64c3814
==== TABLE OF CONTENTS ====
Equidistribution-of-Primitive-Quartic-Torus-Packets-for-Arbitrary-Orders//additive.tex
Equidistribution-of-Primitive-Quartic-Torus-Packets-for-Arbitrary-Orders//conclusion.tex
Equidistribution-of-Primitive-Quartic-Torus-Packets-for-Arbitrary-Orders//exterior.tex
Equidistribution-of-Primitive-Quartic-Torus-Packets-for-Arbitrary-Orders//introduction.tex
Equidistribution-of-Primitive-Quartic-Torus-Packets-for-Arbitrary-Orders//local-fourier.tex
Equidistribution-of-Primitive-Quartic-Torus-Packets-for-Arbitrary-Orders//macros.tex
Equidistribution-of-Primitive-Quartic-Torus-Packets-for-Arbitrary-Orders//main.tex
Equidistribution-of-Primitive-Quartic-Torus-Packets-for-Arbitrary-Orders//ordinary.tex
Equidistribution-of-Primitive-Quartic-Torus-Packets-for-Arbitrary-Orders//packets.tex
Equidistribution-of-Primitive-Quartic-Torus-Packets-for-Arbitrary-Orders//rigidity.tex
Equidistribution-of-Primitive-Quartic-Torus-Packets-for-Arbitrary-Orders//sieve.tex
Equidistribution-of-Primitive-Quartic-Torus-Packets-for-Arbitrary-Orders//references.bib
===========================
==== BEGIN Equidistribution-of-Primitive-Quartic-Torus-Packets-for-Arbitrary-Orders//additive.tex ====
\section{Additive distribution of the conductor weights}
\label{sec:additive}
The factors \(f_p^*\) introduced in Section~\ref{sec:exterior} retain the
local lattice condition of an order at the primes in \(S\). We now average their
product on the real trace plane. The estimate will also permit a residue
class at a small modulus prime to \(S\); this is the form needed to average
the factors at the remaining primes in Section~\ref{sec:sieve}.
Fix a nonzero integer \(m\). In the three real embeddings of \(F\), write
\(z=(z_1,z_2,z_3)\), so that \(\Tr z=z_1+z_2+z_3\). Choose two distinct
indices \(i,j\), a fixed nonnegative function
\(\Phi\in C_c^\infty(\R^2)\) which is at least one on \([-1,1]^2\), and a
fixed nonnegative \(\chi\in C_c^\infty(\R)\) with
\[
\chi(m)=1,\qquad
\supp\chi\subset(m-\tfrac13,m+\tfrac13).
\]
For \(c=(c_i,c_j)\in\R^2\) and positive \(r_i,r_j\), put
\begin{equation}
\phi(z)=
\Phi\left(\frac{z_i-c_i}{r_i},\frac{z_j-c_j}{r_j}\right)\chi(\Tr z),
\qquad
A_\phi=r_i r_j\int_{\R^2}\Phi(u,v)\,du\,dv .
\label{eq:add-box}
\end{equation}
Thus \(A_\phi\) is the integral of \(\phi\) on \(\Tr z=m\), using \(z_i,z_j\)
as coordinates on that plane. Fix \(r_*>0\), keep \(r_i,r_j\le r_*\), and
require the supports of their planar factors to lie in a fixed
bounded region. The supports of all the resulting \(\phi\)'s then lie in a
fixed compact subset \(\Omega_\infty\) of \(\R^3\) on which \(\Tr z\) stays
away from zero. All constants below may depend on \(m,\Phi,\chi\), the
upper bound for the scales, and this bounded region. They do not depend on
the locations \(c\) or on the scales within their specified range.
For \(p\notin S\), the algebra \(F\otimes\Q_p\) is unramified and its
integral trace map \(R_{F,p}\to\Z_p\) is onto. The affine set
\[
\mathcal T_{m,p}=\{z\in R_{F,p}:\Tr z=m\}
\]
is consequently a translate of a free rank-two direct summand of \(R_{F,p}\).
A residue class on this set modulo \(p^a\) means a translate of
\(p^a\ker(\Tr:R_{F,p}\to\Z_p)\). Equivalently, if \(z_p^0\in\mathcal T_{m,p}\)
represents the class, its members satisfy \(z-z_p^0\in p^aR_{F,p}\).
There are \(p^{2a}\) such classes.
\begin{proposition}[Additive distribution]
\label{prop:additive-equidistribution}
There are absolute constants \(\tau,\theta>0\) with the following property.
Let \(K,\OO,F\) be as in Section~\ref{sec:exterior}, put
\(X=[\OO_K:\OO]\sqrt{|\Disc(K)|}\), and take \(S\) as in that section for
the fixed nonzero integer \(m\). In Equation~\eqref{eq:add-box}, choose any
two real partition coordinates and scales
\[
X^{-\tau}\le r_i,r_j\le r_*
\]
whose planar supports stay in the fixed bounded region above. Let
\(1\le l\le X^\theta\) be prime to every prime in \(S\). For each \(p\mid l\)
choose any \(z_p^0\in\mathcal T_{m,p}\), specifying a class modulo
\(p^{v_p(l)}\). Then
\begin{equation}
\sum_{\substack{z\in F,\ \Tr z=m\\
z\in R_{F,p}\ (p\notin S)\\
z-z_p^0\in p^{v_p(l)}R_{F,p}\ (p\mid l)}}
\phi(z)\prod_{p\in S}f_p^*(z)
=
A_\phi\,l^{-2}\prod_{p\in S}d_p(m)
\bigl(1+O(X^{-\theta})\bigr).
\label{eq:add-equidistribution}
\end{equation}
Here \(d_p\) is the trace density of Lemma~\ref{lem:fourier-scalar-density}.
The error is uniform over all the fields, orders, ordered real embeddings,
permitted box locations and scales, moduli \(l\), and chosen residue classes.
The implied constant and the lower threshold for \(X\) depend only on the
fixed real data listed above and on \(m\).
\end{proposition}
\subsection{Finite expansion and Poisson summation}
We first assemble the local inverse transforms into a finite global
expansion. This is the point at which the fact that the cubic algebra is a
field is essential. Let \(\mathfrak d_F\) be the different of \(F\), and
write \(\mathfrak d_p=\mathfrak d_F R_{F,p}\) for its localization.
For a choice of denominator shells at the primes of \(S\), write
\[
\mathfrak h=\prod_{p\in S}\mathfrak h_p,\qquad
H=\prod_{p\in S}H_p=\Nm\mathfrak h,\qquad
G_0=\prod_{p\in S}p^{g_p}.
\]
Local ideals here and below are identified with global ideals supported at
the corresponding primes. The functions \(K_{p,\mathfrak h_p}\) and their
boundary sets \(\partial_{\mathfrak h_p}(z)\) are those of
Lemma~\ref{lem:fourier-shell-estimates}. Set
\[
\mathscr L_{\mathfrak h}=\{p\in S:g_p>0\text{ or }p\le5\}.
\]
\begin{lemma}[Global bounds for denominator profiles]
\label{lem:add-profile-bounds}
For every profile one has
\begin{equation}
G_0\le [R_F:C_F]\le H G_0.
\label{eq:add-profile-indices}
\end{equation}
Suppose that \(z\in F\) is integral outside \(S\), its real image lies in
\(\Omega_\infty\), and
\(\prod_{p\in S}K_{p,\mathfrak h_p}(z)\ne0\). Define
\[
T_\partial=T_\partial(\mathfrak h,z)
=\prod_{\substack{p\in S\setminus\mathscr L_{\mathfrak h}\\
\mathfrak q\in\partial_{\mathfrak h_p}(z)}}
\Nm\mathfrak q ,
\]
where an empty product is one. There is a constant depending only on
\(\Omega_\infty\) such that
\begin{align}
H&\ll D_FG_0^{24}T_\partial,
&T_\partial&\le\left(\prod_{p\in S}p\right)^3,
\label{eq:add-profile-norm}\\
\left|\prod_{p\in S}K_{p,\mathfrak h_p}(z)\right|
&\ll 8^{|S|}T_\partial^{-1/2}.
\label{eq:add-profile-inverse}
\end{align}
All such \(z\) lie in a fractional \(R_F\)-ideal
\(\mathfrak a_{\mathfrak h}\), independent of \(z\), with
\begin{equation}
\Nm(\mathfrak a_{\mathfrak h})^{-1}
\le 30^{12}D_FG_0^{24}.
\label{eq:add-profile-support}
\end{equation}
Set \(C_3=371\). For all sufficiently large \(X\), depending only on the
fixed real data and \(m\), a profile contributing at any such \(z\) has
\(H\le X^{C_3}\). For these \(X\) and every \(z\in F\) integral outside
\(S\) with real image in \(\Omega_\infty\),
\begin{equation}
\prod_{p\in S}f_p^*(z)
=\sum_{\substack{\mathfrak h=(\mathfrak h_p)_{p\in S}\\H\le X^{C_3}}}
\prod_{p\in S}K_{p,\mathfrak h_p}(z).
\label{eq:add-profile-expansion}
\end{equation}
The number of profiles in this sum is \(O_\eta(X^\eta)\) for every fixed
\(\eta>0\), uniformly in the arithmetic data.
\end{lemma}
\begin{proof}
Let \(M_p=\mathfrak h_p\cap C_{F,p}\). The additive lattice identity
\[
[R_{F,p}:C_{F,p}]
=[R_{F,p}:C_{F,p}+\mathfrak h_p]\,[\mathfrak h_p:M_p]
\]
has first factor between \(1\) and
\([R_{F,p}:\mathfrak h_p]=H_p\). The index of \(C_F\) is supported on \(S\)
by Lemma~\ref{lem:ext-order-scale}; multiplying gives
Equation~\eqref{eq:add-profile-indices}.
At \(p\in\mathscr L_{\mathfrak h}\), the local support is contained in
\(p^{-4(g_p+1)}\mathfrak h_p\mathfrak d_p^{-1}\).
At every other \(p\in S\), the support loses at most one power of each
boundary prime \(\mathfrak q\), and loses no power in the other components.
It follows that a supporting \(z\) belongs to
\[
\mathfrak b_{\mathfrak h,z}
=\mathfrak h\mathfrak d_F^{-1}
\prod_{p\in\mathscr L_{\mathfrak h}}
(pR_F)^{-4(g_p+1)}
\prod_{\substack{p\in S\setminus\mathscr L_{\mathfrak h}\\
\mathfrak q\in\partial_{\mathfrak h_p}(z)}}
\mathfrak q^{-1}.
\]
The real trace condition excludes \(z=0\). Since \(F\) is a field, its norm
is nonzero, and containment of the principal ideal \((z)\) in
\(\mathfrak b_{\mathfrak h,z}\) gives
\[
|\Nm_{F/\Q}z|
=\Nm(\mathfrak b_{\mathfrak h,z})
[\mathfrak b_{\mathfrak h,z}:(z)]
\ge \frac{H}
{D_FT_\partial\prod_{p\in\mathscr L_{\mathfrak h}}p^{12(g_p+1)}}.
\]
The left side is bounded above on \(\Omega_\infty\). For \(g_p>0\) we have
\(g_p+1\le2g_p\); the only additional factors, for \(g_p=0\), are at
\(2,3,5\). Hence
\[
\prod_{p\in\mathscr L_{\mathfrak h}}p^{12(g_p+1)}
\le30^{12}G_0^{24}.
\]
This proves the first inequality of Equation~\eqref{eq:add-profile-norm}.
Each boundary prime occurs once, and
\(\sum_{\mathfrak q\mid p}f_{\mathfrak q}\le
\sum_{\mathfrak q\mid p}e_{\mathfrak q}f_{\mathfrak q}=3\);
this proves the bound for \(T_\partial\). Equation~\eqref{eq:add-profile-inverse}
is the product of the boundary bounds of
Lemma~\ref{lem:fourier-shell-estimates}; its absolute implied constant
accounts for the three small primes.
Dropping \(\mathfrak h_p\) in the first support condition enlarges that
support. In the second, the exponent \(v_{\mathfrak q}(\mathfrak h_p)-1\)
at a boundary is nonnegative. Thus every supporting
point belongs to the ideal
\[
\mathfrak a_{\mathfrak h}
=\mathfrak d_F^{-1}
\prod_{p\in\mathscr L_{\mathfrak h}}
(pR_F)^{-4(g_p+1)} ,
\]
which gives Equation~\eqref{eq:add-profile-support}.
Lemma~\ref{lem:ext-order-scale} gives
\(\Delta_F\ll X^{28}\),
\(D_F\le\Delta_F\), and
\([R_F:C_F]\le\Delta_F^{1/2}\). The definition of \(S\) gives
\(\prod_{p\in S}p\ll_m X^2\). Equations
\(\eqref{eq:add-profile-indices}\)--\(\eqref{eq:add-profile-norm}\) therefore
give \(H\ll X^{28+24\cdot14+6}=X^{370}\). The fixed implied constant is
absorbed by \(X^{371}\) for sufficiently large \(X\).
A nonzero field element has nonzero image in every component of
\(F\otimes\Q_p\). Lemma~\ref{lem:fourier-inverse-expansion} therefore
identifies the density \(f_p^*(z)\) with its pointwise finite sum of inverse
shells at all the present \(z\). The norm bound just proved excludes every
product with \(H>X^{371}\), uniformly in \(z\), and proves
Equation~\eqref{eq:add-profile-expansion}. Finally, for any fixed \(u>0\),
Rankin's inequality bounds the number of profiles with \(H\le X^{C_3}\) by
\[
X^{C_3u}\prod_{p\in S}(1-p^{-u})^{-3}.
\]
There are at most three prime-ideal components over each \(p\). The radical
bound on \(S\) implies \(|S|=O(\log X/\log\log X)\), since the product of
\(k\) distinct primes is at least \(k!\). The displayed Euler product is
then \(X^{o(1)}\) for fixed \(u\). Taking \(u\) arbitrarily small proves
the asserted bound for every \(\eta>0\).
\end{proof}
We apply this expansion before imposing Poisson summation. For \(p\notin S\)
let
\[
\eta_{p,l}(z)=
\begin{cases}
\mathbf 1_{z_p^0+p^{v_p(l)}R_{F,p}}(z),&p\mid l,\\
\mathbf 1_{R_{F,p}}(z),&p\nmid l .
\end{cases}
\]
Let \(\eta_{\mathfrak h,l}\) be the finite-adelic test function whose local
factor is \(K_{p,\mathfrak h_p}\) at \(p\in S\) and \(\eta_{p,l}\) elsewhere.
By Lemma~\ref{lem:fourier-shell-estimates}, every \(K_{p,\mathfrak h_p}\)
is supported on elements of integral trace. The factors outside \(S\) are
supported in \(R_{F,p}\) and also have integral trace. Thus any \(z\in F\)
with \(\eta_{\mathfrak h,l}(z)\ne0\) has rational trace integral at every
finite prime, hence an integer. Only \(m\) meets \(\supp\chi\). The finite
profile expansion in Lemma~\ref{lem:add-profile-bounds} consequently shows
that the left side of
Equation~\eqref{eq:add-equidistribution}, denoted by \(\Sigma\), equals
\begin{equation}
\Sigma
=\sum_{\substack{\mathfrak h\\H\le X^{C_3}}}
S_{\mathfrak h},\qquad
S_{\mathfrak h}
=\sum_{z\in F}\phi(z)\eta_{\mathfrak h,l}(z).
\label{eq:add-original-profiles}
\end{equation}
Here \(C_3=371\) is fixed. The preceding lemma makes the profile interchange
finite. Each inner sum also has finitely many terms, since its finite
conditions place \(z\) in a fractional ideal and its real support is compact.
In the local results of Section~\ref{sec:local-fourier}, choose the finite
components of the standard character of \(\mathbb A_\Q/\Q\); they all have
conductor \(\Z_p\). Compose this global character with the field trace.
At a finite place the self-dual measure satisfies
\(\vol(R_{F,p})=|D_F|_p^{1/2}\), while at infinity it is Lebesgue measure
in the three embeddings. The real covolume of \(R_F\) is \(\sqrt{D_F}\)
and \(\prod_p\vol(R_{F,p})=D_F^{-1/2}\). Thus the diagonal copy of \(F\)
has covolume one for these measures. Write
\[
\widehat\phi(A)=\int_{\R^3}\phi(z)\psi_\infty(\Tr(Az))\,dz
\]
for the real transform. Field Poisson summation in this
normalization \cite[Section~4.2, Lemma~4.2.4]{Tate1967} gives
\begin{equation}
S_{\mathfrak h}
=\sum_{A\in F}\widehat\phi(A)
\widehat\eta_{\mathfrak h,l}(A),\qquad
\widehat\eta_{\mathfrak h,l}(A)
=\prod_{p\in S}\gamma_{p,\mathfrak h_p}(A)
\prod_{p\notin S}\widehat\eta_{p,l}(A).
\label{eq:add-poisson}
\end{equation}
Here \(\gamma_{p,\mathfrak h_p}\) is the shell restriction of \(\gamma_p\),
extended by zero, and hats use the chosen character. Each finite test is
Schwartz--Bruhat, so the formula is also ordinary lattice Poisson summation
with congruence classes. At \(p\notin S\), trace duality is unimodular and
\[
\widehat\eta_{p,l}(A)
=p^{-3v_p(l)}
\psi_p(\Tr(Az_p^0))
\mathbf 1_{p^{-v_p(l)}R_{F,p}}(A)
\quad (p\mid l);
\]
for \(p\nmid l\) it is \(\mathbf 1_{R_{F,p}}\).
In particular every such transform is bounded in absolute value by one.
The finite Fourier factors in Equation~\eqref{eq:add-poisson} are supported
on the fractional ideal
\begin{equation}
\Gamma_{l,\mathfrak h}=l^{-1}\mathfrak h^{-1},\qquad P=l^3H,
\qquad
\Nm(\Gamma_{l,\mathfrak h})=P^{-1}.
\label{eq:add-frequency-lattice}
\end{equation}
The following elementary geometry of this ideal will be used both to count
frequencies and to express the local congruences in global integer
coordinates.
\begin{lemma}[Lattice bounds for a cubic ideal]
\label{lem:add-ideal-lattice}
Let \(\mathfrak a\) be a fractional ideal of a totally real cubic field of
discriminant \(D_F\), and write \(\Nm(\mathfrak a)=P^{-1}\) with \(P>0\).
Let \(B_Y=\{x\in\R^3:\|x\|\le Y\}\).
In embedding coordinates it has covolume \(\sqrt{D_F}/P\), first Euclidean
minimum \(\gg P^{-1/3}\), and, for \(Y>0\),
\begin{equation}
\#\{A\in\mathfrak a:\|A\|\le Y\}
\ll 1+YP^{1/3}+Y^2P^{2/3}+\frac{Y^3P}{\sqrt{D_F}} .
\label{eq:add-lattice-count}
\end{equation}
There is a \(\Z\)-basis \(b_1,b_2,b_3\) of \(\mathfrak a\) in which every
\(A=\sum n_i b_i\) with \(\|A\|\le Y\) satisfies
\[
|n_i|\ll YP^{1/3}\quad(1\le i\le3).
\]
All constants here are absolute.
\end{lemma}
\begin{proof}
The covolume formula follows from the discriminant. For \(0\ne A\in
\mathfrak a\), containment \((A)\subset\mathfrak a\) implies
\(|\Nm_{F/\Q}A|\ge \Nm\mathfrak a=P^{-1}\). The product inequality in
three real coordinates gives
\(|\Nm A|\le(\|A\|/\sqrt3)^3\), proving the first-minimum bound.
Let \(\lambda_1\le\lambda_2\le\lambda_3\) be the Euclidean successive
minima. A lattice in fixed dimension has a basis of bounded orthogonality
defect, by Mahler's basis theorem \cite[Section~2, Theorem~1]{Mahler1946}.
For such a basis, order its lengths increasingly and write its defect as
\(\omega=\prod_i\|b_i\|/\covol(\mathfrak a)\ll1\).
Cramer's rule and Hadamard's inequality bound the norm of the \(i\)-th
coordinate functional by \(\omega/\|b_i\|\). Since
\(\|b_i\|\ge\lambda_i\), counting its possible integer coordinates gives
\[
\#(\mathfrak a\cap B_Y)\ll
\prod_{i=1}^3(1+Y/\lambda_i).
\]
Minkowski's second theorem for the Euclidean ball, in the formulation of
\cite[Theorem~4.3]{Widmer2010}, gives
\(\lambda_1\lambda_2\lambda_3\asymp\covol(\mathfrak a)\).
Expanding the last product, using \(\lambda_2\ge\lambda_1\), and inserting
the first-minimum and covolume bounds proves
Equation~\eqref{eq:add-lattice-count}. The same coordinate functional
bound with \(\|b_i\|\ge\lambda_1\) proves the final assertion.
\end{proof}
\subsection{The nonscalar contribution}
Scalar frequencies \(A\in\Q\subset F\) in
Equation~\eqref{eq:add-poisson} record only the trace. The main work is to
show that the remaining frequencies contribute an absolute power saving.
We prove that estimate before evaluating the scalar terms.
\begin{lemma}[Nonscalar contribution]
\label{lem:add-nonscalar}
There are absolute \(\tau,\theta,\epsilon_F>0\), with
\(\epsilon_F>2\tau+3\theta\), such that for \(C_3=371\) and the data of
Proposition~\ref{prop:additive-equidistribution} with these
\(\tau,\theta\),
\begin{equation}
\sum_{\substack{\mathfrak h\\H\le X^{C_3}}}
\left|
\sum_{A\in F\setminus\Q}
\widehat\phi(A)\widehat\eta_{\mathfrak h,l}(A)
\right|
\ll X^{-\epsilon_F}.
\label{eq:add-nonscalar}
\end{equation}
The constants have the same uniformity as in that proposition.
\end{lemma}
\begin{proof}
We use \(\lambda=1/3\). Lemma~\ref{lem:ext-order-scale}, which gives
\(\Delta_F\ge X^{1/2}/2\), implies
\begin{equation}
\Delta_F\ge X^\lambda
\label{eq:add-order-lower}
\end{equation}
for all sufficiently large \(X\).
After bounding the real Fourier tail, we split according to the field
discriminant. If \(D_F\) is large, comparing it with the discriminant of
the order generated by a retained nonscalar frequency after clearing its
denominators forces \(H\) large. Otherwise the lower bound for
\(\Delta_F\) makes the index \([R_F:C_F]\) large. For profiles with small
\(G_0\), this forces \(H\) large; inverse support and boundary decay then
bound the possibly signed original sum \(S_{\mathfrak h}\). Poisson
expresses its full nonscalar Fourier sum as \(S_{\mathfrak h}\) minus the
separately estimated scalar series. For profiles with large \(G_0\), the
inverse depths attached to each frequency and its chosen shifts provide
either sufficient pointwise decay or congruences; in the latter range a
further discriminant comparison supplies the required denominator size.
The positive exponents
\(d,e,e_1,e_2,w,\theta,\tau\) below will be fixed at the end of the proof;
put \(Y=X^w\) and \(s=w+\theta\).
First the frequencies outside \(\|A\|\le Y\) can be discarded. For every
fixed integer \(r\ge0\), integration by parts in the fixed real coordinates
gives, uniformly over the box family,
\begin{equation}
|\widehat\phi(A)|\ll_r
X^{r\tau}(1+\|A\|)^{-r}.
\label{eq:add-real-decay}
\end{equation}
Indeed each derivative of order \(r\) costs at most a constant times
\(X^{r\tau}\), and all supports are in one compact set.
Lemma~\ref{lem:fourier-shell-estimates} and the bounds for \(p\notin S\) give
\(|\widehat\eta_{\mathfrak h,l}(A)|\ll H^{-1}\).
