Representations of Quasi-Split Unitary Groups}, series = {Memoirs of the American Mathematical Society}, volume = {235}, publisher = {American Mathematical Society}, address = {Providence, RI}, year = {2015}, pages = {vi+248}, doi = {10.1090/memo/1108}, eprint = {1206.0882}, archivePrefix = {arXiv}, primaryClass = {math.RT}, url = {https://arxiv.org/abs/1206.0882v5}, note = {Memoir no. 1108; Addendum section 10 locators refer to arXiv version 5, pp. 269--271} } @misc{KMSW2014, author = {Kaletha, Tasho and M{\'i}nguez, Alberto and Shin, Sug Woo and White, Paul-James}, title = {Endoscopic Classification of Representations: Inner Forms of Unitary Groups}, year = {2014}, eprint = {1409.3731}, archivePrefix = {arXiv}, primaryClass = {math.NT}, url = {https://arxiv.org/abs/1409.3731v3}, note = {Preprint, version 3, 220 pp.} } @article{BLGGT2014, author = {Barnet-Lamb, Thomas and Gee, Toby and Geraghty, David and Taylor, Richard}, title = {Local-global compatibility for {$l=p$}, {II}}, journal = {Annales scientifiques de l'{\'E}cole Normale Sup{\'e}rieure}, series = {4}, volume = {47}, number = {1}, year = {2014}, pages = {165--179}, doi = {10.24033/asens.2212}, url = {https://www.numdam.org/articles/10.24033/asens.2212/} } @article{BGHT2011, author = {Barnet-Lamb, Tom and Geraghty, David and Harris, Michael and Taylor, Richard}, title = {A Family of {Calabi--Yau} Varieties and Potential Automorphy {II}}, journal = {Publications of the Research Institute for Mathematical Sciences}, volume = {47}, number = {1}, year = {2011}, pages = {29--98}, doi = {10.2977/PRIMS/31}, url = {https://ems.press/journals/prims/articles/4468} } @article{ChenevierHarris2013, author = {Chenevier, Ga{\"e}tan and Harris, Michael}, title = {Construction of automorphic {Galois} representations, {II}}, journal = {Cambridge Journal of Mathematics}, volume = {1}, number = {1}, year = {2013}, pages = {53--73}, doi = {10.4310/CJM.2013.v1.n1.a2}, url = {https://webusers.imj-prg.fr/~michael.harris/ConstructionII.pdf} } @article{Caraiani2014, author = {Caraiani, Ana}, title = {Monodromy and local-global compatibility for {$l=p$}}, journal = {Algebra \& Number Theory}, volume = {8}, number = {7}, year = {2014}, pages = {1597--1646}, doi = {10.2140/ant.2014.8.1597}, url = {https://msp.org/ant/2014/8-7/ant-v8-n7-p02-s.pdf} } @article{BernsteinZelevinsky1977, author = {Bernstein, I. 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P. Langlands, pp. 203--207; the finite-dimensionality statement is Section 4.3(i), p. 195}, url = {https://www.scribd.com/doc/200645779/Automorphic-Forms-Representations-and-L-Functions-Part-1-A-Borel-W-Casselman} } @misc{AGIKMS2026, author = {Atobe, Hiraku and Gan, Wee Teck and Ichino, Atsushi and Kaletha, Tasho and M{\'i}nguez, Alberto and Shin, Sug Woo}, title = {Local Intertwining Relations and Co-tempered {$A$}-packets of Classical Groups}, year = {2026}, eprint = {2410.13504}, archivePrefix = {arXiv}, primaryClass = {math.NT}, note = {arXiv:2410.13504v3, 24 July 2026; the weighted fundamental lemma dependency is stated in Section 0.4}, url = {https://arxiv.org/abs/2410.13504v3} } @misc{HalleckDubeWFLDraft, author = {Halleck-Dub{\'e}, Connor}, title = {The Weighted Fundamental Lemma for Non-Split Groups}, year = {n.d.