The first-minimum bound also gives the rough count
\(\#(\Gamma_{l,\mathfrak h}\cap B_R)\ll(1+RP^{1/3})^3\).
For \(R\ge Y\ge1\) this is \(O(R^3P)\), since \(P=l^3H\ge1\).
Dyadic summation of Equation~\eqref{eq:add-real-decay}, for \(r>3\), yields
\begin{equation}
\sum_{\substack{A\in F\\\|A\|>Y}}
|\widehat\phi(A)\widehat\eta_{\mathfrak h,l}(A)|
\ll_r X^{r\tau}l^3Y^{3-r}
\le X^{r\tau+3\theta+w(3-r)}\,O_r(1).
\label{eq:add-real-tail}
\end{equation}
This also proves absolute convergence in Equation~\eqref{eq:add-poisson}.
The profile count costs only \(X^{o(1)}\). When bounding Fourier terms
directly, we may therefore restrict to the ball and use the uniform bound
\(|\widehat\phi|\ll1\) there.
Suppose first that \(D_F\ge X^d\). A nonscalar \(A\) generates \(F\), since
the field has degree three. Moreover \(HR_F\subset\mathfrak h\): the
integer \(H=|R_F/\mathfrak h|\) annihilates this finite group. Thus
\(lHA\in R_F\) for \(A\in\Gamma_{l,\mathfrak h}\). The discriminant of the
order \(\Z[lHA]\) is a positive integer multiple of \(D_F\), whereas the
product of the squared differences of its three real conjugates is
\(O((lHY)^6)\). Existence of a nonscalar frequency in the ball therefore
implies
\[
D_F\ll(lHY)^6,\qquad H\gg X^{d/6-s}.
\]
Apply Equation~\eqref{eq:add-lattice-count} to
\(\Gamma_{l,\mathfrak h}\) and multiply it by \(H^{-1}\). The four resulting
terms are bounded by constants times
\begin{equation}
X^{-d/6+s}
+X^{-d/9+5s/3}
+X^{-d/18+7s/3}
+X^{-d/2+3s}.
\label{eq:add-large-field}
\end{equation}
All have a fixed power saving once \(s<d/100\). If the ball contains no
nonscalar frequency there is nothing to count for that profile.
We now assume \(D_F<X^d\). When \(d\le\lambda/3\),
Equation~\eqref{eq:add-order-lower} gives
\[
[R_F:C_F]=(\Delta_F/D_F)^{1/2}\ge X^{\lambda/3}.
\]
Profiles with \(G_0\le X^e\) are best estimated in their original form
\(S_{\mathfrak h}\). Equation~\eqref{eq:add-profile-indices} implies
\(H\ge X^{\lambda/3-e}\). All supporting points lie in
\(\mathfrak a_{\mathfrak h}\), whose reciprocal norm is
\(O(D_FG_0^{24})\) by Equation~\eqref{eq:add-profile-support}.
The first-minimum bound of Lemma~\ref{lem:add-ideal-lattice}, applied in the
fixed real region, gives \(O(D_FG_0^{24})\) possible points.
At each such point, Equations
\(\eqref{eq:add-profile-norm}\)--\(\eqref{eq:add-profile-inverse}\) give
\[
\left|\prod_{p\in S}K_{p,\mathfrak h_p}(z)\right|
\ll X^{o(1)}
\left(\frac{D_FG_0^{24}}{H}\right)^{1/2}.
\]
Here and below \(X^{o(1)}\) denotes \(O_\eta(X^\eta)\) for every fixed
\(\eta>0\), uniformly in the arithmetic data; it includes factors
\(8^{|S|}\). It follows that
\begin{equation}
|S_{\mathfrak h}|
\ll X^{-\lambda/6+(3/2)d+(73/2)e+o(1)}.
\label{eq:add-small-index-profile}
\end{equation}
The use of the original sum has gained the boundary factor from the inverse
transform. It remains to check that its scalar portion is small as well,
so that this bounds the nonscalar portion.
If a rational scalar has denominator \(p^k\) at \(p\), its denominator
ideal in \(R_{F,p}\) is \(p^kR_{F,p}\); its component exponents are
\(e_{\mathfrak q}k\), so its norm is \(p^{3k}\), including at ramified
primes. Thus a scalar in this profile has its \(S\)-part denominator
\(t=H^{1/3}\), or the profile contains no scalar. Its denominator away
from \(S\) divides \(l\), so there are
\(O(1+YlH^{1/3})\) such rationals in the ball. Their absolute contribution
is at most
\begin{equation}
O\bigl(H^{-1}+YlH^{-2/3}\bigr).
\label{eq:add-small-index-scalars}
\end{equation}
By \(H\ge X^{\lambda/3-e}\) this also saves a power for the choices below.
Equation~\eqref{eq:add-poisson} expresses the full dual sum as
\(S_{\mathfrak h}\). Subtracting its scalar frequencies in the ball and
using Equation~\eqref{eq:add-real-tail} for the scalar tail therefore bounds
the full nonscalar sum for these profiles by
Equations~\eqref{eq:add-small-index-profile}--\eqref{eq:add-small-index-scalars}
and the tail. No positivity of an individual inverse shell is needed.
It remains to treat \(D_F<X^d\) and \(G_0>X^e\). For each frequency in
the denominator shell of the profile choose at each \(p\in S\) one of the shifts \(s_p\) of
Lemma~\ref{lem:fourier-shift}, and write
\[
\alpha_p=A+s_p,\qquad \beta_p=\alpha_p^{-1},\qquad
J=\prod_{p\in S}p^{j_p},
\]
where \(j_p\) is the least nonnegative integer with
\(p^{j_p}\beta_p\in C_{F,p}\). All sums involving \(J\) below are over
frequencies in this profile. The pointwise local estimate now reads
\[
|\widehat\eta_{\mathfrak h,l}(A)|
\ll H^{-1}J^{-1/2}.
\]
For frequencies with \(J\ge X^{e_2}\), the rough lattice count immediately
gives
\begin{equation}
\sum_{\substack{\|A\|\le Y\\J\ge X^{e_2}}}
|\widehat\phi(A)\widehat\eta_{\mathfrak h,l}(A)|
\ll \frac{(1+YP^{1/3})^3}{H}\,X^{-e_2/2}
\ll X^{3s-e_2/2}.
\label{eq:add-large-j}
\end{equation}
We used \(H,l,Y\ge1\). This leaves \(J<X^{e_2}\), where the extra factor in
the Fourier bound may be too small. The order discriminant then forces a
large denominator ideal, and the definition of \(j_p\) supplies congruences
among its possible frequencies.
For a nonscalar \(A\) let
\(\operatorname{disc}(A)=\Disc(1,A,A^2)\in\Q^\times\).
The equality \(F=\Q[A]\) remains
\(F\otimes\Q_p=\Q_p[A]\) after tensoring. Shifting by a scalar and
inverting an element with nonzero components preserve the generated
algebra, so \(p^{j_p}\beta_p\) generates \(F\otimes\Q_p\).
The full-rank order \(\Z_p[p^{j_p}\beta_p]\) is contained in \(C_{F,p}\).
The identities
\[
\operatorname{disc}(\beta_p)
=\frac{\operatorname{disc}(A)}
{\Nm_{F_p/\Q_p}(\alpha_p)^4},
\qquad
\operatorname{disc}(p^{j_p}\beta_p)
=p^{6j_p}\operatorname{disc}(\beta_p)
\]
follow by taking the three differences of conjugates and by scaling.
Discriminant divisibility therefore gives
\[
v_p(\operatorname{disc}(A))
-4v_p(\Nm\alpha_p)+6j_p\ge v_p(\Delta_F).
\]
The shift lemma says \(v_p(\Nm\alpha_p)=-\log_pH_p\) for \(p>5\), and
differs from this by at most three at \(2,3,5\).
For \(p\notin S\), \(lA\) is integral and \(C_{F,p}=R_{F,p}\) is unramified,
so \(v_p(\operatorname{disc}(A))\ge-6v_p(l)\).
Combining these valuation bounds for the nonzero rational
\(\operatorname{disc}(A)\), with only an absolute factor from \(2,3,5\),
gives
\begin{equation}
Y^6\gg |\operatorname{disc}(A)|
\gg \Delta_F\,J^{-6}H^{-4}l^{-6}.
\label{eq:add-frequency-discriminant}
\end{equation}
The first inequality uses \(\|A\|\le Y\).
Thus for \(J<X^{e_2}\), Equations
\(\eqref{eq:add-order-lower}\) and \(\eqref{eq:add-frequency-discriminant}\)
imply
\begin{equation}
H\gg X^{\lambda/4-(3/2)(s+e_2)}.
\label{eq:add-denominator-lower}
\end{equation}
In particular, once \(s+e_2<\lambda/12\) with a margin, every nonscalar
frequency under consideration has \(H\ge X^{\lambda/8}\) for large \(X\).
For the remaining count, group frequencies according to their \(j_p\)'s and
the chosen \(s_p\)'s. The number of exponent lists with \(J<X^{e_2}\) is
\(X^{o(1)}\), by the same Rankin argument as for profiles; the bounded lists
of shifts cost \(4^{|S|}=X^{o(1)}\). Groups containing no nonscalar
frequency contribute zero and are omitted. Fix one of the other groups.
Apply the basis
part of Lemma~\ref{lem:add-ideal-lattice} to \(\Gamma_{l,\mathfrak h}\).
All frequencies in the ball have integer coordinates in a cube
\([-L,L]^3\), where we choose \(L\) comparable to the upper bound
\(YP^{1/3}\). This choice, together with
Equation~\eqref{eq:add-denominator-lower}, gives
\begin{equation}
L\asymp YP^{1/3},\qquad
L\gg X^{\lambda/24},\qquad
L^3/H\ll X^{3s}.
\label{eq:add-coordinate-cube}
\end{equation}
At \(p\in S\) choose a generator \(i_p\) of the maximal-order ideal
\(\mathfrak h_p\), and set \(x=i_p(A+s_p)\). Localizing the global basis
of \(\Gamma_{l,\mathfrak h}\) at \(p\), then multiplying it by \(i_p\), gives
a \(\Z_p\)-basis of \(R_{F,p}\), because \(p\nmid l\). Accordingly the
map from the three integer coordinates to \(x\) is an affine
\(\operatorname{GL}_3(\Z_p)\) change of coordinates. This statement holds
also for a product of ramified local fields.
Smith normal form for \(\mathfrak h_p\cap C_{F,p}\) inside \(\mathfrak h_p\)
gives a primitive linear form \(\ell_p:R_{F,p}\to\Z_p\), meaning that its
image is \(\Z_p\), and an integer
\(g_p'\ge g_p/3\) such that
\[
\ell_p\bigl(i_p^{-1}(\mathfrak h_p\cap C_{F,p})\bigr)
\subset p^{g_p'}\Z_p.
\]
Indeed the three elementary-divisor exponents sum to \(g_p\), and we use
the coordinate with largest exponent. If \(p>5\), the shift lemma makes
\(x\) a unit and \(\beta_p=i_px^{-1}\in\mathfrak h_p\). The condition
\(p^{j_p}\beta_p\in C_{F,p}\) implies
\begin{equation}
\ell_p(x^{-1})\equiv0\pmod {p^k}
\quad\text{for every }1\le k\le(g_p'-j_p)_+.
\label{eq:add-inverse-congruence}
\end{equation}
Inversion permutes the units in the finite algebra
\(R_{F,p}/p^kR_{F,p}\). A primitive linear form has \(p^{2k}\) zeros among
the \(p^{3k}\) additive residues. Thus the unit condition and
Equation~\eqref{eq:add-inverse-congruence} retain at most the fraction
\(p^{-k}\) of all residues modulo \(p^k\). Nilpotents in the finite algebra
do not affect this argument.
For \(p=2,3,5\), the shift lemma gives \(x\in R_{F,p}\) and
\(x^{-1}\in p^{-1}R_{F,p}\). Put \(y=p x^{-1}\). Then
\(p^{j_p}i_py=p(p^{j_p}\beta_p)\) still lies in
\(\mathfrak h_p\cap C_{F,p}\), and the congruence in
Equation~\eqref{eq:add-inverse-congruence} holds for \(y\) in place of
\(x^{-1}\). On the permitted locus the inverse identities
\[
y(x)-y(x')=p x^{-1}(x'-x)(x')^{-1},\qquad
x-x'=p^{-1}x\bigl(y(x')-y(x)\bigr)x'
\]
show respectively that \(x\bmod p^{k+1}\) determines \(y\bmod p^k\), and
that a fixed residue of \(y\bmod p^k\) confines \(x\) to one residue modulo
\(p^{k-1}\). The permitted locus has \(v_{\mathfrak q}(x)=0\) on the pole
components and \(0\le v_{\mathfrak q}(x)\le e_{\mathfrak q}\) on the other
components. These conditions are determined modulo \(p^2\), since a
change in \(p^2R_{F,p}\) has component valuation at least \(2e_{\mathfrak q}\).
At modulus \(p^{k+1}\), each of the at most \(p^{2k}\) residues in
the linear kernel therefore allows at most \(p^6\) residues of \(x\).
The retained fraction is at most \(p^{3-k}\), hence \(O(p^{-k})\) with an
absolute constant at these three primes. This proves the required
congruence fraction with only one extra power of a small prime in its
modulus.
Write \(a_p=(g_p'-j_p)_+\). The available congruence powers satisfy
\begin{equation}
\prod_{p\in S}p^{a_p}
\ge G_0^{1/3}/J
> X^{e/3-e_2}.
\label{eq:add-available-congruences}
\end{equation}
We will choose \(e/3-e_2>2e_1\). If every prime with \(a_p>0\) is at most
\(X^{e_1}\), multiply available factors \(p\), one at a time, until their
product \(M_1\) first reaches \(X^{e_1/2}\). Equation
\(\eqref{eq:add-available-congruences}\) ensures this is possible, and
\[
X^{e_1/2}\le M_1\le X^{3e_1/2}<X^{2e_1}.
\]
Impose just the corresponding congruences. The Chinese remainder theorem
combines their retained fractions into \(O(M_1^{-1})\). Their joint modulus
is \(M_1\) times at most \(2\cdot3\cdot5\), because only the small-prime
tests need one extra power. The affine integral coordinate changes above
preserve these fractions in the global coordinates. If
\(2e_1<\lambda/24\), this joint modulus is \(o(L)\) by
Equation~\eqref{eq:add-coordinate-cube}. Counting each admissible residue
class in the cube then gives
\begin{equation}
O(L^3/M_1)=O(L^3X^{-e_1/2})
\label{eq:add-composite-congruence-count}
\end{equation}
possible frequencies.
Otherwise some available prime satisfies \(p>X^{e_1}\); for large \(X\)
it exceeds \(5\). Use its congruence modulo \(p\). Set
\(B=R_{F,p}/pR_{F,p}\), a three-dimensional commutative \(\mathbf F_p\)-algebra,
possibly with nilpotents, and reduce \(\ell_p\) modulo \(p\). Choose an
\(\mathbf F_p\)-basis of \(B\) and let \(x\) be the universal element of
\(B\otimes_{\mathbf F_p}\mathbf F_p[X_1,X_2,X_3]\) in this basis.
Let \(M_x\) be its multiplication matrix and put
\(\mathcal N_B(x)=\det M_x\). This is a nonzero polynomial of degree three,
since \(\mathcal N_B(1)=1\). On the unit open set \(\mathcal N_B\ne0\),
\[
\ell_p(x^{-1})=\frac{P_\ell(x)}{\mathcal N_B(x)},\qquad
P_\ell(x)=\ell_p\bigl(\operatorname{adj}(M_x)1_B\bigr),
\]
where \(P_\ell\) has degree at most two. This numerator is nonzero.
Indeed the affine space parametrizing elements of the vector space \(B\)
has coordinate ring \(\mathbf F_p[X_1,X_2,X_3]\), which is a domain even
when multiplication in \(B\) has nilpotents. The unit open set is dense,
and inversion is an automorphism of that open set. The nonzero linear
polynomial \(\ell_p\) stays nonzero on the open set and after composition
with inversion. The displayed formula, or injectivity of localization at
\(\mathcal N_B\), then forces \(P_\ell\ne0\).
The affine \(\operatorname{GL}_3(\mathbf F_p)\) change from global lattice
coordinates preserves nonvanishing.
Let \(N_L=2\lfloor L\rfloor+1\asymp L\) be the number of possible integers in
one coordinate of the cube. A nonzero polynomial of bounded degree has
\(O(N_L^2)\) zeros on a product of three residue intervals of length \(N_L<p\);
this follows by induction on the number of variables, viewing it as a
polynomial in the last variable and bounding the slices on which all its
coefficients vanish. If \(N_L\ge p\), partition the cube into boxes of side
at most \(p\) and use the same bound there. The number of global integer
coordinates satisfying \(P_\ell=0\bmod p\) is consequently
\begin{equation}
O\bigl(L^3/p+L^2\bigr)
=O\bigl(L^3(X^{-e_1}+X^{-\lambda/24})\bigr).
\label{eq:add-prime-congruence-count}
\end{equation}
This bounds the subset for which \(x\) is a unit as well.
Multiplying Equations
\(\eqref{eq:add-composite-congruence-count}\)--\(\eqref{eq:add-prime-congruence-count}\)
by \(H^{-1}\), and using Equation~\eqref{eq:add-coordinate-cube}, gives
respectively the bounds
\[
O(X^{3s-e_1/2}),\qquad
O(X^{3s-e_1}+X^{3s-\lambda/24})
\]
for the fixed group of \(j_p\)'s and shifts. Their \(X^{o(1)}\) number is
harmless once these exponents are negative with a margin. This completes
the count of nonscalar frequencies in the ball.
We finish by making the parameter choices and all margins explicit. Put
\(\lambda_0=\min(\lambda,1)\) and take
\begin{equation}
\begin{aligned}
d=e&=\frac{\lambda_0}{1000},&
e_1&=\frac e{100},&
e_2&=\frac{e_1}{100},\\
w&=\frac{e_2}{100},&
\theta=\tau&=\frac w{100},&
\epsilon_F&=\frac{e_2}{4}.
\end{aligned}
\label{eq:add-parameters}
\end{equation}
Then \(s=101e_2/10000\),
\[
s<d/100,\quad d<\lambda/3,\quad
e/3-e_2>2e_1,\quad 2e_1<\lambda/24,\quad
s+e_2<\lambda/12.
\]
The saving in Equation~\eqref{eq:add-small-index-profile} is at least
\[
\lambda/6-(3/2)d-(73/2)e
\ge (1/6-0.038)\lambda_0,
\]
and Equation~\eqref{eq:add-small-index-scalars} has a still larger margin
than \(\epsilon_F\), since \(e\le\lambda/1000\).
All four exponents in Equation~\eqref{eq:add-large-field} also have margins
at least \(29d/900>\epsilon_F\), by \(s<d/100\).
Among the remaining ball estimates the smallest
margin is in Equation~\eqref{eq:add-large-j}, namely
\[
e_2/2-3s=\frac{4697}{10000}e_2>\epsilon_F;
\]
the congruence margins \(e_1/2-3s\), \(e_1-3s\), and
\(\lambda/24-3s\) are larger. With \(r=100\),
the exponent in Equation~\eqref{eq:add-real-tail} is
\[
r\tau+3\theta+w(3-r)=-\frac{9597}{10000}e_2.
\]
All \(X^{o(1)}\) costs from profiles, shifts, and exponent lists can thus be
absorbed while retaining \(\epsilon_F=e_2/4\). This proves
Equation~\eqref{eq:add-nonscalar}. Finally
\(2\tau+3\theta=e_2/2000<\epsilon_F\), as asserted.
\end{proof}
\subsection{The scalar term and completion}
\begin{proof}[Proof of Proposition~\ref{prop:additive-equidistribution}]
Use the constants in Equation~\eqref{eq:add-parameters}.
Equations~\eqref{eq:add-original-profiles} and \(\eqref{eq:add-poisson}\),
together with Lemma~\ref{lem:add-nonscalar}, leave the scalar frequencies
with absolute error \(O(X^{-\epsilon_F})\). We first remove the profile
cutoff from those frequencies.
For \(a\in\Q\), let \(t=\prod_{p\in S}p^{\max(0,-v_p(a))}\) be its
\(S\)-part denominator, with \(t=1\) for \(a=0\). As noted above, its
unique profile has \(H=t^3\).
Lemma~\ref{lem:fourier-scalar-density} gives the exact product
\(\prod_{p\in S}\gamma_p(a)=t^{-3}\), including at \(2,3,5\).
The other denominator divides \(l\), so all such scalars with a given \(t\)
lie in \((tl)^{-1}\Z\). The change of real variables
\((z_i,z_j,z_k)\mapsto(z_i,z_j,\Tr z)\) has absolute Jacobian one.
Writing \(\mathbf 1=(1,1,1)\), it gives the exact identity
\[
\widehat\phi(a\mathbf 1)=A_\phi\,\widehat\chi(a).
\]
For any fixed \(r>1\), rapid decay of the fixed function \(\chi\) implies
\(\sum_{n\in\Z}|\widehat\chi(n/(tl))|\ll_r tl\).
The scalars omitted by \(H\le X^{C_3}\) therefore contribute at most
\begin{equation}
\sum_{t>X^{C_3/3}}t^{-3}
\sum_{n\in\Z}|\widehat\phi((n/(tl))\mathbf 1)|
\ll A_\phi\,l\,X^{-C_3/3}.
\label{eq:add-scalar-tail}
\end{equation}
We have enlarged the set of \(t\)'s to all positive integers, which is an
upper bound. With \(C_3=371\), this is \(O(X^{-\epsilon_F})\), since
\(A_\phi\ll1\), \(l\le X^\theta\), and \(371/3-\theta>\epsilon_F\).
The same calculation without the cutoff proves
absolute convergence of the unrestricted scalar series.
That series is one-dimensional Poisson summation for the trace marginals.
At infinity the marginal is \(A_\phi\chi(t)\). At \(p\in S\) it is
\(d_p(t)\mathbf 1_{\Z_p}(t)\). At \(p\mid l\), the onto integral trace map
and two kernel coordinates show that the indicator
\(z_p^0+p^{v_p(l)}R_{F,p}\) has trace density
\[
p^{-2v_p(l)}\mathbf 1_{m+p^{v_p(l)}\Z_p}(t)
\]
with respect to Haar measure of mass one on \(\Z_p\). At every other
finite prime the marginal is \(\mathbf 1_{\Z_p}\).
Their Fourier transforms at a rational \(a\) are exactly the finite factors
in Equation~\eqref{eq:add-poisson} evaluated at the scalar \(a\), because
\(\Tr(az)=a\Tr z\).
For completeness, this one-dimensional application of Poisson does not
require a smoothness assumption on \(d_p\). Its Fourier coefficients on
\(\Q_p/\Z_p\) of denominator \(p^k\) are \(p^{-3k}\), and there are
\(p^k-p^{k-1}\) such characters. Their absolute sum is finite.
Truncating each expansion gives a compactly supported locally constant
density on \(\Z_p\), to which Poisson applies. The densities converge
uniformly by Lemma~\ref{lem:fourier-scalar-density}, and the preceding
absolute scalar bound permits passage to the limit on the Fourier side.
On the original side, the finite marginals force the rational trace to be
an integer. Only \(m\) meets the real cutoff. Hence the unrestricted
scalar contribution is precisely
\[
A_\phi\,l^{-2}\prod_{p\in S}d_p(m).
\]
Lemma~\ref{lem:fourier-scalar-density} bounds the product of \(d_p(m)\)
above and below by positive absolute constants. Also
\(A_\phi\gg X^{-2\tau}\) by Equation~\eqref{eq:add-box}, and
\(l^{-2}\ge X^{-2\theta}\). Dividing the combined absolute error
\(O(X^{-\epsilon_F})\) by the main term therefore costs at most
\(O(X^{2\tau+2\theta})\). Since
\(\epsilon_F>2\tau+3\theta\), the relative error is \(O(X^{-\theta})\).
Every estimate used fixed real derivative and support bounds and constants
independent of the fields, orders, residues, and box locations and scales.
This proves Equation~\eqref{eq:add-equidistribution} with its stated
uniformity.
\end{proof}
==== END Equidistribution-of-Primitive-Quartic-Torus-Packets-for-Arbitrary-Orders//additive.tex ====
==== BEGIN Equidistribution-of-Primitive-Quartic-Torus-Packets-for-Arbitrary-Orders//conclusion.tex ====
\section{Exclusion of the obstruction}\label{sec:conclusion}
We apply Theorem~\ref{thm:exterior-count} to the exterior planes in
Proposition~\ref{prop:rig-obstruction}. Four small exterior coordinates
give two small signed products, whose range in the trace plane has area
of order $r^4$. The integral along a diagonal orbit contributes
logarithmic factors; we will show that they are summable at this scale.
\subsection{The exterior vector integral}
Fix a partition $b_1=ij|kl$ of $\{1,2,3,4\}$, and denote the other two
partitions by $b_2,b_3$. For $m\in\Z\setminus\{0\}$, $L>1$, and
$0<r<1$, define $N_{m,L,r}(\Lambda)$ to be the number of
$v\in\bigwedge^2\Lambda$ satisfying
\begin{equation}\label{eq:end-vector-box}
Q(v)=m,\qquad |v_{ij}|,|v_{kl}|<L,\qquad
|v_{ab}|<r\quad(ab\ne ij,kl).
\end{equation}
This count is finite because $\bigwedge^2\Lambda$ is a lattice in
$\bigwedge^2\R^4$ and the specified box is bounded. To see that it is
Borel, choose a local continuous determinant-one basis around a lattice
with $k$ witnesses and keep their distinct integer exterior labels fixed.
They remain distinct, and their $Q$-values remain $m$ because
$Q((\bigwedge^2g)w)=\det(g)Q(w)$. Their coordinates vary continuously,
so the strict inequalities persist in a neighborhood. Thus
$\{N_{m,L,r}\ge k\}$ is open for every integer $k\ge1$.
Recall that the three real embeddings of $F$ are indexed by
$\mathcal B$. For a nonzero $z\in F$, set
\begin{equation}\label{eq:end-logarithms}
\mathcal L_{L,r}(z)
=\log_+\frac{L^2}{|z_{b_1}|}\,
\log_+\frac{r^2}{|z_{b_2}|}\,
\log_+\frac{r^2}{|z_{b_3}|}.
\end{equation}
All three conjugates of a nonzero element of the field $F$ are nonzero,
so the expression is defined.
In Equation~\eqref{eq:ext-multiplicity}, $M(z)$ sums over
the fixed maximal-order ideal representatives $I$ and the $U^+$-orbits
of nonzero raw $w\in\bigwedge^2_{\Z}I$ with $\Psi(w)/\Nm I=z$. Each
orbit has weight $\prod_pP_p(I,w)$, the average membership of $w$ in
$\bigwedge^2_{\Z}L'$ over the local labels
$L'\in\mathscr L_{\OO}(I)$.
\begin{lemma}[Exterior unfolding]\label{lem:end-unfold}
For every primitive totally real quartic field $K$, order $\OO$, and
ordering of its real embeddings, let $d_U=[U:U^+]$. With the preceding
choices of $b_1,m,L,r$, use the same fixed system of maximal-order ideal
representatives in the packet model and in the definition of $M$. Then
\begin{equation}\label{eq:end-unfold}
\int_{X_4}N_{m,L,r}(\Lambda)\,d\mu_{\OO,\sigma}(\Lambda)
=\frac{4}{d_U\sqrt D\,\kappa}
\sum_{t\in\{m,-m\}}
\sum_{\substack{z\in F\\\Tr z=t}}
M(z)\mathcal L_{L,r}(z).
\end{equation}
The index satisfies $2\le d_U\le16$.
\end{lemma}
\begin{proof}
We use the packet integral in Equation~\eqref{eq:pkt-integration}.
For its maximal-order ideal representative $I$ and local label
$L'\in\mathscr L_{\OO}(I)$, the normalized lattice at the sign vector
$s=(s_1,\ldots,s_4)\in\{\pm1\}^4$ and the logarithmic parameter
$(u_1,\ldots,u_4)\in\R^4$ with $u_1+\cdots+u_4=0$ is
\[
(X\Nm I)^{-1/4}\sigma(L')
\diag(s_1e^{u_1},\ldots,s_4e^{u_4}).
\]
For a raw vector $w\in\bigwedge^2 I$, its normalized exterior coordinates
are
\[
y_{ab}=(X\Nm I)^{-1/2}s_as_b e^{u_a+u_b}w_{ab}.
\]
If $z=\Psi(w)/\Nm I$ and $\epsilon(s)=\prod_a s_a$, then
Equation~\eqref{eq:ext-signed-products} gives
\begin{equation}\label{eq:end-signed-products}
(y_{12}y_{34},-y_{13}y_{24},y_{14}y_{23})
=\epsilon(s)z,\qquad
Q(y)=\epsilon(s)\Tr z.
\end{equation}
There are eight sign vectors for each value of $\epsilon(s)$.
The kernel of the signature map on $U$ is $U^+$, so
$d_U=|\operatorname{sgn}U|\le16$; the signature of $-1$ gives
$d_U\ge2$. After averaging over $L'$, the vector-count integrand in
Equation~\eqref{eq:pkt-integration} is invariant under
$v\mapsto\sigma(e)v$ for $e\in U$, since $L'\mapsto eL'$ permutes the
labels and transports their exterior lattices. Replacing
the fundamental domain for $U$ in $T_\infty$ by one for $U^+$ therefore
multiplies its integral by $d_U$, which we compensate by division by
$d_U$. The latter domain consists of all sign components, with a
fundamental domain for the positive unit logarithms in each component.
For a given raw $w$, its average membership over the local labels is
\[
\frac1{N_{\OO}}\sum_{L'\in\mathscr L_{\OO}(I)}
\mathbf 1_{\{w\in\bigwedge^2L'\}}
=\prod_p P_p(I,w),
\]
the weight in Equation~\eqref{eq:ext-multiplicity}. This weight is constant on its
$U^+$-orbit. That action is free: every embedding coordinate of the
nonzero raw vector $w\in\bigwedge^2_{\Z}I\subset\bigwedge^2_{\Q}K$ is
nonzero by Lemma~\ref{lem:ext-products}. A stabilizing positive unit
$e$ would therefore satisfy $\sigma_a(e)\sigma_b(e)=1$ for every pair
$ab$, forcing $e=1$. Summing the translates of each raw vector thus unfolds the
positive-unit fundamental domain to all of the three-dimensional space
$u_4=-u_1-u_2-u_3$, with multiplicity one. The packet normalization
remaining after the uniform label average is
$h_K\vol(\mathscr F_U)=\sqrt D\,\kappa$ by
Equation~\eqref{eq:pkt-unit-volume}. Hence this unfolding has coefficient
$1/(d_U\sqrt D\,\kappa)$ before the sign sum and the integral in $u$.
All the sums and integrals are nonnegative, so Tonelli's theorem applies.
Choose one coordinate from each opposite pair. The logarithms of the
absolute values of these coordinates are affine functions of
$(u_1,u_2,u_3)$. For the choices $12,13,14$, their linear parts are
\[
u_1+u_2,\qquad u_1+u_3,\qquad -u_2-u_3.
\]
Their matrix has determinant of absolute value two. Choosing the
opposite member of a pair negates its row, and ordering the partitions
only permutes rows, so the same absolute determinant holds for every
choice. If the two entries in an opposite pair have product of absolute
value $|z_b|$ and both must be less than $h>0$ in absolute value, the
logarithm of the chosen entry lies in an interval of length
$\log_+(h^2/|z_b|)$. Therefore, for a raw vector with the required
trace after the sign change, its integral over $(u_1,u_2,u_3)$ is
$\frac12\mathcal L_{L,r}(z)$.
The $U^+$-orbit sum of the weights, including the sum over $I$, is exactly
$M(z)$ by Equation~\eqref{eq:ext-multiplicity}. Equation
\eqref{eq:end-signed-products} allows raw trace $m$ for the eight positive
determinant signs and raw trace $-m$ for the other eight. Multiplying
eight by the Jacobian factor $1/2$ gives the factor $4$ in
Equation~\eqref{eq:end-unfold}. The unit $-1$, although it acts trivially
on exterior vectors, has already been accounted for in the index $d_U$.
\end{proof}
\begin{proposition}[Small exterior boxes]\label{prop:end-tube}
For every fixed $m\in\Z\setminus\{0\}$ and $L>1$, there are constants
$r_0>0$ and $C_{m,L}>0$ with the following property. For every sequence
of the fields, orders, and embedding orderings in Theorem~\ref{thm:main}
with $X_i\to\infty$, and every fixed $0<r<r_0$,
\begin{equation}\label{eq:end-tube}
\limsup_{i\to\infty}
\int_{X_4}N_{m,L,r}(\Lambda)\,
d\mu_{\OO_i,\sigma_i}(\Lambda)
\le C_{m,L}r^4.
\end{equation}
The constants are independent of the sequence, of $r$, and of the choice
of partition $b_1$.
\end{proposition}
\begin{proof}
Take $r_0<1$ so small that $2r_0^2\le |m|/2$. For each
$t\in\{m,-m\}$, fix the coordinate rectangle
\[
\Omega_t=\{z\in\mathcal H_t:|z_{b_2}|,|z_{b_3}|\le1\}.
\]
Now let $0<r<r_0$. In the trace-$t$ sum in
Equation~\eqref{eq:end-unfold}, only points with
$|z_{b_2}|,|z_{b_3}|<r^2$ contribute, and these points lie in $\Omega_t$.
Since their three coordinates have sum $t$,
\[
|z_{b_1}|\ge |m|-2r^2\ge |m|/2.
\]
Thus the first logarithm in Equation~\eqref{eq:end-logarithms} is bounded
in terms of $m,L$ on the contributing points, independently of $r$.
For every $z\in\Omega_t$ with $M(z)>0$, Lemma~\ref{lem:ext-support} gives
$|z_b|\gg_{\Omega_t}X^{-84}$ for each $b$. Hence the last two logarithms
in Equation~\eqref{eq:end-logarithms} are $O(\log X)$, uniformly for
$0<r<1$.
We first remove the points in the two strips
\begin{equation}\label{eq:end-strips}
|z_{b_2}|\le4X^{-\tau}
\quad\text{or}\quad
|z_{b_3}|\le4X^{-\tau}.
\end{equation}
For large $X$, each strip within $\Omega_t$ is covered by a rectangle
of width $8X^{-\tau}$ and length $2$ in the coordinates
$z_{b_2},z_{b_3}$, and these rectangles lie in $\Omega_t$. Both sides
meet the minimum length in
Theorem~\ref{thm:exterior-count} for large $X$. That theorem bounds the
sum of $M(z)$ over their union, divided by $\sqrt D\,\kappa$, by
$O(X^{-\tau})$. The logarithmic bounds just proved show that their
contribution to Equation~\eqref{eq:end-unfold} is
\[
O_{m,L}\bigl(X^{-\tau}(\log X)^2\bigr)=o(1).
\]
The use of the rectangle estimate here controls the weighted mass
directly; it does not require a pointwise bound for $M(z)$.
For the remaining points, separate the signs of $z_{b_2},z_{b_3}$ and
use the dyadic intervals
\[
r^2 2^{-a-1}<|z_{b_2}|\le r^2 2^{-a},\qquad
r^2 2^{-b-1}<|z_{b_3}|\le r^2 2^{-b},
\qquad a,b\ge0.
\]
Each signed dyadic rectangle lies in $\Omega_t$.
Each interval has width half its upper endpoint. If it meets
$|z_{b_j}|>4X^{-\tau}$, that width exceeds $2X^{-\tau}$. Hence every
dyadic rectangle meeting the remaining region satisfies the minimum
side lengths of Theorem~\ref{thm:exterior-count}; any boundary rectangle
may be included in full. Its area is $O(r^4 2^{-a-b})$, while the last
two factors of Equation~\eqref{eq:end-logarithms} are at most
$(a+1)\log2$ and $(b+1)\log2$. Theorem~\ref{thm:exterior-count} gives
\[
\frac1{\sqrt D\,\kappa}
\sum_{z\text{ in that rectangle}} M(z)
\ll_{m,L} r^4 2^{-a-b}.
\]
The sum over the four choices of signs and over all $a,b$ is therefore
at most
\[
C_{m,L}r^4
\sum_{a,b\ge0}(a+1)(b+1)2^{-a-b}
\ll_{m,L}r^4.
\]
All constants are independent of $r$: the rectangles $\Omega_t$ and the
uniform rectangle constant were fixed before choosing $r$. The two
strips tend to zero for each fixed $r$, and $4/d_U\le2$ in
Equation~\eqref{eq:end-unfold}. These observations prove
Equation~\eqref{eq:end-tube} for both traces and all three partitions.
\end{proof}
\subsection{Identification of every probability limit}
\begin{proof}[Proof of Theorem~\ref{thm:main}]
For the sequence in that theorem,
$X_i=\sqrt{|\Disc(\OO_i)|}$ tends to infinity. By
Proposition~\ref{prop:ord-tightness}, the packet probabilities are tight,
and every subsequence has a further weakly convergent subsequence with
an $A_4$-invariant probability limit $\mu$.
Proposition~\ref{prop:ord-balls} gives the local ball bound required by
Corollary~\ref{cor:rig-positive-components}. Hence almost every component in
the $A_4$-ergodic decomposition of $\mu$ has positive entropy for a
diagonal flow. Proposition~\ref{prop:rig-obstruction} reduces the
identification of these components to the exclusion of its three
exterior-plane events.
Fix a partition $b_1$, a nonzero integer $m$, and $L>1$. The event
\[
E_{m,L,r}=\{\Lambda\in X_4:N_{m,L,r}(\Lambda)\ge1\}
\]
is open by the observation following Equation~\eqref{eq:end-vector-box}.
The open-set inequality for weak convergence and
Proposition~\ref{prop:end-tube} now give, along the convergent subsequence,
\[
\mu(E_{m,L,r})
\le\liminf_i\mu_{\OO_i,\sigma_i}(E_{m,L,r})
\le\limsup_i\int N_{m,L,r}\,d\mu_{\OO_i,\sigma_i}
\le C_{m,L}r^4
\]
for every sufficiently small fixed $r>0$. In particular,
\[
\mu\left(\bigcap_{n\ge2}E_{m,L,1/n}\right)
\le \inf_{\substack{n\ge2\\1/n<r_0}} C_{m,L}n^{-4}=0.
\]
Every vector in one of the obstruction planes with $Q\ne0$ has a
nonzero integer $Q$-value by Equation~\eqref{eq:rig-Q-integral}, and its
two entries are less than some integer $L\ge2$ in absolute value.
An exact witness belongs to $E_{m,L,1/n}$ for every $n\ge2$. Thus the
Borel event in Equation~\eqref{eq:rig-events} satisfies
\[
\mathcal E_{b_1}
\subset
\bigcup_{m\in\Z\setminus\{0\}}\ \bigcup_{\substack{L\in\Z\\L\ge2}}
\bigcap_{n\ge2}E_{m,L,1/n}.
\]
The right side is a countable union of null sets, so
$\mu(\mathcal E_{b_1})=0$. The same argument applies to
all three partitions. Integrating these null events over the ergodic
decomposition shows that the proper alternative in
Proposition~\ref{prop:rig-obstruction} has zero total component weight.
Almost every component is therefore $m_4$, and hence $\mu=m_4$.
Every subsequential probability limit is now identified as $m_4$.
Tightness implies that the entire sequence converges weakly to $m_4$,
which gives the asserted convergence against every $f\in C_c(X_4)$.
The same tightness statement supplies, for each $\epsilon>0$, one
compact set carrying at least $1-\epsilon$ of every sufficiently late
packet probability. This is the required no-escape assertion.
\end{proof}
==== END Equidistribution-of-Primitive-Quartic-Torus-Packets-for-Arbitrary-Orders//conclusion.tex ====
==== BEGIN Equidistribution-of-Primitive-Quartic-Torus-Packets-for-Arbitrary-Orders//exterior.tex ====
\section{The cubic resolvent and exterior multiplicities}
\label{sec:exterior}
The obstruction in Section~\ref{sec:rigidity} is detected by an exterior
vector concentrated in one pair of opposite coordinates. We shall count
such vectors through their three signed opposite products. After packet
normalization these products form a triple in a cubic field, and
concentration near one exterior plane confines
two of them to a small rectangle. Our goal is a uniform count in that
rectangle. We first define the count and reduce its multiplicities to local
densities. We then construct the integral quadratic structure that will
control those densities in the next two sections.
\subsection{The three products and the counting problem}
Let \(\widetilde K\subset\R\) be the normal closure of \(K\), and let
\(G=\Gal(\widetilde K/\Q)\) act on the four ordered embeddings of \(K\).
This permutation action is primitive: a nontrivial block would give a field
strictly between \(\Q\) and \(K\). A transitive subgroup of \(S_4\) has
order \(4,8,12\), or \(24\). A subgroup of order \(4\) or \(8\) is contained
in a Sylow \(2\)-subgroup, which preserves a partition into two pairs.
It follows that
\[
G=\mathfrak A_4\quad\hbox{or}\quad G=S_4.
\]
Both groups act transitively on the six two-element subsets of
\(\{1,2,3,4\}\) and on the set of pair partitions
\[
\mathcal B=\{12|34,\ 13|24,\ 14|23\}.
\]
Define
\begin{equation}
F=\left\{(z_b)_{b\in\mathcal B}\in\widetilde K^{\mathcal B}:
g(z_b)=z_{gb}\ \text{for all }g\in G,\ b\in\mathcal B\right\}.
\label{eq:ext-resolvent}
\end{equation}
Evaluation at any partition identifies this algebra with the fixed field
of its stabilizer. Thus \(F\) is a totally real cubic field, and its three
real embeddings are the coordinates in Equation~\eqref{eq:ext-resolvent}.
In particular, \(\Tr_{F/\Q}z=\sum_b z_b\). This is the classical cubic
resolvent from the action on pair partitions; for its relation to cubic
resolvent rings, see \cite[Section~2.3]{Bhargava2004III}. We write
\(R_F=\OO_F\) and \(D_F=|\Disc(F)|\).
Put
\[
V=\bigwedge\nolimits^2_{\Q}K,\qquad
L_V=\bigwedge\nolimits^2_{\Z}\OO.
\]
For \(w=\sum_r x_r\wedge y_r\in V\), its embedding coordinates are
\[
w_{ij}=\sum_r
\bigl(\sigma_i(x_r)\sigma_j(y_r)-\sigma_j(x_r)\sigma_i(y_r)\bigr)
\quad(i<j),\qquad w_{ji}=-w_{ij}.
\]
Choose a basis \(\beta_1,\ldots,\beta_4\) of \(\OO\) whose real embedding
matrix \(A_{\OO}=(\sigma_i(\beta_j))\) has positive determinant. Write
\(\delta_{\OO}=\det A_{\OO}\in\widetilde K\). Its selected real value is
\(X=q\sqrt D\), and
\begin{equation}
\delta_{\OO}^{\,2}=\Disc(\OO)=X^2.
\label{eq:ext-embedding-determinant}
\end{equation}
Define the quadratic expression
\begin{equation}
\Psi(w)=\delta_{\OO}^{-1}
(w_{12}w_{34},-w_{13}w_{24},w_{14}w_{23}).
\label{eq:ext-product-map}
\end{equation}
\begin{lemma}[The product map]
\label{lem:ext-products}
The expression in Equation~\eqref{eq:ext-product-map} defines a rational
quadratic map \(\Psi:V\to F\). In the exterior basis of \(L_V\) induced
by \((\beta_i)\), write \(w=\sum_{i<j}t_{ij}\beta_i\wedge\beta_j\).
Then
\begin{equation}
Q_V(w):=\Tr_{F/\Q}\Psi(w)
=t_{12}t_{34}-t_{13}t_{24}+t_{14}t_{23}.
\label{eq:ext-integral-form}
\end{equation}
If \(w\in V\) is nonzero, all six \(w_{ij}\), and hence all three
components of \(\Psi(w)\), are nonzero. For every extension \(k/\Q\),
\(a\in(K\otimes_\Q k)^\times\), \(c\in k\), and \(w\in V\otimes_\Q k\),
\begin{equation}
\Psi\bigl(c(\bigwedge\nolimits^2 a)w\bigr)
=c^2\Nm_{K\otimes k/k}(a)\Psi(w),
\label{eq:ext-covariance}
\end{equation}
where \(\bigwedge^2a\) is the exterior square of multiplication by \(a\).
\end{lemma}
\begin{proof}
An element \(g\in G\) permutes the rows of \(A_{\OO}\), so it multiplies
\(\delta_{\OO}\) by the sign of that permutation. Permuting the four
indices in the three signed products in
Equation~\eqref{eq:ext-product-map} permutes the partitions and introduces
the same common sign. Division by \(\delta_{\OO}\) gives precisely the
equivariance in Equation~\eqref{eq:ext-resolvent}. This is an identity of
quadratic polynomials in the rational coordinates of \(w\), and therefore
defines a rational quadratic map to \(F\).
The Pfaffian identity for the exterior square of a four-dimensional
linear map is
\[
Q\bigl((\bigwedge\nolimits^2 A_{\OO})t\bigr)
=\det(A_{\OO})Q(t),
\qquad
Q(t)=t_{12}t_{34}-t_{13}t_{24}+t_{14}t_{23}.
\]
It follows either by expanding the six minors or by evaluating
\(\frac12 t\wedge t\) in the top exterior power. This proves
Equation~\eqref{eq:ext-integral-form}. If a coordinate of a rational
\(w\) vanishes, Galois transitivity on the unordered pairs makes all six
coordinates vanish. The exterior embedding map is injective, so \(w=0\).
Finally, multiplication by \(a\) multiplies \(w_{ij}\) by \(a_i a_j\)
after splitting, where \(a_i\) are its four embeddings. Each opposite
product is multiplied by \(\prod_i a_i=\Nm(a)\), and the scalar \(c\)
contributes \(c^2\). This proves Equation~\eqref{eq:ext-covariance}
after splitting and hence over \(k\).
\end{proof}
We relate these products to the packet model of Section~\ref{sec:packets}.
Let \(I\) be one of its maximal-order ideal-class representatives, let
\(L\subset I\) be one of its local labels, and let
\(v=(v_i)\in T_\infty\). If \(w\in\bigwedge^2_{\Z}L\), the corresponding
exterior vector in the normalized lattice has coordinates
\[
\widehat w_{ij}=(X\Nm I)^{-1/2}w_{ij}v_i v_j .
\]
Since \(\prod_i|v_i|=1\), the number
\(\epsilon(v)=\prod_i v_i\) is \(1\) or \(-1\). Thus
\begin{equation}
(\widehat w_{12}\widehat w_{34},
-\widehat w_{13}\widehat w_{24},
\widehat w_{14}\widehat w_{23})
=\epsilon(v)\,z(I,w),\qquad
z(I,w):=\frac{\Psi(w)}{\Nm I}.
\label{eq:ext-signed-products}
\end{equation}
In particular \(Q(\widehat w)=\epsilon(v)\Tr z(I,w)\). This records
exactly how the diagonal signs act on the products.