}, howpublished = {Author-hosted draft, 185 PDF pages}, note = {Undated author draft, retrieved 24 September 2026; Theorems 16.5 and 17.7, Section 1.14. The cited copy has SHA-256 \nolinkurl{6ce24e401622ad7a0ec09cb2ffdc77d474869e8f98496f953227928fd71cd316}}, url = {https://math.berkeley.edu/~chd/expo/WFL.pdf} } @book{Lan2018Ordinary, author = {Lan, Kai-Wen}, title = {Compactifications of {PEL}-Type {Shimura} Varieties and {Kuga} Families with Ordinary Loci}, publisher = {World Scientific Publishing Co.}, address = {Singapore}, year = {2018}, isbn = {9789813207325}, doi = {10.1142/10374}, url = {https://www.kwlan.org/articles/cpt-ram-ord.pdf}, note = {Result locators refer to the undated author preprint (missing the preface), consulted 19 September 2026} } @article{CoatesWiles1977, author = {Coates, John and Wiles, Andrew}, title = {On the conjecture of {Birch} and {Swinnerton-Dyer}}, journal = {Inventiones Mathematicae}, volume = {39}, year = {1977}, pages = {223--251}, doi = {10.1007/BF01402975}, url = {https://link.springer.com/article/10.1007/BF01402975}, note = {Theorem 1, p. 223} } @article{Kato2004, author = {Kato, Kazuya}, title = {{$p$}-adic {Hodge} theory and values of zeta functions of modular forms}, journal = {Ast{\'e}risque}, volume = {295}, year = {2004}, pages = {117--290}, url = {https://www.numdam.org/item/AST_2004__295__117_0/}, note = {Theorem 14.2 and Corollary 14.3, pp. 235--236} } @article{SkinnerUrban2014, author = {Skinner, Christopher and Urban, Eric}, title = {The {Iwasawa} Main Conjectures for {$\mathrm{GL}_2$}}, journal = {Inventiones Mathematicae}, volume = {195}, number = {1}, year = {2014}, pages = {1--277}, doi = {10.1007/s00222-013-0448-1}, url = {https://www.math.columbia.edu/~urban/eurp/MC.pdf}, note = {Theorem 2(b), also Theorem 3.6.11(b), for the rank-zero converse} } @article{Zhang2014, author = {Zhang, Wei}, title = {{Selmer} groups and the indivisibility of {Heegner} points}, journal = {Cambridge Journal of Mathematics}, volume = {2}, number = {2}, year = {2014}, pages = {191--253}, doi = {10.4310/CJM.2014.v2.n2.a2}, url = {https://math.mit.edu/~wz2113/math/online/Kconj.pdf}, note = {Theorem 1.4 for the converse over the rationals} } @article{CastellaWan2024, author = {Castella, Francesc and Wan, Xin}, title = {{Perrin-Riou}'s main conjecture for elliptic curves at supersingular primes}, journal = {Mathematische Annalen}, volume = {389}, number = {3}, year = {2024}, pages = {2595--2636}, doi = {10.1007/s00208-023-02711-w}, url = {https://web.math.ucsb.edu/~castella/Perrin-Riou-print.pdf}, note = {Theorem A and Theorem 6.11; locators refer to the published text} } @article{CGLS2022, author = {Castella, Francesc and Grossi, Giada and Lee, Jaehoon and Skinner, Christopher}, title = {On the anticyclotomic {Iwasawa} theory of rational elliptic curves at {Eisenstein} primes}, journal = {Inventiones Mathematicae}, volume = {227}, number = {2}, year = {2022}, pages = {517--580}, doi = {10.1007/s00222-021-01072-y}, eprint = {2008.02571}, archivePrefix = {arXiv}, primaryClass = {math.NT}, url = {https://arxiv.org/abs/2008.02571v2}, note = {Theorem E, also Theorem 5.2.1; arXiv version 2 dated 14 September 2021} } @misc{KellerYinSemistable2024, author = {Keller, Timo and Yin, Mulun}, title = {On the anticyclotomic {Iwasawa} theory of newforms at {Eisenstein} primes of semistable