Lemma~\ref{lem:ext-products} and the normalization above show that every
lattice in a packet lies outside each \(\mathcal E_b\). To prove that weak
limits give these events zero mass, we need quantitative control of exterior
vectors near \(V_b\); the count below supplies the arithmetic input for that
control.
For a rational prime \(p\), fix a generator \(I_p=\xi_pR_p\) as in the
packet model, and give \(R_p^\times\) its Haar probability. For
\(0\ne w\in\bigwedge^2_{\Z}I\), define
\begin{equation}
P_p(I,w)=\int_{R_p^\times}
\mathbf1_{\bigwedge^2_{\Z_p}(\xi_p t\OO_p)}(w)\,d^\times t .
\label{eq:ext-label-probability}
\end{equation}
Changing \(\xi_p\) by an \(R_p\)-unit leaves this probability unchanged.
For \(p\nmid q\), it is \(1\). Hence \(\prod_p P_p(I,w)\) is a finite
product. Multiplication by \(U^+\) preserves these probabilities by
translation of the local unit variables. For \(e\in U^+\), the algebraic
norm \(\Nm_{K/\Q}(e)=1\), so Equation~\eqref{eq:ext-covariance} also
shows that \(z(I,w)\) is unchanged. We use \(U^+\) to unfold the positive
real parameters on each choice of signs.
Its action on nonzero exterior vectors is free: if a totally positive
unit \(e\) fixes \(w\), then \(e_i e_j=1\) for every pair by
Lemma~\ref{lem:ext-products}; the four positive \(e_i\) must all be \(1\).
For \(z\in F\), define the nonnegative weighted multiplicity
\begin{equation}
M(z)=
\sum_{\substack{I,\;
w\in(\bigwedge^2_{\Z}I\setminus\{0\})/U^+\\
z(I,w)=z}}
\prod_p P_p(I,w).
\label{eq:ext-multiplicity}
\end{equation}
The sum is initially allowed to have value \(+\infty\); the local bounds
below prove its finiteness on the nonzero trace slices used here.
It depends on \(K,\OO\), the ordering of
the embeddings, and the chosen system of maximal-order ideal
representatives, which we suppress from the notation. In fact replacing
\(I\) by \(aI\) and transporting \(w\) by \(\bigwedge^2a\) multiplies
that class's products by \(\operatorname{sgn}(\Nm_{K/\Q}a)\), since
\(\Nm(aI)=|\Nm_{K/\Q}a|\Nm I\). We keep the representative system fixed.
A term of positive
weight admits a local label at every prime. These labels patch to a
global \(L\), and local membership implies \(w\in\bigwedge^2_{\Z}L\).
Equation~\eqref{eq:ext-signed-products}, with positive signs, then shows
that \(\Tr z(I,w)\) is the integral value of \(Q\) on a unimodular
exterior lattice. Thus
\begin{equation}
M(z)>0\quad\Longrightarrow\quad
z\in F^\times,\qquad \Tr_{F/\Q}z\in\Z.
\label{eq:ext-trace-support}
\end{equation}
For \(m\in\Z\setminus\{0\}\), put
\[
\mathcal H_m=\{(y_b)_{b\in\mathcal B}\in\R^{\mathcal B}:
\textstyle\sum_b y_b=m\}.
\]
A rectangle in \(\mathcal H_m\) means the inverse image of a product of
two intervals under any two distinct partition coordinates. Its area
is the product of those interval lengths. The following estimate is
the arithmetic input needed to exclude the obstruction.
\begin{theorem}[Planar exterior count]
\label{thm:exterior-count}
There is an absolute \(\tau>0\) with the following property. Fix
\(m\in\Z\setminus\{0\}\) and a bounded set
\(\Omega\subset\mathcal H_m\). For every primitive totally real quartic
field \(K\), every order \(\OO\subset K\), every ordering of its real
embeddings, and every chosen system of fractional \(R\)-ideal
representatives for the ordinary classes, let \(M\) be defined by
Equation~\eqref{eq:ext-multiplicity}.
For all sufficiently large \(X\), every rectangle
\(\mathcal R\subset\Omega\) whose two side lengths are at least
\(X^{-\tau}\) satisfies
\begin{equation}
\sum_{\substack{z\in F,\ \Tr z=m\\z\in\mathcal R}}M(z)
\ll_{m,\Omega}\kappa\sqrt D\,\operatorname{area}(\mathcal R).
\label{eq:ext-planar-count}
\end{equation}
The implied constant and the threshold for \(X\) are uniform in the
field, the order, the embedding ordering, the ideal representative system,
the position of the rectangle, and its side lengths and aspect ratio.
\end{theorem}
Here and below we identify \(z\in F\) with its real triple. To see the
scale of this statement, suppose \(Q(\widehat w)=m\) and the four
coordinates outside \(12|34\) have absolute value less than \(r\).
The signed products of \(\widehat w\) then have the form
\((m-y_2-y_3,y_2,y_3)\) with \(|y_2|,|y_3|<r^2\). Their region has area
\(O(r^4)\). Equation~\eqref{eq:ext-signed-products} allows either trace
\(m\) or \(-m\) for \(z(I,w)\), so the theorem is used at both fixed
traces. Sections~\ref{sec:local-fourier}--\ref{sec:sieve} prove
Theorem~\ref{thm:exterior-count}; the remainder of this section supplies
its local and algebraic inputs.
\subsection{The local product fibers}
Fix \(m\ne0\) for the rest of the section. Choose once a sufficiently
large absolute \(p_0\), and put
\begin{equation}
S=\{p:p\mid Dqm\}\ \cup\ \{p:p\le p_0\}.
\label{eq:ext-prime-set}
\end{equation}
For fixed \(m\),
\(\prod_{p\in S}p\ll_m Dq\le X^2\). At primes in \(S\) we shall keep
the exact additive density of the products. Outside \(S\), integral
valuations give a simpler bound.
The following comparison explains what a fiber remembers about an exterior
vector. For \(k=\Q\) or \(k=\Q_p\), put \(K_k=K\otimes_\Q k\) and
\(V_k=V\otimes_\Q k\), and take embedding coordinates over a splitting
field of the etale algebra \(K_k\).
\begin{lemma}[Comparison of two product fibers]
\label{lem:ext-comparison}
Let \(w,w'\in V_k\) have all six embedding coordinates nonzero. Suppose
that their coordinate ratios \(r_{ij}=w'_{ij}/w_{ij}\) satisfy
\[
r_{12}r_{34}=r_{13}r_{24}=r_{14}r_{23}=s\in k^\times.
\]
There exist \(a\in K_k^\times\) and \(c\in k^\times\) such that
\[
w'=c(\bigwedge\nolimits^2 a)w,\qquad c^2\Nm_{K_k/k}(a)=s.
\]
The pair \((a,c)\) is unique up to
\((a,c)\mapsto(da,d^{-2}c)\), \(d\in k^\times\).
\end{lemma}
\begin{proof}
Over the splitting field set
\[
c_0=\frac{r_{12}r_{13}}{r_{23}},\qquad
(a_1,a_2,a_3,a_4)=
\left(1,\frac{r_{23}}{r_{13}},\frac{r_{23}}{r_{12}},
\frac{r_{14}r_{23}}{r_{12}r_{13}}\right).
\]
The three equal opposite products give \(r_{ij}=c_0a_i a_j\) for every
pair. If two such factorizations are divided, all products of two of
the four coordinatewise quotients are equal. Comparing pairs that
share one index shows that the four quotients are equal. Hence the
projective tuple \([a_1:a_2:a_3:a_4]\) is uniquely determined.
The ratios \(r_{ij}\) are Galois-equivariant without alternating signs.
Uniqueness makes the projective tuple equivariant for the permutation
action on the four embeddings. Express it in a \(k\)-basis of \(K_k\).
The resulting projective coefficient vector is fixed by the usual
Galois action. Dividing by one nonzero coefficient makes every
coefficient fixed, hence in \(k\). It gives \(a\in K_k^\times\), since
all four embedding components are nonzero. The remaining coefficient
\(c\) in \(r_{ij}=c a_i a_j\) is Galois-fixed as well. This proves
existence over \(k\), including when \(K_k\) is a product of nonsplit
fields. The projective uniqueness proves the stated ambiguity.
Multiplying two opposite equalities gives \(c^2\Nm(a)=s\).
\end{proof}
For a prime \(p\), \(F_p^\times\) denotes the elements nonzero in every
field factor of \(F_p\). In the local formulas below, \(\Nm(a)\) denotes
the algebraic norm \(\Nm_{K_p/\Q_p}(a)\). For a global pair in
Equation~\eqref{eq:ext-multiplicity}, choose a representative
\(0\ne w\in\bigwedge^2_\Z I\) of its \(U^+\)-orbit and put
\[
x=(\bigwedge\nolimits^2\xi_p)^{-1}w,\qquad
u=\frac{\Nm_{K_p/\Q_p}(\xi_p)}{\Nm I}.
\]
The valuations of the numerator and denominator in \(u\) are equal:
both are the norm valuation of \(I_p\). Thus \(u\in\Z_p^\times\), and
Equation~\eqref{eq:ext-covariance} gives \(u\Psi(x)=z(I,w)\).
Inverting the unit variable in Equation~\eqref{eq:ext-label-probability}
gives
\begin{equation}
P_p(I,w)=
\Pr_{a\in R_p^\times}\bigl[(\bigwedge\nolimits^2a)x\in L_{V,p}\bigr].
\label{eq:ext-unit-membership}
\end{equation}
These representatives lead us to consider, for arbitrary \(z\in F_p^\times\),
the full local fiber
\[
\mathscr F_p(z)=
\{(x,u)\in V_p\times\Z_p^\times:u\Psi(x)=z\}.
\]
Define the abelian local group and its unit subgroup by
\begin{align}
\mathscr G_p
&=\{(a,c)\in K_p^\times\times\Q_p^\times:
c^2\Nm(a)\in\Z_p^\times\}/
\{(d,d^{-2}):d\in\Q_p^\times\}, \notag\\
\mathscr U_p
&=\operatorname{image}(R_p^\times\times\Z_p^\times)
\ \subset\mathscr G_p.
\label{eq:ext-local-group}
\end{align}
The quotient in the first line means that \(d\) acts by
\((a,c)\mapsto(da,d^{-2}c)\). The action on the fiber is
\begin{equation}
(a,c)\cdot(x,u)=
\left(c(\bigwedge\nolimits^2a)x,\,
\frac{u}{c^2\Nm(a)}\right).
\label{eq:ext-fiber-action}
\end{equation}
Lemma~\ref{lem:ext-comparison}, with \(s=u/u'\), shows that this action
is free and transitive when the fiber is nonempty. The explicit ratios
in its proof also give analytic local inverses, so this is an analytic
identification of the fiber with \(\mathscr G_p\). The group
\(\mathscr U_p\) is compact and open. The kernel of its unit
parametrization is exactly \(d\in\Z_p^\times\): if both \(a\) and \(da\)
are \(R_p\)-units, then \(v_p(d)=0\). In particular this assertion
holds at ramified primes.
Choose a reference in a nonempty fiber and use it to identify the fiber
with \(\mathscr G_p\). For a coset
\(\mathfrak c\in\mathscr G_p/\mathscr U_p\), let
\(\alpha_p(\mathfrak c;z)\) be the fraction of that coset for which
\(x\in L_{V,p}\), using Haar probability on \(\mathscr U_p\).
We call the coset accepted when \(\alpha_p(\mathfrak c;z)>0\).
The fraction is independent of the representative of the coset. The sum of
these fractions is independent of the reference. For an empty fiber
we declare this sum to be zero.
A scalar \(c\in\Z_p^\times\) preserves \(L_{V,p}\). The pushforward of
the product Haar probability on \(R_p^\times\times\Z_p^\times\) is
Haar probability on \(\mathscr U_p\). Hence the representative constructed
from \((I,w)\) occupies a coset \(\mathfrak c\) for which
\(P_p(I,w)=\alpha_p(\mathfrak c;z(I,w))\).
We now measure the sum over all local cosets. Fix an additive character
\(\psi_p\) of \(\Q_p\) with conductor \(\Z_p\), and let \(dz\) be the
self-dual Haar measure on \(F_p\) for the trace pairing. Thus
\begin{equation}
\vol_{dz}(R_{F,p})=|D_F|_p^{1/2}.
\label{eq:ext-self-dual-measure}
\end{equation}
For every \(p\), consider the law of the random variable
\begin{equation}
u\Psi(x),\qquad x\ \text{uniform additive in }L_{V,p},\qquad
u\ \text{uniform in }\Z_p^\times,
\label{eq:ext-density-law}
\end{equation}
with the two choices independent. The next lemma proves that this law
has a density with respect to \(dz\); we denote its locally constant
version on \(F_p^\times\) by \(f_p^*(z)\). Only the densities for
\(p\in S\) will enter the multiplicity bound.
\begin{lemma}[Local density and disintegration]
\label{lem:ext-density}
The law in Equation~\eqref{eq:ext-density-law} has a density, with a
locally constant version on \(F_p^\times\). For every \(z\in F_p^\times\),
only finitely many cosets are accepted, and their fractions satisfy the
exact identity
\begin{equation}
\sum_{\mathfrak c\in\mathscr G_p/\mathscr U_p}
\alpha_p(\mathfrak c;z)
=
|D|_p^{-1/2}
\frac{1-p^{-1}}
{\displaystyle\prod_{\mathfrak p\mid p,\ \mathfrak p\subset R}
(1-\Nm(\mathfrak p)^{-1})}
f_p^*(z).
\label{eq:ext-density-disintegration}
\end{equation}
The sum and the density are both zero if \(\mathscr F_p(z)\) is empty.
The identity holds at every prime, including primes dividing \(q\) and
the small ramified primes.
\end{lemma}
\begin{proof}
Let \(\Phi_p(x,u)=u\Psi(x)\). If \(\Phi_p(x,u)\in F_p^\times\), all six
split coordinates of \(x\) are nonzero. After splitting, the derivative
in \(x\) is the derivative of three products on three disjoint
two-dimensional planes. It has rank three, and therefore has rank
three over \(\Q_p\). The map is a submersion there. The remaining
input set is a finite union of proper algebraic zero loci and has
additive measure zero. Local change of variables gives an absolutely
continuous pushforward on the submersion locus, hence a density for
the full law. To obtain a locally constant version at a fixed
\(z\in F_p^\times\), cover its inverse image in the compact set
\(L_{V,p}\times\Z_p^\times\) by finitely many submersion charts,
refined to disjoint compact open pieces. The compact complement has
image avoiding a sufficiently small neighborhood of \(z\).
On that neighborhood each chart contributes a locally constant
density. This also treats an empty inverse image.
Give \(V_p\) its additive measure for which \(L_{V,p}\) has mass one,
before restricting to that lattice. We first compute the fiber form for
the product input measure after disintegration against \(dz\), and then
the mass of \(\mathscr U_p\) for that form. In a \(\Q_p\) submersion chart,
the relative absolute four-form is obtained by evaluating the input top
form on four fiber basis vectors and three lifts of a target basis, then
dividing by the \(dz\)-form on that target basis. We may compute this
identity after extension to a splitting field, still evaluating on \(\Q_p\) tangent
vectors and using the fixed extension of \(|p|_p=p^{-1}\). We use the
same interpretation for the scalar quotient below.
Over a splitting field, the exterior embedding matrix has determinant
\(\delta_{\OO}^{\,3}\). Consequently its measure form in the six
embedding coordinates is
\[
|\delta_{\OO}|_p^{-3}\left|\bigwedge_{i<j}dx_{ij}\right|.
\]
Likewise \(dz\) is the absolute embedding-coordinate form
\(\left|\bigwedge_{b\in\mathcal B}dz_b\right|\). To verify the latter
normalization, evaluate this form on an \(R_{F,p}\)-basis. The square
of its embedding determinant is the discriminant, so its absolute
value is \(|D_F|_p^{1/2}\), as in
Equation~\eqref{eq:ext-self-dual-measure}.
After splitting, compute the relative form by retaining
\(x_{12},x_{13},x_{14},u\) and substituting the three product equations
for \(x_{34},x_{24},x_{23}\). The three change-of-variable factors are
\(|\delta_{\OO}/(u x_{12})|_p\),
\(|\delta_{\OO}/(u x_{13})|_p\), and
\(|\delta_{\OO}/(u x_{14})|_p\). They cancel
\(|\delta_{\OO}|_p^{-3}\), since \(u\) is a unit. Haar probability on
\(\Z_p^\times\) is \((1-p^{-1})^{-1}|du/u|\). The resulting fiber form is
\begin{equation}
\frac{1}{1-p^{-1}}
\left|\frac{dx_{12}}{x_{12}}\wedge
\frac{dx_{13}}{x_{13}}\wedge
\frac{dx_{14}}{x_{14}}\wedge\frac{du}{u}\right|.
\label{eq:ext-fiber-form}
\end{equation}
The fiber action preserves this measure. Indeed its determinant on
\(V_p\) is \(c^6\Nm(a)^3=(c^2\Nm(a))^3\), of absolute value one,
and it multiplies \(u\) by a unit while fixing \(z\).
It remains to compute the mass of \(\mathscr U_p\) in
Equation~\eqref{eq:ext-fiber-form}. Put
\(\ell_i=da_i/a_i\) and \(s=dc/c\) over the splitting field. The
pullbacks of its four logarithmic differentials under
Equation~\eqref{eq:ext-fiber-action} are
\[
\ell_1+\ell_2+s,\quad \ell_1+\ell_3+s,\quad
\ell_1+\ell_4+s,\quad -\ell_1-\ell_2-\ell_3-\ell_4-2s.
\]
The vertical scalar direction \(d\) has vector
\((1,1,1,1,-2)\). Taking the first \(\ell_i\) as its coordinate and
using the slice \(\ell_1=0\), the matrix on the four remaining
differentials is
\[
\begin{pmatrix}
1&0&0&1\\
0&1&0&1\\
0&0&1&1\\
-1&-1&-1&-2
\end{pmatrix},
\qquad \det=1.
\]
The vertical coordinate has coefficient \(1\). Equivalently, the
wedge of the four displayed pullbacks is the contraction of
\((\bigwedge_i\ell_i)\wedge s\) by the vertical vector, up to sign.
Evaluating this identity on four \(\Q_p\) tangent lifts of a quotient basis
gives the same absolute quotient form for nonsplit \(K_p\). On a patch
where a \(\Q_p\)-linear coefficient \(\lambda(a)\) is nonzero, normalizing
it to its value at a reference point gives a genuine \(\Q_p\) section;
after shrinking a unit-group patch, the normalizing scalar is a unit.
Since \(\lambda(da)=d\lambda(a)\), the vertical logarithmic derivative is
\(1\). Thus the contraction is the quotient of the parameter form by
\(|dd/d|\), with quotient Jacobian \(1\), including at \(p=2\).
For an \(R_p\)-basis, the multiplicative embedding form on \(K_p^\times\)
is the additive basis form multiplied by its embedding determinant
and divided by \(\Nm(a)\). On \(R_p^\times\) the latter is a unit.
The mass of this unit set is consequently
\[
|D|_p^{1/2}
\prod_{\mathfrak p\mid p,\ \mathfrak p\subset R}
(1-\Nm(\mathfrak p)^{-1}).
\]
The \(c\)-unit mass and the scalar-kernel \(d\)-unit mass are both
\(1-p^{-1}\), and cancel when taking the quotient. Haar integration
on the compact unit groups assembles the local sections. Including the
factor \((1-p^{-1})^{-1}\) in Equation~\eqref{eq:ext-fiber-form}, the
mass of each \(\mathscr U_p\)-coset is
\begin{equation}
\frac{|D|_p^{1/2}}{1-p^{-1}}
\prod_{\mathfrak p\mid p,\ \mathfrak p\subset R}
(1-\Nm(\mathfrak p)^{-1}).
\label{eq:ext-unit-coset-mass}
\end{equation}
This calculation uses the actual image of the unit group, so it
requires no surjectivity assertion about the norm on units.
The accepted part of the fiber is the inverse image of \(z\) in the
compact set \(L_{V,p}\times\Z_p^\times\). Since \(\mathscr U_p\) is
open, this compact set meets only finitely many of its cosets.
Integrating its indicator in the fiber multiplies the mass in
Equation~\eqref{eq:ext-unit-coset-mass} by
\(\sum_{\mathfrak c}\alpha_p(\mathfrak c;z)\). The result is the
density \(f_p^*(z)\). Solving for the sum proves
Equation~\eqref{eq:ext-density-disintegration}.
\end{proof}
In particular the order index has disappeared from the prefactor in
Equation~\eqref{eq:ext-density-disintegration}: its contribution through
\(\delta_{\OO}\) cancels in the three products. Its effect remains
inside the density of the lattice \(L_{V,p}\). The discriminant in the
prefactor is \(D\) because the translating parameters are the
maximal-order units \(R_p^\times\).
\subsection{Global multiplicity and the unramified factors}
The local identity counts every local coset. We next bound how many
global pairs can occupy one assignment of those cosets.
\begin{lemma}[Global pairs in one assignment of cosets]
\label{lem:ext-global-cosets}
Fix \(z\in F^\times\). Choose a reference in each nonempty local fiber
\(\mathscr F_p(z)\). For any assignment
\(\mathfrak c_p\in\mathscr G_p/\mathscr U_p\), at most \(32\) pairs
\[
I,\qquad w\in(\bigwedge^2_{\Z}I\setminus\{0\})/U^+,\qquad z(I,w)=z
\]
occupy the assigned coset at every prime.
\end{lemma}
\begin{proof}
For two such pairs, Lemma~\ref{lem:ext-comparison} over \(\Q\) gives
\(w'=c(\bigwedge^2a)w\) with \(a\in K^\times\), \(c\in\Q^\times\), since
the common opposite ratio is \(\Nm(I')/\Nm(I)\). Relative to the local
generators \(\xi_p,\xi'_p\), their local comparison is represented by
\((a\xi_p/\xi'_p,c)\). Equality of the local cosets means that some
\(d_p\in\Q_p^\times\) satisfies
\[
d_pa\xi_p/\xi'_p\in R_p^\times,\qquad d_p^{-2}c\in\Z_p^\times.
\]
In particular \(v_p(c)=2v_p(d_p)\) is even at every finite prime.
Therefore \(c=\epsilon s^2\) with \(s\in\Q^\times\) and
\(\epsilon\in\{1,-1\}\). Absorbing \(s\) into \(a\) makes
\(a\xi_p/\xi'_p\) a unit for every \(p\). Hence \(I'=aI\) as fractional
maximal-order ideals. The chosen representatives of the ordinary ideal
classes then give \(I'=I\) and \(a\in R^\times=U\).
There are at most \([U:U^+]\le16\) possibilities for the unit action
modulo \(U^+\), distinguished by the four real signs, and two
possibilities for \(\epsilon\). This proves the bound \(32\).
\end{proof}
At \(p\notin S\), both local algebras are unramified,
\(\OO_p=R_p\), and \(\delta_{\OO}\) is a unit in a splitting field.
Let \(\varphi_p\) be Frobenius acting on the four embeddings and on
\(\mathcal B\). Its orbits on \(\mathcal B\) correspond to primes
\(\mathfrak q\mid p\) of \(F\), of residue degrees
\(f_{\mathfrak q}\). The map from two-element subsets to partitions
sends \(ij\) to \(ij|kl\). Above a Frobenius orbit of partitions its
pair labels form either two orbits or one. Define
\[
\epsilon_{\mathfrak q}=
\begin{cases}
1,&\text{if the pair labels form two orbits},\\
-1,&\text{if they form one orbit},
\end{cases}
\qquad
b_{\mathfrak q}(j)=
\begin{cases}
j+1,&\epsilon_{\mathfrak q}=1,\\
\mathbf1_{\{2\mid j\}},&\epsilon_{\mathfrak q}=-1
\end{cases}
=\sum_{r=0}^j\epsilon_{\mathfrak q}^{\,r}.
\]
For \(z\in R_{F,p}\cap F_p^\times\), put
\begin{equation}
j_{\mathfrak q}=v_{\mathfrak q}(z),\qquad
b_p(z)=\prod_{\mathfrak q\mid p}b_{\mathfrak q}(j_{\mathfrak q}),
\qquad n_p(z)=\max_{\mathfrak q\mid p}j_{\mathfrak q}.
\label{eq:ext-unramified-factor}
\end{equation}
Here \(v_{\mathfrak q}\) is the integral uniformizer valuation; it has
\(v_{\mathfrak q}(p)=1\) at these primes. For probability statements
one may set \(n_p=\infty\), \(b_p=0\) when a component vanishes, a set
of Haar measure zero.
\begin{lemma}[Unramified local data]
\label{lem:ext-unramified-data}
Let \(p\notin S\). A local fiber with an accepted coset has
\(z\in R_{F,p}\), and for \(z\in R_{F,p}\cap F_p^\times\),
\begin{equation}
\sum_{\mathfrak c\in\mathscr G_p/\mathscr U_p}
\alpha_p(\mathfrak c;z)\le b_p(z).
\label{eq:ext-unramified-cosets}
\end{equation}
One has \(b_p=1\) when \(n_p=0\), and \(0\le b_p\le(1+n_p)^3\).
Give the affine lattice
\[
\{z\in R_{F,p}:\Tr_{F_p/\Q_p}z=m\}
\]
its Haar probability. With \(t=p^{-1}\) and
\(f_{\mathfrak p}\) the residue degrees of the primes of \(K\) above \(p\),
\begin{align}
\E_p b_p
&=(1-t)(1-t^3)
\prod_{\mathfrak p\mid p}(1-t^{f_{\mathfrak p}})^{-1}
=1+\frac{a_K(p)-1}{p}+O(p^{-2}), \label{eq:ext-unramified-mean}\\
\P_p(n_p\ge k)&\le3p^{-k}\qquad(k\ge1). \label{eq:ext-unramified-tail}
\end{align}
The error constant is absolute. In any integral affine coordinates
on the trace lattice, \(\{n_p\ge k\}\) is determined modulo \(p^k\),
and \(b_p\mathbf1_{\{n_p=k\}}\) is determined modulo \(p^{k+1}\)
for every finite \(k\ge0\).
\end{lemma}
\begin{proof}
An accepted coset has a representative \(x\in L_{V,p}=\bigwedge^2 R_p\).
Use the extended valuation with \(v_p(p)=1\) on a splitting field.
Its six valuations \(k_{ij}=v_p(x_{ij})\) are nonnegative integers,
invariant under Frobenius. Since \(u\) and \(\delta_{\OO}\) are units,
\[
k_{ij}+k_{kl}=j_{\mathfrak q}
\quad\text{for the partition }ij|kl\text{ over }\mathfrak q .
\]
This proves integrality of \(z\). If the pair labels over
\(\mathfrak q\) have two Frobenius orbits, one of their two valuations
can be chosen from \(0,\ldots,j_{\mathfrak q}\). If there is one orbit,
the two valuations are equal, which requires even \(j_{\mathfrak q}\).
The number of records is therefore at most \(b_p(z)\).
Two representatives with the same six valuations have comparison
\((a,c)\) satisfying
\[
v_p(c)+v_p(a_i)+v_p(a_j)=0\qquad(i<j).
\]
Subtraction makes all four \(v_p(a_i)\) equal to a common \(r\), and
\(v_p(c)=-2r\). Because \(K_p\) is unramified, \(r\in\Z\).
Replacing \(a\) by \(p^{-r}a\) and \(c\) by \(p^{2r}c\) puts both in
the unit subgroup. Thus a record specifies at most one coset, also
for nonsplit unramified factors. The subgroup \(\mathscr U_p\)
preserves \(\bigwedge^2R_p\), so its coset fraction is either \(0\)
or \(1\). This proves Equation~\eqref{eq:ext-unramified-cosets}.
The bounds at depth zero and at general depth follow immediately
from the definition and the at most three factors.
The trace from an unramified integer ring to \(\Z_p\) is onto, as is
seen after reduction from the nonzero finite-field trace. If
\(F_p\) has more than one factor, the trace-\(m\) coset therefore
projects uniformly onto each chosen factor: the other factors can
supply any required trace. More generally, for integers
\(r_{\mathfrak q}\ge0\), the probability that
\(v_{\mathfrak q}(z)\ge r_{\mathfrak q}\) for all \(\mathfrak q\) is
\[
\begin{cases}
t^{\sum_{\mathfrak q}f_{\mathfrak q}r_{\mathfrak q}},
&\min_{\mathfrak q}r_{\mathfrak q}=0,\\
0,&\min_{\mathfrak q}r_{\mathfrak q}>0.
\end{cases}
\]
The first line follows by projecting onto the constrained factors
while a factor with \(r_{\mathfrak q}=0\) supplies the trace.
The second uses \(p\nmid m\). It also covers the case that \(F_p\)
is a field.
Expanding \(b_{\mathfrak q}(j)=\sum_{r=0}^j\epsilon_{\mathfrak q}^r\)
and summing the absolutely convergent geometric series gives
\begin{equation}
\E_p b_p=
\frac{1-(\prod_{\mathfrak q}\epsilon_{\mathfrak q})t^3}
{\prod_{\mathfrak q}(1-\epsilon_{\mathfrak q}t^{f_{\mathfrak q}})}.
\label{eq:ext-geometric-mean}
\end{equation}
The five Frobenius types give the following data. A pair
\((f,\epsilon)\) in the second column is attached to one prime of \(F\).
\[
\begin{array}{c|c|c|c}
\text{cycle type on four labels}
& (f_{\mathfrak q},\epsilon_{\mathfrak q})
& \prod_{\mathfrak p\mid p}(1-t^{f_{\mathfrak p}})
& a_K(p)-1\\ \hline
1^4 &(1,+),(1,+),(1,+)&(1-t)^4&3\\
2\,1^2 &(1,+),(2,+)&(1-t)^2(1-t^2)&1\\
2^2 &(1,+),(1,-),(1,-)&(1-t^2)^2&-1\\
3\,1 &(3,+)&(1-t)(1-t^3)&0\\
4 &(1,-),(2,-)&1-t^4&-1
\end{array}
\]
In every row \(\prod_{\mathfrak q}\epsilon_{\mathfrak q}=1\) and
\[
\prod_{\mathfrak q}(1-\epsilon_{\mathfrak q}t^{f_{\mathfrak q}})
=\frac{\prod_{\mathfrak p\mid p}(1-t^{f_{\mathfrak p}})}{1-t}.
\]
Substitution in Equation~\eqref{eq:ext-geometric-mean} proves the exact
mean. Its linear term is the number of degree-one primes of \(K\)
minus one, which is \(a_K(p)-1\); the five fixed rows give the uniform
error term.
If \(F_p\) is a field, \(n_p\ge1\) would make its trace divisible by
\(p\), and is impossible. Otherwise the uniform projection above
gives
\(\P_p(v_{\mathfrak q}(z)\ge k)=p^{-f_{\mathfrak q}k}\).
Taking the union over at most three factors proves
Equation~\eqref{eq:ext-unramified-tail}. Finally, the trace map is
onto, so the affine trace lattice has two integral coordinates.
Projection to each unramified factor modulo \(p^{k+1}\) is determined
by these coordinates modulo \(p^{k+1}\). The valuations truncated
at \(k+1\), and hence the asserted threshold and finite-depth
functions, have the stated congruence dependence. This remains true
on the component-vanishing set with the convention above.
\end{proof}
\begin{proposition}[Multiplicities from local densities]
\label{prop:ext-multiplicity-majorant}
For \(z\in F\) with \(\Tr z=m\ne0\), a positive value of \(M(z)\)
requires \(z\) to be integral outside \(S\). For such integral \(z\),
\begin{equation}
M(z)\le32
\prod_{p\in S}
\left(
|D|_p^{-1/2}
\frac{1-p^{-1}}
{\displaystyle\prod_{\mathfrak p\mid p,\ \mathfrak p\subset R}
(1-\Nm(\mathfrak p)^{-1})}
f_p^*(z)\right)
\prod_{p\notin S}b_p(z).
\label{eq:ext-local-majorant}
\end{equation}
All products in this assertion are finite at every contributing \(z\).
\end{proposition}
\begin{proof}
The field \(F\) and the condition \(\Tr z=m\ne0\) imply
\(z\in F^\times\). At every prime the set of accepted cosets is
finite by Lemma~\ref{lem:ext-density}. At \(p\notin S\),
Lemma~\ref{lem:ext-unramified-data} proves the integrality requirement
and bounds the local sum by \(b_p(z)\). The set
\[
E_z=S\cup\{p:z\notin R_{F,p}^\times\}
\]
is finite, since a nonzero global \(z\) is a unit at all but finitely
many primes. At \(p\notin E_z\), \(b_p=1\), and each coset fraction is
\(0\) or \(1\), so there is at most one accepted coset, of fraction
\(1\). If some prime has no accepted coset, then \(M(z)=0\) and the
claimed inequality is immediate. Otherwise the accepted coset is
unique with fraction \(1\) outside \(E_z\), and each prime inside the
finite set \(E_z\) has finitely many accepted cosets. Thus there are
only finitely many assignments of accepted cosets, and all local sums
outside \(E_z\) equal \(1\).
Choose local references. Equation~\eqref{eq:ext-unit-membership}
identifies the weight of a global pair at each prime with its coset
fraction \(\alpha_p\). By Lemma~\ref{lem:ext-global-cosets}, at most
\(32\) global pairs occupy any assignment of accepted cosets.
Summing over the finite set of assignments, including ones not
realized globally, bounds \(M(z)\) by \(32\) times the product of the
local sums of \(\alpha_p\). Use
Equation~\eqref{eq:ext-density-disintegration} at \(p\in S\) and
Equation~\eqref{eq:ext-unramified-cosets} outside \(S\).
This proves Equation~\eqref{eq:ext-local-majorant} and finiteness.
\end{proof}
The unramified mean in Equation~\eqref{eq:ext-unramified-mean} has the
Euler factor needed for the residue \(\kappa\). The remaining problem
in Theorem~\ref{thm:exterior-count} is to average the densities
\(f_p^*\) for \(p\in S\) before inserting these unramified factors.
For a frequency \(\alpha\in F_p\), testing the law \(u\Psi(x)\) against
the character \(z\mapsto\psi_p(\Tr_{F_p/\Q_p}(\alpha z))\) produces the
quadratic phase \(u\Tr_{F_p/\Q_p}(\alpha\Psi(x))\).
We express these forms through a single unimodular bilinear form and an
action of \(F\). The elements of \(F\) whose operators preserve \(L_V\)
will then form the order that controls the local Fourier analysis.
\subsection{The integral quadratic structure and its order}
Let \(B_V\) be the polarization of \(Q_V\) without division by two:
\[
B_V(x,y)=Q_V(x+y)-Q_V(x)-Q_V(y).
\]
Over a splitting field, write \(V_b\) for the plane spanned by the
two coordinate axes belonging to the partition \(b\), the same
coordinate planes used over \(\R\) in Section~\ref{sec:rigidity}.
For an equivariant triple \(\alpha=(\alpha_b)\in F\), let \(\alpha_b\)
scale both coordinate axes of \(V_b\).
\begin{lemma}[Self-adjoint quadratic structure]
\label{lem:ext-quadratic-structure}
The plane-scalar action descends to a rational action
\(\rho:F\to\End_{\Q}(V)\) for which \(V\) has rank two over \(F\).
The form \(Q_V\) in an exterior \(\Z\)-basis of \(L_V\) is the three
split pairs in Equation~\eqref{eq:ext-integral-form}; in particular
\(B_V\) is unimodular on \(L_V\). The action \(\rho\) is self-adjoint
for \(B_V\), and for \(\alpha\in F\),
\begin{align}
Q_\alpha(x):=\Tr_{F/\Q}(\alpha\Psi(x))
&=\tfrac12 B_V(\rho(\alpha)x,x), \label{eq:ext-polar-operator}\\
B_\alpha(x,y)&=B_V(\rho(\alpha)x,y), \notag\\
\det_{\Q}\rho(\alpha)&=\Nm_{F/\Q}(\alpha)^2,\qquad
\Psi(\rho(\alpha)x)=\alpha^2\Psi(x). \label{eq:ext-rank-two}
\end{align}
Here \(B_\alpha\) is the polarization of \(Q_\alpha\).
The same assertions hold after base change to every \(\Q_p\).
\end{lemma}
\begin{proof}
Galois permutes the planes \(V_b\), including possible alternating
signs on their coordinate axes, and permutes the scalars \(\alpha_b\)
in the same way. The plane-scalar action therefore commutes with
Galois and descends. After splitting there are three planes of
dimension two, proving the rank assertion. Equation~\eqref{eq:ext-integral-form}
shows that the matrix of \(B_V\) pairs the three opposite pairs by
unit off-diagonal entries, with the displayed signs, so it is
unimodular. On each plane, multiplying both coordinates by
\(\alpha_b\) is self-adjoint for its split pairing. The contribution
of that plane to \(\frac12B_V(\rho(\alpha)x,x)\) is
\(\alpha_b\Psi_b(x)\), which proves
Equation~\eqref{eq:ext-polar-operator}. Polarization gives the next
identity. The determinant and the last quadratic identity follow
by multiplying, respectively, the two equal eigenvalues and the
two coordinate scalars on each plane. All these are rational
identities, hence persist under base change.
\end{proof}
Define the stabilizer order and its discriminant by
\begin{equation}
C_F=\{\alpha\in F:\rho(\alpha)L_V\subset L_V\},\qquad
\Delta_F=\Disc(C_F).
\label{eq:ext-stabilizer-order}
\end{equation}
It is a ring containing \(1\) and a full \(\Z\)-lattice: this follows
from the rationality of \(\rho\) by clearing denominators on a basis
of \(F\). It is contained in \(R_F\), since an \(\alpha\) whose
matrix on \(L_V\) is integral is a root of that integral monic
characteristic polynomial. Thus it is an order.
The stabilizer condition is a finite list of matrix-entry integrality
conditions in a lattice basis, so it commutes with completion:
\begin{equation}
C_{F,p}=C_F\otimes_\Z\Z_p
=\{\alpha\in F_p:\rho(\alpha)L_{V,p}\subset L_{V,p}\}.
\label{eq:ext-local-stabilizer}
\end{equation}
The following comparison ensures that its discriminant tends to infinity
with the discriminant of \(\OO\), even when the field \(K\) stays fixed.
\begin{lemma}[Scale of the stabilizer order]
\label{lem:ext-order-scale}
There is an absolute constant \(C\) such that
\begin{equation}
\frac12 X^{1/2}\le
\Delta_F=D_F[R_F:C_F]^2\le C X^{28}.
\label{eq:ext-order-scale}
\end{equation}
More precisely, with \(d=\Disc(\OO)=X^2\),
\[
d^2R_F\subset C_F\subset R_F,\qquad D_F\ll D^2.
\]
In particular \(C_{F,p}=R_{F,p}\) for \(p\nmid Dq\).
\end{lemma}
\begin{proof}
Let \(E=\bigwedge^2 A_{\OO}\) be the exterior embedding matrix.
Its entries are integral in every local splitting ring, and
\(\det E=\delta_{\OO}^{\,3}\). Since \(d=\delta_{\OO}^{\,2}\),
\[
d^2 E^{-1}=\delta_{\OO}\operatorname{adj}(E)
\]
also has integral entries. For \(\alpha\in R_{F,p}\), its
plane-scalar matrix over the splitting ring is integral.
Consequently
\(d^2\rho(\alpha)=d^2E^{-1}\diag(\alpha_b,\alpha_b)E\)
is integral over the splitting ring. Its entries
are in \(\Q_p\), so they lie in \(\Z_p\). This proves
\(d^2R_F\subset C_F\) and \([R_F:C_F]\le d^6\).
If \(p\nmid D\), the etale algebra \(R_p\) splits over an unramified
extension of \(\Z_p\); its splitting field and cubic resolvent are
unramified. Hence every prime dividing \(D_F\) divides \(D\).
For \(p>3\), a cubic algebra is tamely ramified and its discriminant
exponent is at most \(2\). At \(2\) and \(3\), the finitely many
local extensions of degree at most three have bounded discriminant
exponents \cite[Proposition~7.64 and Remark~7.65]{MilneANT}.
It follows that \(D_F\ll D^2\). The discriminant-index formula now
gives
\[
\Delta_F=D_F[R_F:C_F]^2\ll D^2d^{12}\le X^{28}.
\]
The containment \(d^2R_F\subset C_F\) also proves equality of the
local orders away from \(Dq\).
For the reverse inequality, we first explain why a projection
denominator controls \(\Disc(\OO)\). Write \(\operatorname{tr}\) for
matrix trace, put \(\mathscr E=\End_\Z(\OO)\), and let
\(\mathscr W_K\subset\End_\Q(K)\) be the rational subspace of
multiplication matrices. Its intersection
\(\mathscr L=\mathscr W_K\cap\mathscr E\) is saturated. It is
multiplication by \(\OO\): if \(\alpha\OO\subset\OO\), then
\(\alpha=\alpha\cdot1\in\OO\), and the converse follows because
\(\OO\) is a ring. The pairing
\((T,S)\mapsto\operatorname{tr}(TS)\) is unimodular on
\(\mathscr E\), since its matrix units are dual. Its restriction
to \(\mathscr W_K\) is the nondegenerate field trace pairing.
Let \(\mathcal E_K\) be the orthogonal projection onto
\(\mathscr W_K\).
For the restricted pairing one has
\begin{equation}
\mathcal E_K(\mathscr E)=\mathscr L^\vee.
\label{eq:ext-projected-dual}
\end{equation}
One inclusion follows by pairing a projected integral matrix with
\(\mathscr L\). Conversely, an integer functional on the primitive
sublattice \(\mathscr L\) extends to one on \(\mathscr E\).
Unimodularity represents the extension by an element of
\(\mathscr E\); its orthogonal projection represents the original
functional on \(\mathscr L\). Nondegeneracy identifies that
projection with the prescribed element of \(\mathscr L^\vee\).
Consequently, if a positive integer \(N\) satisfies
\(N\mathcal E_K(\mathscr E)\subset\mathscr E\), then
\[
\mathscr L^\vee\subset N^{-1}\mathscr L,\qquad
X^2=\Disc(\OO)=[\mathscr L^\vee:\mathscr L]\le N^4.
\]
It remains to produce the denominator \(N=2\Delta_F\).
Choose trace-dual bases \(a_i\) of \(C_F\) and \(b_i\) of \(C_F^\vee\),
where
\[
C_F^\vee=\{\beta\in F:\Tr_{F/\Q}(\beta C_F)\subset\Z\}.
\]
The adjugate of the trace Gram matrix gives
\(\Delta_F b_i\in C_F\). Define on \(\End_\Q(V)\)
\begin{equation}
\mathcal E_F(T)=\sum_{i=1}^3\rho(a_i)T\rho(b_i).
\label{eq:ext-centralizer-projection}
\end{equation}
For two embeddings \(\sigma,\tau\) of \(F\), the dual-basis identity is
\[
\sum_i\sigma(a_i)\tau(b_i)=
\begin{cases}1,&\sigma=\tau,\\0,&\sigma\ne\tau.\end{cases}
\]
Thus, after splitting, \(\mathcal E_F\) extracts exactly the three
diagonal blocks \(\End(V_b)\). It is the orthogonal projection
for the matrix trace pairing, and
\begin{equation}
\Delta_F\mathcal E_F(\End_\Z(L_V))\subset\End_\Z(L_V).
\label{eq:ext-centralizer-denominator}
\end{equation}
This denominator conclusion uses the trace-dual lattice only; it
does not require that lattice to be an invertible \(C_F\)-ideal.
We transfer this projection to matrices on \(K\) explicitly.
Let
\[
j(T)(v\wedge w)=Tv\wedge w+v\wedge Tw
\qquad(T\in\End_\Q(K)).
\]
There is a natural contraction
\(\mathfrak c:\End_\Q(\bigwedge^2K)\to\End_\Q(K)\).
For clarity, in a basis \(e_1,\ldots,e_4\) write
\[
S(e_j\wedge e_l)=\sum_{i<k}S^{ik}_{jl}e_i\wedge e_k\qquad(j<l),
\]
and extend these coefficients antisymmetrically in \(i,k\) and in
\(j,l\), with zero coefficients when either pair repeats. Then
\[
\mathfrak c(S)^i_j=\sum_{k=1}^4 S^{ik}_{jk}.
\]
This is the contraction of
\(\bigwedge^2K\otimes\bigwedge^2K^\vee\) in one covariant and one
contravariant index, so it is independent of the basis. Both \(j\)
and \(\mathfrak c\) are integral in the \(\OO\) and exterior
\(\OO\)-bases. Expanding their coefficients gives
\begin{equation}
\mathfrak c(j(T))=2T+\operatorname{tr}(T)\id_K.
\label{eq:ext-contraction}
\end{equation}
Indeed an off-diagonal coefficient of \(T\) occurs for the two
remaining indices \(k\); a diagonal coefficient gives
\(\sum_{k\ne i}(T^i_i+T^k_k)=2T^i_i+\operatorname{tr}(T)\).
In embedding coordinates, an off-diagonal entry of \(j(T)\) changes
one index of a two-element subset. Such subsets belong to distinct
opposite-pair planes. It follows that
\(\mathcal E_F(j(T))=j(\diag T)\) over the splitting field.
Equation~\eqref{eq:ext-contraction} now proves the formula
\begin{equation}
\mathcal E_K(T)=
\tfrac12\bigl(\mathfrak c(\mathcal E_F(j(T)))-\operatorname{tr}(T)\id_K\bigr)
\label{eq:ext-multiplication-projection}
\end{equation}
for the projection defined above: its right side is \(\diag T\)
over the splitting field. A Galois-equivariant diagonal tuple is
the tuple of embeddings of an element of \(K\), so these diagonal
matrices are precisely \(\mathscr W_K\). Diagonal extraction is
orthogonal for matrix trace. Finally,
Equation~\eqref{eq:ext-centralizer-denominator} and integrality of
\(j\) and \(\mathfrak c\) give
\[
2\Delta_F\mathcal E_K(\End_\Z(\OO))\subset\End_\Z(\OO).
\]
Taking \(N=2\Delta_F\) in the preceding lattice bound gives
\[
X^2\le(2\Delta_F)^4.
\]
This is the lower inequality in Equation~\eqref{eq:ext-order-scale}.
\end{proof}
For \(x\in L_{V,p}\) and \(\alpha\in C_{F,p}\),
Equation~\eqref{eq:ext-polar-operator} gives two integral pairings:
\[
\Tr\Psi(x)=Q_V(x)\in\Z_p,\qquad
2\Tr(\alpha\Psi(x))=B_V(\rho(\alpha)x,x)\in\Z_p.
\]
They remain integral after multiplying \(\Psi(x)\) by a \(\Z_p\)-unit.
Thus the products pair integrally with the scalar \(1\) and with
\(2C_{F,p}\). Define the order and its trace dual
\[
C_F^\sharp=\Z+2C_F,\qquad
(C_F^\sharp)^\vee
=\{z\in F:\Tr_{F/\Q}(zC_F^\sharp)\subset\Z\}.