reduction}, year = {2024}, eprint = {2402.12781}, archivePrefix = {arXiv}, primaryClass = {math.NT}, url = {https://arxiv.org/abs/2402.12781v2}, note = {Version 2, 30 October 2024; Theorem E, Theorem 4.1.1 and Corollary 5.2.2} } @article{BCS2025, author = {Burungale, Ashay and Castella, Francesc and Skinner, Christopher}, title = {Base Change and {Iwasawa} Main Conjectures for {$\mathrm{GL}_2$}}, journal = {International Mathematics Research Notices}, volume = {2025}, number = {8}, year = {2025}, eid = {rnaf082}, doi = {10.1093/imrn/rnaf082}, eprint = {2405.00270}, archivePrefix = {arXiv}, primaryClass = {math.NT}, url = {https://web.math.ucsb.edu/~castella/BCS-print.pdf}, note = {Article rnaf082, 15 pp.; Theorems 1.1.2(a), 1.2.2(a) and the converse application on p. 4} } @article{BurungaleTian2026, author = {Burungale, Ashay A. and Tian, Ye}, title = {A rank zero {$p$}-converse to a theorem of {Gross--Zagier}, {Kolyvagin} and {Rubin}}, journal = {Annals of Mathematics}, series = {2}, volume = {203}, number = {1}, year = {2026}, pages = {1--13}, doi = {10.4007/annals.2026.203.1.1}, eprint = {2506.03465}, archivePrefix = {arXiv}, primaryClass = {math.NT}, url = {https://annals.math.princeton.edu/2026/203-1/p01}, note = {Theorem 1.1; arXiv version 2 dated 11 October 2025} } @article{Ribet1976, author = {Ribet, Kenneth A.}, title = {A modular construction of unramified {$p$}-extensions of {$\mathbf{Q}(\mu_p)$}}, journal = {Inventiones Mathematicae}, volume = {34}, year = {1976}, pages = {151--162}, doi = {10.1007/BF01403065}, url = {https://math.berkeley.edu/~ribet/Articles/invent_34.pdf} } @article{Wiles1990, author = {Wiles, Andrew}, title = {The {Iwasawa} conjecture for totally real fields}, journal = {Annals of Mathematics}, series = {2}, volume = {131}, number = {3}, year = {1990}, pages = {493--540}, doi = {10.2307/1971468}, url = {https://annals.math.princeton.edu/1990/131-3/p02} } @article{Weil1964, author = {Weil, Andr{\'e}}, title = {Sur certains groupes d'op{\'e}rateurs unitaires}, journal = {Acta Mathematica}, volume = {111}, year = {1964}, pages = {143--211}, doi = {10.1007/BF02391012}, url = {https://doi.org/10.1007/BF02391012} } @article{Rallis1984Injectivity, author = {Rallis, Stephen}, title = {Injectivity properties of liftings associated to {Weil} representations}, journal = {Compositio Mathematica}, volume = {52}, number = {2}, year = {1984}, pages = {139--169}, url = {https://www.numdam.org/item/CM_1984__52_2_139_0/} } @article{CastellaLiuWan2022, author = {Castella, Francesc and Liu, Zheng and Wan, Xin}, title = {{Iwasawa--Greenberg} main conjecture for nonordinary modular forms and {Eisenstein} congruences on {$\mathrm{GU}(3,1)$}}, journal = {Forum of Mathematics, Sigma}, volume = {10}, year = {2022}, eid = {e110}, note = {Article e110, 90 pp.; theorem locators follow the published version}, doi = {10.1017/fms.2022.95}, url = {https://web.math.ucsb.edu/~castella/CLW-print.pdf} } @misc{CastellaLiuWanAddendum, author = {Castella, Francesc and Liu, Zheng and Wan, Xin}, title = {Addendum to: {Iwasawa--Greenberg} main conjecture for nonordinary modular forms and {Eisenstein} congruences on {$\mathrm{GU}(3,1)$}}, howpublished = {Author-hosted addendum}, note = {Undated in the inspected text; 6 pp.; accessed 24 September 2026}, url = {https://web.math.ucsb.edu/~castella/CLW-addendum.pdf} } @misc{BHTY2026, author = {Burungale, Ashay and He, Wei and Tian, Ye and Ye, Xiangdong}, title = {Mod {$\ell$} non-vanishing of self-dual {Hecke} {$L$}-values over {CM} fields and applications}, year = {2026}, eprint = {2508.19706}, archivePrefix = {arXiv}, note = {Version 2, 13 March 2026; Theorem 1.1, Lemma 4.12 and Corollary 4.20}, url = {https://arxiv.org/abs/2508.19706v2} } @article{Shin2024WeakTransfer, 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3.8 is the reduction cited as [W4] in AGIKMS, Section 0.4} } @incollection{Katz1981SerreTate, author = {Katz, Nicholas M.}, title = {{Serre--Tate} local moduli}, booktitle = {Surfaces Alg{\'e}briques: S{\'e}minaire de G{\'e}om{\'e}trie Alg{\'e}brique d'Orsay 1976--78}, editor = {Giraud, Jean and Illusie, Luc and Raynaud, Michel}, series = {Lecture Notes in Mathematics}, volume = {868}, publisher = {Springer}, address = {Berlin--Heidelberg}, year = {1981}, pages = {138--202}, doi = {10.1007/BFb0090648}, url = {https://web.math.princeton.edu/~nmk/old/serretatelocmod.pdf}, note = {Theorem 2.1(1), (2), (4), pp. 148--149; relevant construction on pp. 149--150} } @article{ColmezFontaine2000, author = {Colmez, Pierre and Fontaine, Jean-Marc}, title = {Construction des repr{\'e}sentations {$p$}-adiques semi-stables}, journal = {Inventiones Mathematicae}, volume = {140}, year = {2000}, pages = {1--43}, doi = {10.1007/s002220000042}, url = {https://www.imo.universite-paris-saclay.fr/~fontaine/CF.pdf}, note = {Theorem and page locators follow the 37-page author version: Theorem A, p. 2; Proposition 4.2, p. 23; unnumbered corollary following Theorem 4.3, p. 24} } @article{BernsteinDeligneKazhdan1986, author = {Bernstein, J. and Deligne, P. and Kazhdan, D.}, title = {Trace {Paley--Wiener} theorem for reductive {$p$}-adic groups}, journal = {Journal d'Analyse Math\'{e}matique}, volume = {47}, year = {1986}, pages = {180--192}, doi = {10.1007/BF02792538}, url = {https://www.math.tau.ac.il/~bernstei/Publication_list/publication_texts/Bernst_Deligne_Kazhdan.pdf}, note = {Section 2.3, Decomposition Theorem, p. 182, recalled there from Bernstein--Deligne, reference [1], Section 2.10} } @incollection{CluckersHalesLoeser2011, author = {Cluckers, Raf and Hales, Thomas and Loeser, Fran{\c{c}}ois}, title = {Transfer principle for the fundamental lemma}, booktitle = {On the Stabilization of the Trace Formula}, editor = {Clozel, Laurent and Harris, Michael and Labesse, Jean-Pierre and Ng{\^o}, Bao-Ch{\^a}u}, series = {Stabilization of the Trace Formula, Shimura Varieties, and Arithmetic Applications}, volume = {1}, publisher = {International Press}, address = {Somerville, MA}, year = {2011}, pages = {309--347}, url = {https://webusers.imj-prg.fr/~francois.loeser/transfer_fl_2010_10_14.pdf}, note = {Theorem locators follow the author version dated 14 October 2010} } @article{Ngo2010FundamentalLemma, author = {Ng{\^o}, Bao Ch{\^a}u}, title = {Le lemme fondamental pour les alg{\`e}bres de {Lie}}, journal = {Publications Math{\'e}matiques de l'IH{\'E}S}, volume = {111}, year = {2010}, pages = {1--169}, doi = {10.1007/s10240-010-0026-7}, url = {https://www.math.uchicago.edu/~ngo/LF.pdf} } @article{HidaTilouine1993, author = {Hida, Haruzo and Tilouine, Jacques}, title = {Anti-cyclotomic {Katz} {$p$}-adic {$L$}-functions and congruence modules}, journal = {Annales scientifiques de l'{\'E}cole Normale Sup{\'e}rieure}, series = {4}, volume = {26}, number = {2}, year = {1993}, pages = {189--259}, url = {https://www.numdam.org/article/ASENS_1993_4_26_2_189_0.pdf} } @article{GeisserSuzuki2022, author = {Geisser, Thomas H. and Suzuki, Takashi}, title = {Special values of {$L$}-functions of one-motives over function fields}, journal = {Journal f{\"u}r die reine und angewandte Mathematik}, volume = {793}, year = {2022}, pages = {281--304}, doi = {10.1515/crelle-2022-0081}, eprint = {2009.14504}, archivePrefix = {arXiv}, url = {https://arxiv.org/abs/2009.14504v3}, note = {The section and proposition locators also apply to arXiv version 3} } @article{ChaudouardLaumon2012II, author = {Chaudouard, Pierre-Henri and Laumon, G{\'e}rard}, title = {Le lemme fondamental pond{\'e}r{\'e}. {II}. {\'E}nonc{\'e}s cohomologiques}, journal = {Annals of Mathematics}, series = {2}, volume = {176}, number = {3}, year = {2012}, pages = {1647--1781}, doi = {10.4007/annals.2012.176.3.6}, url = {https://annals.math.princeton.edu/wp-content/uploads/annals-v176-n3-p06-p.pdf} } @article{AdamsJohnson1987, author = {Adams, Jeffrey and Johnson, Joseph F.}, title = {Endoscopic groups and packets of non-tempered representations}, journal = {Compositio Mathematica}, volume = {64}, number = {3}, year = {1987}, pages = {271--309}, url = {https://www.numdam.org/item/CM_1987__64_3_271_0.pdf} } @article{ClozelDelorme1990, author = {Clozel, Laurent and Delorme, Patrick}, title = {Le th{\'e}or{\`e}me de {Paley--Wiener} invariant pour les groupes de {Lie} r{\'e}ductifs. {II}}, journal = {Annales scientifiques de l'{\'E}cole Normale Sup{\'e}rieure}, series = {4}, volume = {23}, number = {2}, year = {1990}, pages = {193--228}, doi = {10.24033/asens.1602}, url = {https://www.numdam.org/item/ASENS_1990_4_23_2_193_0.pdf} } @incollection{Labesse2011CM, author = {Labesse, Jean-Pierre}, title = {Changement de base {CM} et s{\'e}ries discr{\`e}tes}, booktitle = {On the Stabilization of the Trace Formula}, editor = {Clozel, Laurent and Harris, Michael and Labesse, Jean-Pierre and Ng{\^o}, Bao-Ch{\^a}u}, series = {Stabilization of the Trace Formula, Shimura Varieties, and Arithmetic Applications}, volume = {1}, publisher = {International Press}, address = {Somerville, MA}, year = {2011}, pages = {429--470}, url = {https://www.imj-prg.fr/fa/bpFiles/Labesse2.pdf}, note = {Corollary 5.3 and its sign property (*) are on pp. 42--44 of the linked author version} } @article{MoeglinWaldspurger1989, author = {M{\oe}glin, Colette and Waldspurger, Jean-Loup}, title = {Le spectre r{\'e}siduel de {$\mathrm{GL}(n)$}}, journal = {Annales scientifiques de l'{\'E}cole Normale Sup{\'e}rieure}, series = {4}, volume = {22}, number = {4}, year = {1989}, pages = {605--674}, doi = {10.24033/asens.1595}, url = {https://www.numdam.org/article/ASENS_1989_4_22_4_605_0.pdf} } @article{SalamancaRiba1988, author = {Salamanca Riba, Susana}, title = {On the unitary dual of some classical {Lie} groups}, journal = {Compositio Mathematica}, volume = {68}, number = {3}, year = {1988}, pages = {251--303}, url = {https://www.numdam.org/item/CM_1988__68_3_251_0.pdf} } ==== END The-Selmer-converse-for-elliptic-curves-at-every-prime//references.bib ====