\]
This trace dual gives the integral support for the additive count.
The same support will bound the total depth of the unramified factors
and prevent a nonzero real product coordinate from being too small.
\begin{lemma}[Trace-dual support and product size]
\label{lem:ext-support}
One has
\begin{equation}
[C_F:C_F^\sharp]=4,\qquad
\Delta_F^\sharp:=\Disc(C_F^\sharp)=16\Delta_F\ll X^{28},
\label{eq:ext-sharp-discriminant}
\end{equation}
and \(\Tr:(C_F^\sharp)^\vee\to\Z\) is onto.
For \(p\notin S\), \(C_{F,p}^\sharp=(C_{F,p}^\sharp)^\vee=R_{F,p}\).
For every prime \(p\), the density \(f_p^*\) on \(F_p^\times\) is
supported on \((C_{F,p}^\sharp)^\vee\). Consequently
\[
M(z)>0\ \Longrightarrow\ z\in(C_F^\sharp)^\vee.
\]
The following stronger containment is used to average the majorant in
Proposition~\ref{prop:ext-multiplicity-majorant}: if \(z\in F^\times\) is
integral outside \(S\) and \(\prod_{p\in S}f_p^*(z)>0\), then
\(z\in(C_F^\sharp)^\vee\).
Fix \(m\ne0\) and a bounded
\(\Omega\subset\mathcal H_m\). Every
\(z\in(C_F^\sharp)^\vee\cap\Omega\) satisfies
\begin{align}
&\Delta_F^\sharp z\in R_F,\qquad
1\le|\Nm_{F/\Q}(\Delta_F^\sharp z)|\ll_\Omega X^{84},
\label{eq:ext-norm-bound}\\
&\prod_{p\notin S}p^{n_p(z)}
\le|\Nm_{F/\Q}(\Delta_F^\sharp z)|,\qquad
|z_b|\gg_\Omega X^{-84}\quad(b\in\mathcal B).
\label{eq:ext-depth-coordinate-bound}
\end{align}
The constants are uniform in all arithmetic data and in the point
of \(\Omega\).
\end{lemma}
\begin{proof}
Since \(C_F\) is an order, \(C_F\cap\Q=\Z\), so \(1\) is primitive
in its lattice. A basis \(1,a_2,a_3\) of \(C_F\) gives the basis
\(1,2a_2,2a_3\) of \(C_F^\sharp\). This proves the index and
discriminant assertions in Equation~\eqref{eq:ext-sharp-discriminant}.
The functional taking \(1\) to \(1\) and the other two basis vectors
to \(0\) is represented in the trace dual. Hence trace from that
dual is onto \(\Z\). Lemma~\ref{lem:ext-order-scale}, and \(2\in S\),
give \(C_{F,p}^\sharp=R_{F,p}\) outside \(S\).
The two integrality identities preceding the lemma show that the
law in Equation~\eqref{eq:ext-density-law} is supported on
\((C_{F,p}^\sharp)^\vee\). For a contributing global pair, use its local
representation \(z=u\Psi(x)\) and Equation~\eqref{eq:ext-fiber-action}
to translate \(x\) into \(L_{V,p}\), adjusting \(u\) by the inverse
norm of that unit. This proves local trace-dual membership
at every prime. Intersecting the local lattices proves the stated
global support. For the second support assertion, the density factors
give this membership at \(p\in S\). At \(p\notin S\), the order is
\(R_{F,p}\), whose trace pairing is unimodular because \(F_p\) is
unramified; thus \(R_{F,p}^\vee=R_{F,p}\), and the assumed integrality
gives membership there as well.
The adjugate of the trace Gram matrix of \(C_F^\sharp\) gives
\(\Delta_F^\sharp(C_F^\sharp)^\vee\subset C_F^\sharp\subset R_F\).
If \(z\in\Omega\), then \(\Tr z=m\ne0\), so \(z\ne0\); since \(F\)
is a field, all its embeddings are nonzero. The norm
\(\Nm(\Delta_F^\sharp z)\) is consequently a nonzero integer. The
boundedness of \(\Omega\) and \(\Delta_F^\sharp\ll X^{28}\) give
\[
1\le|\Nm(\Delta_F^\sharp z)|
=(\Delta_F^\sharp)^3\prod_{b\in\mathcal B}|z_b|
\ll_\Omega X^{84}.
\]
At \(p\notin S\), \(z\) is integral and the local algebra is
unramified. Hence
\[
n_p(z)\le\sum_{\mathfrak q\mid p}
f_{\mathfrak q}v_{\mathfrak q}(z)
\le v_p\bigl(\Nm(\Delta_F^\sharp z)\bigr).
\]
The product of these inequalities gives the depth bound because
\(\Nm(\Delta_F^\sharp z)\) is an integer. Finally, if the coordinates of
\(\Omega\) have absolute value at most \(B\), with \(B\ge1\), then
for each \(b\)
\[
|z_b|=\frac{|\Nm(\Delta_F^\sharp z)|}
{(\Delta_F^\sharp)^3\prod_{b'\ne b}|z_{b'}|}
\ge\frac{1}{(\Delta_F^\sharp)^3 B^2}\gg_\Omega X^{-84}.
\]
This proves Equation~\eqref{eq:ext-depth-coordinate-bound}.
\end{proof}
==== END Equidistribution-of-Primitive-Quartic-Torus-Packets-for-Arbitrary-Orders//exterior.tex ====
==== BEGIN Equidistribution-of-Primitive-Quartic-Torus-Packets-for-Arbitrary-Orders//introduction.tex ====
\section{Introduction}\label{sec:introduction}
Ideal classes in a totally real number field give compact orbits of a
diagonal group on a space of lattices. A natural arithmetic question is
whether the finite collections of orbits obtained in this way become
uniformly distributed as their discriminants grow. We prove this for the
complete packets attached to invertible ideals of every order in a primitive
totally real quartic field. The order may have arbitrarily large index in
the ring of integers.
Let
\[
X_4=\mathrm{SL}_4(\Z)\backslash\mathrm{SL}_4(\R)
\]
be the space of covolume-one row lattices in $\R^4$, and let $m_4$ be its
invariant probability measure. Put
\[
H_4=\{u\in\R^4:u_1+u_2+u_3+u_4=0\},\qquad
A_4=\{\diag(e^{u_1},e^{u_2},e^{u_3},e^{u_4}):u\in H_4\}.
\]
The group $A_4$ acts on $X_4$ by right multiplication. Its Haar measure
is $du_1\,du_2\,du_3$, with $u_4=-u_1-u_2-u_3$.
A quartic field $K$ is \emph{primitive} if it has no field strictly between
$\Q$ and $K$. Let $K$ be primitive and totally real, let $\OO\subseteq\OO_K$
be an order, and fix an ordering $\sigma=(\sigma_1,\ldots,\sigma_4)$ of its
real embeddings. Here an order is a full-rank subring containing $1$.
A fractional $\OO$-ideal $I$ is \emph{proper} if its multiplier order
$\{a\in K:aI\subseteq I\}$ is $\OO$, and it is \emph{invertible} if
$IJ=\OO$ for some fractional $\OO$-ideal $J$. The ordinary ideal class
group $\Pic(\OO)$ consists of proper invertible fractional $\OO$-ideals
modulo $I\sim aI$ for $a\in K^\times$.
For such an ideal put
\begin{equation}\label{eq:intro-lattice}
\Lambda_{I,\sigma}
=\covol(\sigma(I))^{-1/4}\sigma(I)\in X_4.
\end{equation}
A covolume-one lattice is represented here by any positively oriented
basis; two such bases differ by an element of $\mathrm{SL}_4(\Z)$.
Let $W_{\mathrm{all}}$ be the sixteen diagonal matrices with entries in
$\{1,-1\}$. Choose one ideal in each ordinary class and form the finite
set of distinct orbits
\[
\mathcal P_{\OO,\sigma}
=\{\Lambda_{I,\sigma}wA_4:I\text{ a class representative},\
w\in W_{\mathrm{all}}\}.
\]
A matrix in $W_{\mathrm{all}}$ acts on the underlying lattice, so a sign
matrix of determinant $-1$ also gives a point of $X_4$. Using all signs
makes the set independent of the chosen ordinary ideal representatives.
Every orbit is compact by the unit theorem; Section~\ref{sec:packets}
proves these assertions and describes the associated measure.
For an orbit $\mathcal Y\in\mathcal P_{\OO,\sigma}$, let $\nu_{\mathcal Y}$
be its invariant probability and let $\vol(\mathcal Y)$ be its volume for
the specified Haar measure. Define
\begin{equation}\label{eq:intro-measure}
\mu_{\OO,\sigma}
=\frac{\displaystyle\sum_{\mathcal Y\in\mathcal P_{\OO,\sigma}}
\vol(\mathcal Y)\nu_{\mathcal Y}}
{\displaystyle\sum_{\mathcal Y\in\mathcal P_{\OO,\sigma}}
\vol(\mathcal Y)}.
\end{equation}
Each distinct orbit occurs once in these sums.
\begin{theorem}\label{thm:main}
Let $(K_i,\OO_i,\sigma_i)$ be any sequence in which $K_i$ is a primitive
totally real quartic field, $\OO_i$ is an order in $K_i$, and $\sigma_i$ is
an ordering of its real embeddings. If $|\Disc(\OO_i)|\to\infty$, then
\[
\int_{X_4} f\,d\mu_{\OO_i,\sigma_i}
\longrightarrow \int_{X_4} f\,dm_4
\qquad\text{for every }f\in C_c(X_4).
\]
There is no escape of mass: for every $\varepsilon>0$ there is a compact
$\mathcal K\subset X_4$ such that
$\mu_{\OO_i,\sigma_i}(\mathcal K)\ge1-\varepsilon$ for all sufficiently
large $i$.
\end{theorem}
In particular, the field may remain fixed while the order index tends to
infinity. The packet in Theorem~\ref{thm:main} uses every class in
$\Pic(\OO)$, so its ideals are locally free of rank one over the specified
order. The proof uses this complete class average in both of its counting
arguments.
\subsection{History and context}
For real quadratic fields, compact diagonal orbits are closed geodesics on
the modular surface. Linnik's ergodic method and Skubenko's theorem for
positive discriminants established arithmetic distribution under the
condition $\left(\frac d p\right)=1$ for the discriminant $d$ at a fixed
odd prime $p$ \cite{Linnik1968,Skubenko1962}.
Using Iwaniec's estimates for half-integral weight forms \cite{Iwaniec1987},
Duke removed that restriction for the geodesic collections associated with
positive fundamental discriminants \cite[Theorem~1]{Duke1988}.
Chelluri extended the corresponding statement to the unit tangent bundle,
as recorded in \cite[Section~1.2, p.~253]{ELMV2012}.
Einsiedler, Lindenstrauss, Michel, and Venkatesh later gave a proof using
entropy for all positive nonsquare order discriminants
\cite[Theorem~2.3]{ELMV2012}.
In higher degree the diagonal group has higher rank, and the arithmetic
packets lie in spaces of higher-dimensional lattices. Einsiedler,
Lindenstrauss, Michel, and Venkatesh developed the discriminant and volume
framework connecting separation of periodic torus orbits to entropy
\cite{ELMV2009}. Their cubic theorem proves equidistribution of packets
that include those arising from nonmaximal totally real cubic orders
\cite[Theorem~1.4]{ELMV2011}. The local homothety class of a lattice is
part of their packet data; a multiplier order alone need not specify it
\cite[Section~5.5]{ELMV2011}. Their higher-dimensional discussion also
identifies the possibility of proper intermediate homogeneous limits
\cite[Section~1.6.3]{ELMV2011}. The theorem of Einsiedler, Katok, and
Lindenstrauss makes a diagonal-ergodic probability with positive entropy
homogeneous \cite[Theorem~1.3]{EKL2006}; in prime dimension at least three
its homogeneous conclusion is already Haar \cite[Corollary~1.4]{EKL2006}.
For each fixed prime $p\ge5$, Lemke Oliver, Thorner, and Zaman proved
equidistribution for the orbit collections attached to orders in totally
real degree-$p$ fields as the order discriminant grows, outside an
exceptional set containing $O_{p,\varepsilon}(Q^\varepsilon)$ fields of
field discriminant at most $Q$ \cite[Theorem~2.3]{LOTZ2024}.
The separate preprint \cite[Theorem~1.1]{PrimeDegreePackets} treats every
packet in each fixed prime degree at least five and permits arbitrary local module types.
We adapt its ordinary-vector method in the argument below.
For quartic packets, Khayutin obtained entropy bounds for probability
weak-$*$ limits of maximal-order packet measures with two-transitive Galois
groups, including the primitive quartic types $\mathfrak A_4$ and $S_4$
\cite[Theorem~1.1]{Khayutin2019Tori}. Wieser and Yang obtain entropy bounds
for certain maximal-order quartic packets whose fields have a quadratic
subfield, under further hypotheses \cite{WieserYang2026}. The present
argument must establish tightness and positive entropy in almost every
diagonal-ergodic component for arbitrary orders, and then exclude the
proper homogeneous limits allowed in degree four.
The arithmetic use of a sieve to rule out intermediate algebraic limits
has a precedent in Khayutin's work on joint distributions of CM points
\cite{Khayutin2019CM}. Local Fourier analysis for nonmaximal orders already
appears in the cubic work of Einsiedler, Lindenstrauss, Michel, and Venkatesh
\cite[Proposition~9.9]{ELMV2011}. Here the object being transformed is a
specific quadratic product map on an exterior lattice. The resulting
estimate controls arbitrary conductor depth in a cubic-resolvent count,
before the remaining multiplicative weights are inserted by a sieve.
\subsection{The two counts and the conductor estimate}
Write
\[
\begin{gathered}
R=\OO_K,\qquad D=|\Disc(K)|,\qquad q=[R:\OO],\\
X=q\sqrt D=\sqrt{|\Disc(\OO)|},\qquad
\kappa=\Res_{s=1}\zeta_K(s).
\end{gathered}
\]
We also write $a_K(n)$ for the number of nonzero integral $R$-ideals of
norm $n$. Both counting arguments preserve the factor $\sqrt D\,\kappa$
that normalizes the packet after its finite local labels are averaged.
Sections~\ref{sec:packets} and~\ref{sec:ordinary} treat ordinary lattice
vectors. An invertible $\OO$-ideal is locally a unit translate of $\OO$
inside a maximal-order ideal. Averaging those translates gives an exact
unfolding into a weighted sum over integral $R$-ideals. A local moment
bound controls the order index, while Stark's exceptional-zero result and
the absence of a quadratic subfield retain the residue $\kappa$ in the
ideal count. Counting short vectors proves tightness. Counting vectors
in boxes away from the coordinate hyperplanes gives an $O(r^4)$ bound for
radius-$r$ neighborhoods in $X_4$, uniformly on each compact set of centers
in every probability limit. For a diagonal flow with distinct weights,
a two-sided tube is covered by $O(r^{-3})$ such neighborhoods with centers
in a fixed compact enlargement, where $r$ decreases exponentially with the
tube length. The resulting exponential decay also holds for normalized
averages over sets of $A_4$-ergodic components with positive measure,
because they are dominated by a multiple of the limit measure. This forces
positive entropy in almost every component.
Section~\ref{sec:rigidity} applies measure classification and identifies
the remaining obstruction. A proper positive-entropy component has two
coordinate blocks of dimension two. Each lattice in its supporting orbit
then has a vector in its exterior square supported on the two opposite
coordinates $ij$ and $kl$, with both coefficients nonzero. There are three
choices of the partition $ij|kl$.
For an exterior vector $v=(v_{ij})_{i<j}$, the positive diagonal action
preserves the three signed products
\[
(v_{12}v_{34},\,-v_{13}v_{24},\,v_{14}v_{23}).
\]
Their sum is an integer on the exterior square of a unimodular lattice.
For the arithmetic lattices in our packets, Section~\ref{sec:exterior}
places the products in the three embeddings of a cubic resolvent field
$F$. Fixing their sum at a nonzero integer $m$ puts them on a trace-$m$
plane. If the four exterior coordinates outside $ij,kl$ have size at most
$r$, two product coordinates have size at most $r^2$. Thus the geometric
obstruction is tested by a rectangle of area $O(r^4)$ on that plane.
Theorem~\ref{thm:exterior-count} supplies the corresponding arithmetic
rectangle bound, with the multiplicity of every product and the packet
normalization included.
The main new work lies in proving that bound uniformly in $\OO$. At primes
dividing $Dq$, the difficult local factors in the multiplicity are given by
the density of the pushforward of Haar measure on the exterior square of the
order lattice, with a unit scalar, under the quadratic product map.
Its exact fiber measure accounts for ramification and for the order index.
An element of $F$ acts on each opposite-coordinate plane by the value of the
corresponding real embedding; the resulting action descends to the rational
exterior space $\bigwedge^2_{\Q}K$. The elements preserving
$\bigwedge^2_{\Z}\OO$ form an order $C_F$. Its discriminant is bounded
below by a fixed positive power of $X$ and above by a fixed power of $X$,
even if the field $F$ remains fixed.
Section~\ref{sec:local-fourier} analyzes these densities through their
quadratic Fourier transforms. Denominator ideals organize the frequencies;
the stabilizer order controls the decay of the Fourier coefficients and the
support and size of the corresponding inverse transforms. The local proof
is expressed in terms of the quadratic lattice, its action by a cubic
\'etale algebra, and its stabilizer order, a formulation suitable for
analogous cubic-resolvent counts. In Section~\ref{sec:additive}, Poisson
summation gives the trace density from scalar frequencies. Complementary
estimates control the total nonscalar contribution using discriminant
separation, bounds for the Fourier and inverse transforms, and congruences
supplied by inverse quadratic forms. The estimate is uniform for small
residue moduli and for shrinking rectangles.
At the remaining primes, elementary valuation records give a local factor
with mean $1+(a_K(p)-1)/p+O(p^{-2})$. Section~\ref{sec:sieve} averages
these factors by a weighted Selberg sieve. The subtraction of $1$ in the
local mean removes the logarithm in the prime-sum bound, leaving $\kappa$.
Finally, Section~\ref{sec:conclusion} unfolds exterior vectors along the
diagonal group. The fiber lengths are logarithmic and summable over the
rectangles above. The resulting $O(r^4)$ bound excludes every proper
homogeneous component and proves Theorem~\ref{thm:main}.
\subsection{Conventions}
The notation $\Nm_{E/k}$ denotes the algebraic field norm. Ideal norms
$\Nm(\mathfrak a)$ are positive absolute norms, extended multiplicatively
to fractional ideals; absolute values are written explicitly for field
norms when needed. For a rational prime $p$,
a subscript $p$ denotes tensoring a field with $\Q_p$ or a lattice with
$\Z_p$. For a prime $\mathfrak p$ or $\mathfrak q$ of a local field we use
the integer valuation with value $1$ on a uniformizer, and write $e$ and $f$
for its ramification index and residue degree. A normalized extension of
$v_p$ has $v_p(p)=1$, and $|p|_p=p^{-1}$. Ideal norms include the residue
degrees. We use $\mathfrak p$ for primes of $K$ and $\mathfrak q$ for primes
of $F$ when the two occur together.
All logarithms are natural, and $\log_+t=\max(0,\log t)$ for $t>0$.
An upper bound $X^{o(1)}$ means $O_\eta(X^\eta)$ for every fixed $\eta>0$.
Unless stated otherwise, constants are uniform in $K$, $\OO$, and the
ordering of the embeddings. In the exterior estimates a nonzero trace $m$
and a bounded real region are fixed before $X$ tends to infinity; permitted
dependence on them is stated in the relevant results.
==== END Equidistribution-of-Primitive-Quartic-Torus-Packets-for-Arbitrary-Orders//introduction.tex ====
==== BEGIN Equidistribution-of-Primitive-Quartic-Torus-Packets-for-Arbitrary-Orders//local-fourier.tex ====
\section{Local Fourier estimates}\label{sec:local-fourier}
The exterior multiplicity bound in Section~\ref{sec:exterior} uses an
additive density at each prime in $S$. We now estimate its Fourier transform.
Scalar frequencies detect only the trace and have an exact, order-independent
value. For the remaining frequencies we record both decay and the support of
the inverse transform on a fixed denominator shell. The support is needed
because the denominator at a prime dividing the order index can be arbitrarily
large.
\subsection{Quadratic data and the trace marginal}
Fix a rational prime $p$. We first specify the local hypotheses used in the
argument. Let
\[
E=\prod_{\nu=1}^r E_\nu,\qquad R_E=\prod_{\nu=1}^r R_\nu
\]
be a cubic \'etale $\Q_p$-algebra and its maximal order. Here $E_\nu$ is a
field, $R_\nu$ is its integer ring, and $r\le3$. Write
$\mathfrak q_\nu=(\pi_\nu)$ for the maximal ideal of $R_\nu$,
$v_\nu(\pi_\nu)=1$, $e_\nu=v_\nu(p)$, and
$q_\nu=|R_\nu/\mathfrak q_\nu|=p^{f_\nu}$. Thus
\[
\sum_\nu e_\nu f_\nu=3,
\qquad |\Nm_{E_\nu/\Q_p}a|_p=q_\nu^{-v_\nu(a)}.
\]
Let $\mathfrak d_\nu=\mathfrak q_\nu^{d_\nu}$ be the different of
$E_\nu/\Q_p$, and put $\mathfrak d_E=\prod_\nu\mathfrak d_\nu$.
Let $W$ be a free $E$-module of rank two, with its unital action denoted by
$\rho:E\longrightarrow\End_{\Q_p}(W)$. Suppose that $B$ is a nondegenerate
symmetric $\Q_p$-bilinear form satisfying
\[
B(\rho(a)x,y)=B(x,\rho(a)y)\qquad(a\in E,\ x,y\in W).
\]
Let $L$ be a full $\Z_p$-lattice in $W$ which is self-dual for $B$:
\[
L=\{y\in W:B(y,L)\subset\Z_p\}.
\]
Assume also that $Q_1(x)=B(x,x)/2$ is integral on $L$. For the scalar
calculation we shall use the stronger property that in some integral
coordinates on $L$,
\begin{equation}\label{eq:fourier-split-form}
Q_1(x_1,y_1,x_2,y_2,x_3,y_3)
=x_1y_1-x_2y_2+x_3y_3.
\end{equation}
The division by $2$ takes place in $\Q_p$. Equation~\eqref{eq:fourier-split-form}
asserts the required integrality also when $p=2$. The shell estimates below
use only $Q_1(L)\subset\Z_p$; the exact split expression is used for the
scalar marginal.
For $a\in E$, set $Q_a(x)=B(\rho(a)x,x)/2$. The nondegenerate trace pairing
on $E$ determines a unique quadratic map $\Psi_E:W\longrightarrow E$ by
\begin{equation}\label{eq:fourier-polar-identity}
\Tr_{E/\Q_p}(a\Psi_E(x))=Q_a(x)
=\frac12 B(\rho(a)x,x)\qquad(a\in E).
\end{equation}
Define the stabilizer order and the auxiliary order
\[
\mathcal C_L=\{a\in E:\rho(a)L\subset L\},
\qquad \mathcal C_L^\sharp=\Z_p+2\mathcal C_L.
\]
Indeed $\mathcal C_L$ is an order in $R_E$. It is a subring containing $1$,
and a sufficiently large power of $p$ times $R_E$ preserves $L$. Conversely,
an element preserving $L$ satisfies the monic characteristic polynomial of
its integral matrix on $L$. The action is faithful, so the element is
integral over $\Z_p$ and belongs to $R_E$.
In the application, $E=F_p$, $W=V_p$, $L=L_{V,p}$, $B=B_V$, and
$\mathcal C_L=C_{F,p}$. Lemma~\ref{lem:ext-quadratic-structure} supplies
all the stated hypotheses and identifies $\Psi_E$ with the base change of
$\Psi$. We use the abstract notation temporarily to include every product
algebra and every ramification type in one argument.
Choose an additive character $\psi_p$ of $\Q_p$ with conductor $\Z_p$.
We give $W$ the self-dual Haar measure for $(x,y)\mapsto\psi_p(B(x,y))$;
it gives $L$ mass $1$. On $E$ use the self-dual measure for
$(A,z)\mapsto\psi_p(\Tr_{E/\Q_p}(Az))$, denoted by $dz$ or $dA$ according
to the variable. If $\mathfrak a$ is a fractional $R_E$-ideal, its
annihilator for this pairing is
$\mathfrak a^\perp=\mathfrak a^{-1}\mathfrak d_E^{-1}$. In particular
\begin{equation}\label{eq:fourier-self-dual-volume}
\vol(R_E)=\prod_\nu q_\nu^{-d_\nu/2}\le1.
\end{equation}
To see the normalization, $R_E^\perp=\mathfrak d_E^{-1}$ and
$\vol(R_E)\vol(R_E^\perp)=1$, whereas
$[\mathfrak d_E^{-1}:R_E]=\prod_\nu q_\nu^{d_\nu}$.
For $E=F_p$, Equation~\eqref{eq:fourier-self-dual-volume} reads
$\vol(R_{F,p})=|D_F|_p^{1/2}$.
Let $\mu_p$ be the law of
\begin{equation}\label{eq:fourier-density-law}
u\Psi_E(x),\qquad x\in L\text{ with additive probability},
\qquad u\in\Z_p^\times\text{ with Haar probability},
\end{equation}
with $x$ and $u$ independent. This is the law defining $f_p^*$ in
Lemma~\ref{lem:ext-density}. The following regularity fact specifies the
version of that density which we shall invert pointwise.
\begin{lemma}\label{lem:fourier-density}
The measure $\mu_p$ has a density $f_p^*$ with respect to $dz$. It has a
finite locally constant version at every $z=(z_\nu)\in E$ with
$z_\nu\ne0$ for all $\nu$. This version vanishes outside
\[
(\mathcal C_L^\sharp)^\vee
=\{z\in E:\Tr_{E/\Q_p}(z\mathcal C_L^\sharp)\subset\Z_p\}.
\]
In particular $\Tr_{E/\Q_p}(u\Psi_E(x))\in\Z_p$ on the domain of
Equation~\eqref{eq:fourier-density-law}.
\end{lemma}
\begin{proof}
Define an $E$-valued bilinear form $b_E$ by
\[
\Tr_{E/\Q_p}(a b_E(x,y))=B(\rho(a)x,y)\qquad(a\in E).
\]
Self-adjointness of the action makes $b_E$ symmetric and $E$-bilinear.
It is nondegenerate because $B$ is nondegenerate. Equation~\eqref{eq:fourier-polar-identity}
then gives $\Psi_E(x)=b_E(x,x)/2$ and
$d\Psi_{E,x}(h)=b_E(x,h)$. On the $\nu$-th field factor, a nonzero
component of $\Psi_E(x)$ implies that the component $x_\nu$ of $x$ is
nonzero. The nondegenerate form on the rank-two $E_\nu$-space makes
$h_\nu\mapsto b_E(x_\nu,h_\nu)$ a nonzero, hence onto, $E_\nu$-linear
functional. It follows that $\Psi_E$ is a $\Q_p$-submersion at every
preimage of an output with nonzero components. Multiplication by a unit
$u$ is invertible on $E$, so the map $(x,u)\mapsto u\Psi_E(x)$ is a
submersion at the same points.
For each $\nu$ the polynomial $\Psi_{E,\nu}$ is nonzero: a nondegenerate
symmetric form in characteristic zero has a nonzero quadratic form. Choose
a nonzero $\Q_p$-coordinate polynomial of this component. Its zero set in
the open lattice $L$ has Haar measure zero, by the usual one-variable root
bound followed by induction and Fubini. Off the union of
these null sets, the local submersion theorem gives absolute continuity of
the pushforward. For a fixed output with nonzero components, its preimage
in the compact set $L\times\Z_p^\times$ is compact. Cover this preimage
by finitely many compact-open submersion charts, refined to disjoint
compact-open pieces within those charts. The image of the remaining compact
set avoids a sufficiently small neighborhood of the output. After shrinking
that neighborhood, integration in the fiber coordinates of the pieces gives
a finite constant density on it. This proves the asserted
pointwise version, including value zero when the neighborhood misses the
image.
Finally, for $c\in\mathcal C_L$ and $r\in\Z_p$,
\[
\Tr_{E/\Q_p}((r+2c)\Psi_E(x))
=rQ_1(x)+B(\rho(c)x,x)\in\Z_p\qquad(x\in L).
\]
Multiplication by $u\in\Z_p^\times$ preserves this inclusion. The image
of the compact domain is therefore contained in the displayed trace dual,
which proves the support and trace assertions.
\end{proof}
We use the Fourier convention
\[
\widehat f(A)=\int_E f(z)\psi_p(\Tr_{E/\Q_p}(Az))\,dz,
\qquad
\check g(z)=\int_E g(A)\psi_p(-\Tr_{E/\Q_p}(Az))\,dA.
\]
The transform of the measure in Equation~\eqref{eq:fourier-density-law} is
\begin{equation}\label{eq:fourier-gamma}
\gamma_p(A)=\widehat{f_p^*}(A)
=\E_{u\in\Z_p^\times}\int_L\psi_p(uQ_A(x))\,dx.
\end{equation}
\begin{lemma}[The scalar trace density]\label{lem:fourier-scalar-density}
Suppose that the integral scalar form is exactly
$Q_1=x_1y_1-x_2y_2+x_3y_3$ as in
Equation~\eqref{eq:fourier-split-form}. Regard $a\in\Q_p$ as a scalar in
$E$. In the quartic application this scalar form is $Q_1=Q_V$. Then, for
every prime $p$, including $p=2$,
\begin{equation}\label{eq:fourier-scalar-coefficients}
\gamma_p(a)=
\begin{cases}
1,&a\in\Z_p,\\
p^{-3k},&v_p(a)=-k<0.
\end{cases}
\end{equation}
The trace of the law in Equation~\eqref{eq:fourier-density-law} has density
$d_p(t)\mathbf1_{\Z_p}(t)$ for additive measure giving $\Z_p$ mass $1$.
This density is continuous on $\Z_p$ and satisfies
\begin{equation}\label{eq:fourier-scalar-density-bound}
|d_p(t)-1|\le\frac1{p(p+1)}\qquad(t\in\Z_p).
\end{equation}
In particular $5/6\le d_p(t)\le7/6$, and for every finite set of primes
$S$ and every choice $t_p\in\Z_p$,
\[
\prod_{p\in S}d_p(t_p)\asymp1
\]
with absolute upper and lower constants.
\end{lemma}
\begin{proof}
For fixed $u\in\Z_p^\times$ and $v_p(a)=-k<0$, integration in the second
coordinate gives
\[
\int_{\Z_p^2}\psi_p(ua xy)\,dx\,dy
=\int_{\Z_p}\mathbf1_{\{ua x\in\Z_p\}}\,dx=p^{-k}.
\]
For $a\in\Z_p$ the integral is $1$. The sign of $a$ has no effect, so the
three independent pairs in Equation~\eqref{eq:fourier-split-form} give
Equation~\eqref{eq:fourier-scalar-coefficients}, even before averaging $u$.
This calculation uses the integral split expression and does not divide by
$2$ in $\Z_p$.
The scalar coefficient is the Fourier coefficient of the trace law, since
$\Tr(a z)=a\Tr(z)$. That law is supported on $\Z_p$ by
Lemma~\ref{lem:fourier-density}. The character group of $\Z_p$ is
$\Q_p/\Z_p$, and it has $p^k-p^{k-1}$ characters of exact denominator
$p^k$. Hence its nonconstant Fourier coefficients have total absolute sum
\[
\sum_{k\ge1}(p^k-p^{k-1})p^{-3k}
=\frac{1-p^{-1}}{p^2-1}=\frac1{p(p+1)}.
\]
Fourier inversion on the compact group $\Z_p$ is therefore uniformly
convergent and gives the continuous density and
Equation~\eqref{eq:fourier-scalar-density-bound}. The lower bound is
positive at every prime, and $\sum_p1/(p(p+1))<\infty$. Taking logarithms
of the upper and lower bounds proves the last assertion.
\end{proof}
\subsection{Denominator shells and quadratic transformation}
For $A=(A_\nu)\in E$, its denominator ideal is
\[
\mathfrak h_p(A)=\prod_\nu\mathfrak q_\nu^{h_\nu(A)},\qquad
h_\nu(A)=\max\{0,-v_\nu(A_\nu)\},
\]
where $v_\nu(0)=+\infty$. For an integral $R_E$-ideal
$\mathfrak h_p=\prod_\nu\mathfrak q_\nu^{h_\nu}$, its shell is thus
\[
\mathscr S_{\mathfrak h_p}
=\prod_{h_\nu=0}R_\nu\ \times\!
\prod_{h_\nu>0}\pi_\nu^{-h_\nu}R_\nu^\times.
\]
The integral components of a shell may be zero. Define
\begin{equation}\label{eq:fourier-shell-indices}
H_p=\Nm(\mathfrak h_p)=\prod_\nu q_\nu^{h_\nu},\qquad
p^{g_p}=[\mathfrak h_p:\mathfrak h_p\cap\mathcal C_L].
\end{equation}
The shell is a compact open subset of $\mathfrak h_p^{-1}$. We shall shift
its frequencies by integral scalars so that the quadratic form is
nondegenerate, without changing the integral in
Equation~\eqref{eq:fourier-gamma}.
\begin{lemma}[A scalar shift on a shell]\label{lem:fourier-shift}
Put $\iota_p=1$ for $p\le5$ and $\iota_p=0$ for $p>5$. For every
$A\in\mathscr S_{\mathfrak h_p}$ there is an $s\in\{0,1,2,3\}$ such that
$\alpha=A+s\in E^\times$ and
\[
v_\nu(\alpha_\nu)=-h_\nu\quad(h_\nu>0),\qquad
0\le v_\nu(\alpha_\nu)\le e_\nu\iota_p\quad(h_\nu=0).
\]
Writing $\beta=\alpha^{-1}$, one has
\begin{equation}\label{eq:fourier-shift-bounds}
\beta\in p^{-\iota_p}\mathfrak h_p,
\qquad p^{-3\iota_p}H_p
\le |\Nm_{E/\Q_p}\alpha|_p\le H_p.
\end{equation}
In particular the norm is exactly $H_p$ when $p>5$. If
\begin{equation}\label{eq:fourier-j-definition}
j_p=j_p(A,s)=\min\{j\ge0:p^j\beta\in\mathcal C_L\},
\end{equation}
then $j_p$ is finite and $j_p\le g_p+\iota_p$. Finally, for any generator
$i\in E^\times$ of the ideal $\mathfrak h_p$, the element $x=i(A+s)$ is in
$R_E$, is a unit of $R_E$ if $p>5$, and satisfies
$x^{-1}\in p^{-\iota_p}R_E$.
\end{lemma}
\begin{proof}
An integral scalar shift leaves every pole valuation $-h_\nu<0$ unchanged.
For an integral component and $p>5$, the condition that $A_\nu+s$ fail to be
a unit forbids at most one residue $s\in\mathbf F_p$: it forbids none if
the residue of $-A_\nu$ does not belong to the scalar subfield. There are
at most three components, and $0,1,2,3$ are distinct modulo $p$. Some choice
therefore makes all integral components units.
For $p\le5$, normalize the extension of $v_p$ by $v_p(p)=1$, so that on
$E_\nu$ it is $v_\nu/e_\nu$. Distinct elements $s,s'$ of the list satisfy
$v_p(s-s')\le1$. The ultrametric inequality shows that for a fixed integral
component at most one candidate can satisfy $v_p(A_\nu+s)>1$. Again there
are at most three components, so a candidate remains for which all their
valuations are at most $1$. Thus $A_\nu+s$ is nonzero in every integral
component.
The bounds on $\beta$ follow componentwise. In the norm, the possible
loss from the integral components is at most
$p^{-\iota_p\sum_\nu e_\nu f_\nu}=p^{-3\iota_p}$, proving
Equation~\eqref{eq:fourier-shift-bounds}. The group
$\mathfrak h_p/(\mathfrak h_p\cap\mathcal C_L)$ has order $p^{g_p}$ and is
killed by $p^{g_p}$. Thus
\begin{equation}\label{eq:fourier-order-containment}
p^{g_p}\mathfrak h_p\subset\mathcal C_L.
\end{equation}
Multiplying the containment for $\beta$ by $p^{g_p+\iota_p}$ proves the
claim about $j_p$. Finally $v_\nu(i_\nu)=h_\nu$. The components of
$i\alpha$ consequently have valuation zero on the pole factors and valuation
between zero and $e_\nu\iota_p$ on the others. This proves the assertions
about $x$ and $x^{-1}$.
\end{proof}
For the shift in Lemma~\ref{lem:fourier-shift},
$Q_{A+s}=Q_A+sQ_1$ and $Q_1(L)\subset\Z_p$. Thus for every fixed unit $u$
\begin{equation}\label{eq:fourier-scalar-shift-invariance}
\int_L\psi_p(uQ_{A+s}(x))\,dx
=\int_L\psi_p(uQ_A(x))\,dx.
\end{equation}
The following quadratic transformation turns the norm of $A+s$ into decay,
and leaves an integral involving its inverse.
\begin{lemma}[Quadratic Fourier transformation]\label{lem:fourier-quadratic-transform}
Let $W_0$ be a finite-dimensional $\Q_p$-space with a nondegenerate symmetric
form $B_0$, and let $L_0\subset W_0$ be a full $\Z_p$-lattice self-dual for
$B_0$. Use the self-dual Haar
measure, which gives $L_0$ mass $1$. For an invertible $B_0$-self-adjoint
operator $P$, put $q_P(x)=B_0(Px,x)/2$. There is a phase $\omega(P)$ with
$|\omega(P)|=1$ such that
\begin{equation}\label{eq:fourier-general-quadratic}
\int_{L_0}\psi_p(q_P(x))\,dx
=\omega(P)|\det P|_p^{-1/2}
\int_{L_0}\psi_p(-q_{P^{-1}}(x))\,dx.
\end{equation}
The normalized phase is invariant under linear isometry of the quadratic
form: $\omega(T^*PT)=\omega(P)$ for every invertible $T$, where $T^*$ is
the $B_0$-adjoint.
For the local data above, $\alpha\in E^\times$, $\beta=\alpha^{-1}$ and
$u\in\Z_p^\times$, this gives
\begin{equation}\label{eq:fourier-quadratic-transform}
\int_L\psi_p(uQ_\alpha(x))\,dx
=\omega_p(u,\alpha)|\Nm_{E/\Q_p}\alpha|_p^{-1}
\int_L\psi_p(-u^{-1}Q_\beta(x))\,dx,
\qquad |\omega_p(u,\alpha)|=1.
\end{equation}
For every $r\in E^\times$ one has
$\omega_p(u,\alpha r^2)=\omega_p(u,\alpha)$.
All assertions hold at $p=2$.
\end{lemma}
\begin{proof}
We use Weil's Fourier formula for a quadratic character
\cite[Chapter~I, \S14, Theorem~2, p.~161]{Weil1964}.
We record its normalization for the transform
\[
\mathcal F_{B_0}h(y)=\int h(x)\psi_p(B_0(x,y))\,dx.
\]
Completing the square gives
\[
q_P(x)+B_0(x,y)=q_P(x+P^{-1}y)-q_{P^{-1}}(y).
\]
Let $B_N=p^{-N}L_0$ and $c_N=\int_{B_N}\psi_p(q_P(x))\,dx$. On any fixed
compact set of $y$, translation by $P^{-1}y$ preserves $B_N$ for all
sufficiently large $N$. The transform of
$\mathbf1_{B_N}\psi_p(q_P)$ on that set is therefore
$c_N\psi_p(-q_{P^{-1}})$. If this identity is paired with a
Schwartz--Bruhat function, its left side stabilizes with $N$, since the
Fourier transform of the test function has compact support. Choosing a
test function whose integral against $\psi_p(-q_{P^{-1}})$ is nonzero shows
that $c_N$ stabilizes as well. This proves the distributional shape of the
formula.
Differencing the absolute square of $c_N$ and integrating in one variable
gives
\[
|c_N|^2=\vol(B_N)
\int_{B_N\cap P^{-1}B_N^\perp}\psi_p(q_P(h))\,dh.
\]
For sufficiently large $N$, the intersection is $P^{-1}p^NL_0$ and
$q_P$ is integral on it. The right side then equals
$\vol(p^{-N}L_0)\vol(P^{-1}p^NL_0)=|\det P|_p^{-1}$.
Hence the stabilized constant is $\omega(P)|\det P|_p^{-1/2}$ with unit
phase. The transform of $\mathbf1_{L_0}$ is $\mathbf1_{L_0}$, so pairing
the distribution identity with this function gives
Equation~\eqref{eq:fourier-general-quadratic}.
Under $q_{T^*PT}=q_P\circ T$, changing variables multiplies the Gaussian
constant by $|\det T|_p^{-1}$ and its normalizing determinant square root by
$|\det T|_p$. Their product, the phase, is unchanged; this is also the
isometry invariance in Weil~\cite[Chapter~II, \S25, p.~173]{Weil1964}.
For $P=u\rho(\alpha)$ its inverse is $u^{-1}\rho(\beta)$. Rank two over
$E$ gives $\det\rho(\alpha)=\Nm_{E/\Q_p}(\alpha)^2$, and $|u|_p=1$.
This proves Equation~\eqref{eq:fourier-quadratic-transform}. Taking
$T=\rho(r)$ proves the last phase identity. The polar operator of $q_P$ is
$P$, because its polarization is $B_0(Px,y)$. In particular this proof
never assumes that $2$ is a unit of $\Z_p$.
\end{proof}
\subsection{The inverse transform on a shell}
Extend the restriction of $\gamma_p$ to a shell by zero, and denote its
inverse transform by
\begin{equation}\label{eq:fourier-shell-transform}
\gamma_{p,\mathfrak h_p}(A)
=\mathbf1_{\mathscr S_{\mathfrak h_p}}(A)\gamma_p(A),
\qquad
K_{p,\mathfrak h_p}(z)
=\int_E\gamma_{p,\mathfrak h_p}(A)
\psi_p(-\Tr_{E/\Q_p}(Az))\,dA.
\end{equation}
The next lemma gives the estimates for these functions which will be used
in Poisson summation. Its quantitative support assertions separate the case where
the denominator ideal is contained in the stabilizer order.
\begin{lemma}[Intrinsic shell estimates]\label{lem:fourier-shell-estimates}
Let $E$ be the cubic algebra above, let $W$ have rank two over $E$ with
self-adjoint action for a nondegenerate symmetric form $B$, and let $L$ be
self-dual for $B$, with $Q_1(L)\subset\Z_p$. Use the stabilizer order
$\mathcal C_L=\{a\in E:\rho(a)L\subset L\}$ and $\Psi_E,\gamma_p$ from
Equations~\eqref{eq:fourier-polar-identity} and \eqref{eq:fourier-gamma}.
Fix an integral ideal
$\mathfrak h_p=\prod_\nu\mathfrak q_\nu^{h_\nu}$, and use
$H_p,g_p$ from Equation~\eqref{eq:fourier-shell-indices}.
The zero-extended function $\gamma_{p,\mathfrak h_p}$ and its inverse
$K_{p,\mathfrak h_p}$ are Schwartz--Bruhat functions. Moreover,
\[
\supp K_{p,\mathfrak h_p}\subset(\mathcal C_L^\sharp)^\vee
\subset\{z\in E:\Tr_{E/\Q_p}z\in\Z_p\}.
\]
For
$A\in\mathscr S_{\mathfrak h_p}$ choose $s$ and $j_p=j_p(A,s)$ as in
Lemma~\ref{lem:fourier-shift}. Then
\begin{equation}\label{eq:fourier-shell-decay}
|\gamma_p(A)|\le p^{3\iota_p}H_p^{-1}p^{-j_p/2},
\qquad
\|K_{p,\mathfrak h_p}\|_\infty
\le p^{3\iota_p}\vol(R_E)\le p^{3\iota_p}.
\end{equation}
Here $\iota_p=\mathbf1_{\{p\le5\}}$, so the first constant is $1$ at
every $p>5$ and is at most $125$ at the three remaining primes.
If $g_p>0$ or $p\le5$, then
\begin{equation}\label{eq:fourier-inverse-support}
\supp K_{p,\mathfrak h_p}
\subset p^{-4(g_p+1)}\mathfrak h_p\mathfrak d_E^{-1}.
\end{equation}
If $g_p=0$ and $p>5$, then support instead requires, for each $\nu$,
\begin{equation}\label{eq:fourier-boundary-support}
z_\nu\in
\begin{cases}
\mathfrak d_\nu^{-1},&h_\nu=0,\\
\mathfrak q_\nu^{h_\nu-1}\mathfrak d_\nu^{-1},&h_\nu>0.
\end{cases}
\end{equation}
For a $z$ satisfying these support conditions, define its set of boundary
primes by
\[
\partial_{\mathfrak h_p}(z)
=\{\mathfrak q_\nu:h_\nu>0,
\ z_\nu\notin\mathfrak q_\nu^{h_\nu}\mathfrak d_\nu^{-1}\}.
\]
In this case
\begin{equation}\label{eq:fourier-boundary-decay}
|K_{p,\mathfrak h_p}(z)|
\le 8\vol(R_E)
\prod_{\mathfrak q_\nu\in\partial_{\mathfrak h_p}(z)}q_\nu^{-1/2}.
\end{equation}
Every constant is uniform in the lattice, the order $\mathcal C_L$, the
shell, and the field factors and their ramification.
\end{lemma}
\begin{proof}
For $c\in\mathcal C_L^\sharp$, Equation~\eqref{eq:fourier-polar-identity}
shows that $Q_c(L)\subset\Z_p$. Consequently $\gamma_p(A+c)=\gamma_p(A)$.
Each denominator shell is invariant under translation by $R_E$: adding an
integral element preserves every negative component valuation and preserves
the set of integral components. Since $\mathcal C_L^\sharp\subset R_E$,
the zero-extended shell restriction retains this period. Its inverse is
therefore supported in the annihilator $(\mathcal C_L^\sharp)^\vee$, whose
elements have integral trace because $1\in\mathcal C_L^\sharp$.
The order $\mathcal C_L^\sharp$ is an open lattice, and the shell is compact
open. Thus the zero extension in Equation~\eqref{eq:fourier-shell-transform}
is Schwartz--Bruhat, as is its Fourier inverse.
For a fixed unit $u$ put
$I_\beta(u)=\int_L\psi_p(-u^{-1}Q_\beta(x))\,dx$.
Polarization and additive orthogonality on the self-dual lattice give
\[
|I_\beta(u)|^2
=\int_{\substack{h\in L\\\rho(\beta)h\in L}}
\psi_p(-u^{-1}Q_\beta(h))\,dh.
\]
Indeed, in the difference $Q_\beta(y+h)-Q_\beta(y)$ the term linear in $y$
is $B(\rho(\beta)h,y)$, whose integral is the indicator of
$\rho(\beta)h\in L$. The absolute value of the last display is at most the
relative mass of $L\cap\rho(\beta)^{-1}L$. If
$p^{a_1},\ldots,p^{a_6}$ are the Smith elementary divisors of
$\rho(\beta)$ on $L$, then
\[
j_p=\max\{0,-\min_i a_i\},\qquad
[L:L\cap\rho(\beta)^{-1}L]
=p^{\sum_i\max(0,-a_i)}\ge p^{j_p}.
\]
The equality for $j_p$ is exactly the definition of $\mathcal C_L$.
It follows that $|I_\beta(u)|\le p^{-j_p/2}$. Apply
Equations~\eqref{eq:fourier-scalar-shift-invariance} and
\eqref{eq:fourier-quadratic-transform}, then use
Equation~\eqref{eq:fourier-shift-bounds} and average over $u$. This proves
the first bound in Equation~\eqref{eq:fourier-shell-decay}. The shell is
contained in $\mathfrak h_p^{-1}$, which has volume
$H_p\vol(R_E)$. Integration of the first bound, with $p^{-j_p/2}\le1$,
proves the second.
We next prove Equation~\eqref{eq:fourier-inverse-support}. Put
$N=4(g_p+1)$ and take $t\in p^N\mathfrak h_p^{-1}$. For $A$ in the shell
choose $s$ as in Lemma~\ref{lem:fourier-shift}, and write
$\alpha=A+s$, $\beta=\alpha^{-1}$. The component valuations of $t$ are
at least $-h_\nu+e_\nu N$. Translation by $t$ therefore preserves the
shell, and the same $s$ still has the valuation properties of that lemma.
Set $\zeta=t\beta$. Equation~\eqref{eq:fourier-shift-bounds} gives
\[
\zeta\in p^{N-\iota_p}R_E,\qquad
\alpha+t=\alpha(1+\zeta),\qquad
(\alpha+t)^{-1}-\beta
=-\frac{t\beta^2}{1+\zeta}
\in p^{N-2\iota_p}\mathfrak h_p.
\]
Here $1+\zeta$ is a unit. It is a square unit in each factor when $p$ is
odd, by Hensel's lemma on principal units. When $p=2$, one has
$\zeta\in8R_E$. In the $\nu$-th field the Hensel inequality for
$Y^2-(1+\zeta_\nu)$ at $Y=1$ is
$v_\nu(\zeta_\nu)\ge3e_\nu>2e_\nu=2v_\nu(2)$, so it is a square unit
there as well, including for ramified dyadic factors.
Moreover $N-2\iota_p\ge g_p+v_p(2)$. Equation~\eqref{eq:fourier-order-containment}
therefore places the inverse difference in $2\mathcal C_L$. Its $Q$-value
on $L$ is integral by Equation~\eqref{eq:fourier-polar-identity}.
In Equation~\eqref{eq:fourier-quadratic-transform}, the inverse integral is
unchanged by this difference. The norm is unchanged by the unit $1+\zeta$,
and the phase is unchanged because that unit is a square. The original
integral, for each fixed $u$, is consequently unchanged under $A\mapsto A+t$.
So is its unit average. Translation by $p^N\mathfrak h_p^{-1}$ preserves
the shell as a set, hence preserves its zero-extended function. The inverse
transform is supported on the annihilator of this group, namely
$p^{-N}\mathfrak h_p\mathfrak d_E^{-1}$. This proves
Equation~\eqref{eq:fourier-inverse-support} with the absolute exponent $4$.
It remains to prove the sharper assertions for $g_p=0$ and $p>5$. The
equality $g_p=0$ says $\mathfrak h_p\subset\mathcal C_L$. For each factor
with $h_\nu=0$, its component idempotent $1_\nu$ and its whole component
ring $1_\nu R_\nu$ lie in $\mathfrak h_p$. The self-adjoint projections
$\rho(1_\nu)$ preserve $L$ and split off their lattices orthogonally.
The complementary projection $1-\sum_{h_\nu=0}\rho(1_\nu)$ also preserves
$L$. Each resulting summand is self-dual, because the splitting is
orthogonal and $L$ is self-dual.
On an $h_\nu=0$ summand, an integral $A_\nu$ preserves its lattice, so
$Q_{A_\nu}$ is integral there since $2$ is a unit. That factor of the
integral in Equation~\eqref{eq:fourier-gamma} is $1$. On the combined
summand of all $h_\nu>0$, the frequency $A_+$ is invertible. Its inverse,
extended by zero on the other factors, lies in
$\mathfrak h_p\subset\mathcal C_L$. Its inverse quadratic integral in
Equation~\eqref{eq:fourier-general-quadratic} is therefore also $1$.
On that summand the square root of the polar determinant has absolute value
$H_p$, because each positive factor still has rank two. Applying the formula
therefore shows that the remaining frequency function is $H_p^{-1}$ times a
unit-average of Gaussian phases. Its absolute value is at most $H_p^{-1}$.
If there are no positive
$h_\nu$, this conclusion has $H_p=1$ and the empty summand contributes $1$.
Let $\chi_\nu$ be the quadratic character of
$(R_\nu/\mathfrak q_\nu)^\times$ for a factor with $h_\nu>0$. At odd
residue characteristic every principal unit is a square, so unit square
classes are exactly the two residue square classes. Phase invariance in
Lemma~\ref{lem:fourier-quadratic-transform} shows that the bounded unit-average
just obtained depends only on the tuple of these classes. This use of phase
invariance concerns the ambient quadratic form; a unit need not preserve
the lattice. Fourier expansion on the product of the two-element
square-class groups expresses the shell function as a sum of at most
$2^{|\{\nu:h_\nu>0\}|}\le8$ terms of the form
\[
c_{\boldsymbol\varepsilon}H_p^{-1}
\prod_{h_\nu=0}\mathbf1_{R_\nu}(A_\nu)
\prod_{h_\nu>0}
\mathbf1_{\pi_\nu^{-h_\nu}R_\nu^\times}(A_\nu)
\chi_\nu(\overline{\pi_\nu^{h_\nu}A_\nu})^{\varepsilon_\nu},
\qquad |c_{\boldsymbol\varepsilon}|\le1,
\]
where $\varepsilon_\nu\in\{0,1\}$. The bound for the coefficients follows
from the normalized finite Fourier expansion of a function of absolute
value at most $1$. It also covers any coupling of components caused by
the common average over $u$.
We compute the inverse of each factor. Put
$\psi_\nu=\psi_p\circ\Tr_{E_\nu/\Q_p}$ and use the component self-dual
measure. For $h_\nu=0$, the integral over $R_\nu$ is
$\vol(R_\nu)\mathbf1_{\mathfrak d_\nu^{-1}}(z_\nu)$.
For $h=h_\nu>0$, the normalized inverse of the trivial unit character is
\[
q_\nu^{-h}\int_{\pi_\nu^{-h}R_\nu^\times}
\psi_\nu(-Az)\,dA
=\vol(R_\nu)\left(
\mathbf1_{\mathfrak q_\nu^h\mathfrak d_\nu^{-1}}(z)
-q_\nu^{-1}\mathbf1_{\mathfrak q_\nu^{h-1}\mathfrak d_\nu^{-1}}(z)
\right).
\]
This follows by subtracting the integral over $\pi_\nu^{-h+1}R_\nu$ from
that over $\pi_\nu^{-h}R_\nu$ and using their annihilators. For the quadratic
unit character, its function of $a=\pi_\nu^hA$ is invariant under addition
by $\mathfrak q_\nu$. This forces the normalized inverse to vanish outside
$\mathfrak q_\nu^{h-1}\mathfrak d_\nu^{-1}$. It also vanishes on
$\mathfrak q_\nu^h\mathfrak d_\nu^{-1}$, where the additive character on
$R_\nu$ is trivial and the sum of $\chi_\nu$ is zero. At the intervening
boundary the additive character descends to a nontrivial character $\eta$
of the residue field $k_\nu$, and the normalized integral is
$\vol(R_\nu)q_\nu^{-1}$ times its quadratic Gauss sum.
For completeness, if $\chi$ is the quadratic character of a finite field
$k$ of odd order $q$ and $\eta$ is a nontrivial additive character, then
\[
\left|\sum_{a\in k^\times}\chi(a)\eta(a)\right|^2
=\sum_{t\in k^\times}\chi(t)
\sum_{b\in k^\times}\eta((t-1)b)
=(q-1)-\sum_{\substack{t\in k^\times\\t\ne1}}\chi(t)=q.
\]
Thus the quadratic factor has magnitude
$\vol(R_\nu)q_\nu^{-1/2}$ at the boundary. The trivial factor there has
the smaller magnitude $\vol(R_\nu)q_\nu^{-1}$. Multiplying these
component estimates, using $H_p=\prod_{h_\nu>0}q_\nu^{h_\nu}$ and
$\prod_\nu\vol(R_\nu)=\vol(R_E)$, and summing at most eight terms proves
Equations~\eqref{eq:fourier-boundary-support} and
\eqref{eq:fourier-boundary-decay}. The annihilator computation retains the
different in every ramified factor.
\end{proof}
The density need not be locally constant on the component-vanishing locus.
For the nonvanishing outputs that occur in the global field, the shell
inverses nevertheless give a finite pointwise expansion.
\begin{lemma}[Pointwise inverse expansion]\label{lem:fourier-inverse-expansion}
If $z=(z_\nu)\in E$ has $z_\nu\ne0$ for every $\nu$, then
\begin{equation}\label{eq:fourier-inverse-expansion}
f_p^*(z)=\sum_{\mathfrak h_p}K_{p,\mathfrak h_p}(z),
\end{equation}
where the sum is over integral $R_E$-ideals and only finitely many summands
are nonzero.
\end{lemma}
\begin{proof}
Let $B_n=p^{-n}R_E$. Fubini and additive orthogonality give
\[
\int_{B_n}\gamma_p(A)\psi_p(-\Tr_{E/\Q_p}(Az))\,dA
=\vol(B_n)\,\mu_p(z+p^n\mathfrak d_E^{-1}).
\]
The self-dual normalizations give
\[
B_n^\perp=p^n\mathfrak d_E^{-1},\qquad
\vol(B_n)\vol(B_n^\perp)=1.
\]
Thus the right side is the average density on that neighborhood. By
Lemma~\ref{lem:fourier-density} it equals $f_p^*(z)$ for sufficiently large
$n$. The ball $B_n$ is the disjoint union of the finitely many shells with
$h_\nu\le ne_\nu$.
There can in fact be only finitely many nonzero shell inverses at this $z$.
The injection
$\mathfrak h_p/(\mathfrak h_p\cap\mathcal C_L)\hookrightarrow R_E/\mathcal C_L$
gives $g_p\le v_p([R_E:\mathcal C_L])$. In
Equation~\eqref{eq:fourier-inverse-support}, a nonzero value requires
\[
h_\nu\le v_\nu(z_\nu)+d_\nu+4e_\nu(g_p+1).
\]
In Equation~\eqref{eq:fourier-boundary-support}, it requires
$h_\nu\le v_\nu(z_\nu)+d_\nu+1$ whenever $h_\nu>0$.
All right sides are bounded for this $z$, because its component valuations
are finite. The shell sum therefore stabilizes, and the preceding ball
inversion proves Equation~\eqref{eq:fourier-inverse-expansion}.
\end{proof}
The exact scalar coefficients in Lemma~\ref{lem:fourier-scalar-density}
supply the trace marginal in the next section. The nonscalar estimate uses
the pointwise decay in Equation~\eqref{eq:fourier-shell-decay}, the inverse
supports in Equations~\eqref{eq:fourier-inverse-support} and
\eqref{eq:fourier-boundary-support}, and the boundary bound in
Equation~\eqref{eq:fourier-boundary-decay}. In the case $g_p=0$ and $p>5$,
the last upper bound contains a factor $q_\nu^{-1/2}$ for each actual boundary
component where the support permits one fewer power of $\mathfrak q_\nu$.
Once the real boxes are specified, Section~\ref{sec:additive} combines the
supports with the nonzero field norm to restrict the denominator profiles in
Poisson summation.
==== END Equidistribution-of-Primitive-Quartic-Torus-Packets-for-Arbitrary-Orders//local-fourier.tex ====
==== BEGIN Equidistribution-of-Primitive-Quartic-Torus-Packets-for-Arbitrary-Orders//macros.tex ====
\newcommand{\Q}{\mathbb Q}
\newcommand{\R}{\mathbb R}
\newcommand{\Z}{\mathbb Z}
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\newcommand{\E}{\mathbb E}
\renewcommand{\P}{\mathbb P}
\newcommand{\OO}{\mathcal O}
\newcommand{\eps}{\varepsilon}
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\DeclareMathOperator{\Gal}{Gal}
\DeclareMathOperator{\Disc}{Disc}
\DeclareMathOperator{\Res}{Res}
\DeclareMathOperator{\covol}{covol}
\DeclareMathOperator{\vol}{vol}
\DeclareMathOperator{\diag}{diag}
\DeclareMathOperator{\rank}{rank}
\DeclareMathOperator{\End}{End}
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\DeclareMathOperator{\supp}{supp}
\DeclareMathOperator{\id}{id}
\newtheorem{theorem}{Theorem}[section]
\newtheorem{proposition}[theorem]{Proposition}
\newtheorem{lemma}[theorem]{Lemma}
\newtheorem{corollary}[theorem]{Corollary}
\theoremstyle{definition}
\newtheorem{definition}[theorem]{Definition}
\theoremstyle{remark}
\newtheorem{remark}[theorem]{Remark}
\numberwithin{equation}{section}
==== END Equidistribution-of-Primitive-Quartic-Torus-Packets-for-Arbitrary-Orders//macros.tex ====
==== BEGIN Equidistribution-of-Primitive-Quartic-Torus-Packets-for-Arbitrary-Orders//main.tex ====
\documentclass[11pt]{article}
\usepackage[T1]{fontenc}
\usepackage{lmodern,amsmath,amssymb,amsthm,mathtools}
\usepackage[mathscr]{eucal}
\usepackage[margin=1.05in]{geometry}
\usepackage{microtype,graphicx,xcolor,booktabs}
\usepackage{hyperref}
\hypersetup{colorlinks=true,linkcolor=blue!45!black,citecolor=blue!45!black,urlcolor=blue!45!black,pdftitle={Equidistribution of primitive quartic torus packets for arbitrary orders},pdfauthor={OpenAI}}
\input{macros}
\title{Equidistribution of primitive quartic torus packets\\for arbitrary orders}
\author{OpenAI}
\date{October 5, 2026}
\begin{document}
\maketitle
\begin{abstract}
Let $K$ range over totally real quartic fields with no proper intermediate field, and let $\OO$ be any order in $K$. We prove that the packets of periodic diagonal orbits attached to invertible $\OO$-ideal classes, weighted by orbit volume, equidistribute with no escape of mass as $|\Disc(\OO)|$ tends to infinity. The proof combines measure rigidity with a cubic-resolvent estimate that remains uniform at primes dividing the order index.
\end{abstract}
\tableofcontents
\input{introduction}
\input{packets}
\input{ordinary}
\input{rigidity}
\input{exterior}
\input{local-fourier}
\input{additive}
\input{sieve}
\input{conclusion}
\bibliographystyle{plain}
\bibliography{references}
\end{document}
==== END Equidistribution-of-Primitive-Quartic-Torus-Packets-for-Arbitrary-Orders//main.tex ====
==== BEGIN Equidistribution-of-Primitive-Quartic-Torus-Packets-for-Arbitrary-Orders//ordinary.tex ====
\section{Ordinary vectors and probability limits}\label{sec:ordinary}
We first control vectors in the lattices themselves. A count near the
origin will prevent escape into the cusp, and a count near a vector with
no zero coordinate will bound the mass of small neighborhoods in \(X_4\).
The latter bound is the input to the entropy argument in
Section~\ref{sec:rigidity}. We follow the method of
\cite[Sections~2--5]{PrimeDegreePackets}, proving the estimates with
constants uniform over primitive totally real quartic fields, their orders,
and the orderings of their embeddings. The absence of a quadratic
subfield supplies the residue estimate needed in this degree.
\subsection{Containment probabilities and unfolding}
For a rational prime \(p\), write
\[
R_p=\prod_{\mathfrak p\mid p}R_{\mathfrak p}.
\]
Here \(R_{\mathfrak p}\) is the integer ring of the completion
\(K_{\mathfrak p}\). Let \(v_{\mathfrak p}\) be its integer valuation,
normalized to be \(1\) on a uniformizer, and put
\[
e_{\mathfrak p}=v_{\mathfrak p}(p),\qquad
f_{\mathfrak p}=[R/\mathfrak p:\Z/p\Z].
\]
Thus \(\Nm\mathfrak p=p^{f_{\mathfrak p}}\) and
\(\sum_{\mathfrak p\mid p}e_{\mathfrak p}f_{\mathfrak p}=4\);
see \cite[Theorem~3.34]{MilneANT}. For \(p\mid q\) and a tuple
\(\boldsymbol l_p=(l_{\mathfrak p})_{\mathfrak p\mid p}\) of
nonnegative integers, call the set of \(x\in R_p\) with
\(v_{\mathfrak p}(x_{\mathfrak p})=l_{\mathfrak p}\) in every component
the valuation shell \(\boldsymbol l_p\). Choose \(x\) in this shell
and define
\begin{equation}\label{eq:ord-local-probability}
P_p(\boldsymbol l_p)
=\P_{t_p\in R_p^\times}\{x\in t_p\OO_p\},
\end{equation}
where the probability is normalized unit Haar measure. The definition
is independent of \(x\): the unit group acts transitively on each set
with specified component valuations.
For a nonzero integral \(R\)-ideal \(\mathfrak a\), set
\begin{equation}\label{eq:ord-ideal-weight}
P(\mathfrak a)=
\prod_{p\mid q}
P_p\bigl((v_{\mathfrak p}(\mathfrak a))_{\mathfrak p\mid p}\bigr).
\end{equation}
An empty product, including the case \(q=1\), is \(1\). If
\(0\ne b\in I\), its image has no zero local component and
\((b)I^{-1}\) is integral. In the coordinates
\(I_p=\xi_pR_p\), the valuations of \(\xi_p^{-1}b\) are those of
\((b)I^{-1}\). Lemma~\ref{lem:pkt-local-labels} and the independence
of the local choices therefore give
\begin{equation}\label{eq:ord-membership}
\P_{L\in\mathscr L_{\OO}(I)}\{b\in L\}
=P((b)I^{-1}).
\end{equation}
For a nonnegative Borel function \(f\) on \(\R^4\), write
\[
E_f(\Lambda)=\sum_{0\ne x\in\Lambda}f(x).
\]
This is a Borel function on \(X_4\): in a local basis chart it is a
countable sum of the Borel functions obtained by evaluating \(f\) at
the integer linear combinations of that basis.
For \(z>0\), define the integral over the surface with absolute
coordinate product \(z\) by
\begin{equation}\label{eq:ord-fiber}
\mathcal V_f(z)=
\sum_{\omega\in\{\pm1\}^4}
\int_{\R^3}
f\bigl(\omega_1e^{u_1},\omega_2e^{u_2},\omega_3e^{u_3},
\omega_4z e^{-u_1-u_2-u_3}\bigr)\,du_1\,du_2\,du_3 .
\end{equation}
The signs and logarithmic measure are exactly those of \(T_\infty\).
\begin{proposition}[Ordinary vector unfolding]\label{prop:ord-unfolding}
For every nonnegative Borel \(f\) on \(\R^4\),
\begin{equation}\label{eq:ord-unfolding}
\int_{X_4}E_f\,d\mu_*
=\frac1{\sqrt D\,\kappa}
\sum_{\substack{\mathfrak a\subset R\\\mathfrak a\ne0}}
P(\mathfrak a)\,
\mathcal V_f\bigl(\Nm\mathfrak a/X\bigr).
\end{equation}
Both sides are allowed to be infinite.
\end{proposition}
\begin{proof}
This is the unfolding of \cite[Proposition~2.2]{PrimeDegreePackets}
with the packet normalization of Proposition~\ref{prop:pkt-model}.
Average the vector sum over \(L\) in
Equation~\eqref{eq:pkt-integration}, and use
Equation~\eqref{eq:ord-membership}. The factor \(N_{\OO}\) cancels,
and the left side becomes
\[
\frac1{h_K\vol(\mathscr F_U)}
\sum_I\int_{\mathscr F_U}
\sum_{0\ne b\in I}P((b)I^{-1})f(c_I\sigma(b)v)\,dv .
\]
For fixed \(I\), the map \(b\mapsto(b)I^{-1}\) takes \(I\setminus\{0\}\)
onto the integral ideals in the inverse class of \(I\). Each fiber
is one free \(U\)-orbit. Indeed, two generators of the same principal
ideal differ by a unique unit. The coefficient \(P((b)I^{-1})\) is
constant on that orbit. Summing its unit translates and integrating
over \(\mathscr F_U\) gives
\[
\int_{\mathscr F_U}\sum_{\epsilon\in U}
f(c_I\sigma(\epsilon b)v)\,dv
=\int_{T_\infty}f(c_I\sigma(b)v)\,dv .
\]
The translates tile \(T_\infty\), including its sign components.
All changes in the order of integration and summation are justified
by Tonelli's theorem.
The absolute product of the coordinates of \(c_I\sigma(b)\) is
\[
c_I^4|\Nm_{K/\Q}(b)|
=\frac{\Nm((b)I^{-1})}{X}.
\]
Translation of the first three logarithmic coordinates and relabeling
of the signs therefore identify the last integral with
\(\mathcal V_f(\Nm((b)I^{-1})/X)\), with Jacobian one. Each integral
ideal belongs to the inverse class of exactly one chosen \(I\).
Equation~\eqref{eq:pkt-unit-volume} now gives the coefficient in
Equation~\eqref{eq:ord-unfolding}.
\end{proof}
\subsection{The mass and tail of the local weights}
The ideal sum in Equation~\eqref{eq:ord-unfolding} need not have simple
coefficients at primes dividing \(q\). We will use its exact total
mass over valuation shells and a positive moment of their norms.
Both statements are the order case of
\cite[Lemma~3.1]{PrimeDegreePackets}; the argument below explains why
they are uniform even when the local order has large index.
For \(p\mid q\), put
\[
q_p=[R_p:\OO_p],\qquad
N_p(\boldsymbol l_p)=p^{\sum_{\mathfrak p\mid p}
f_{\mathfrak p}l_{\mathfrak p}},
\qquad
Z_p=\prod_{\mathfrak p\mid p}(1-p^{-f_{\mathfrak p}})^{-1}.
\]
For a collection \(\boldsymbol l=(\boldsymbol l_p)_{p\mid q}\), write
\[
d(\boldsymbol l)=\prod_{p\mid q}N_p(\boldsymbol l_p),\qquad
P(\boldsymbol l)=\prod_{p\mid q}P_p(\boldsymbol l_p),\qquad
Z_q=\prod_{p\mid q}Z_p .
\]
Thus \(d(\boldsymbol l)\) is the norm of the part of an integral ideal
supported above primes dividing \(q\), and \(\prod_{p\mid q}q_p=q\).
\begin{lemma}[Local mass and moment]\label{lem:ord-shells}
With the preceding notation,
\begin{equation}\label{eq:ord-shell-mass}
\sum_{\boldsymbol l}\frac{P(\boldsymbol l)}{d(\boldsymbol l)}
=\frac{Z_q}{q}.
\end{equation}
For every \(\eta>0\) there is a constant \(C_\eta\), independent of
\(K,\OO\), such that
\begin{equation}\label{eq:ord-shell-moment}
\sum_{\boldsymbol l}
P(\boldsymbol l)d(\boldsymbol l)^{-1+1/8}
\le C_\eta\,\frac{q^\eta}{q}.
\end{equation}
In particular there is an absolute constant \(C\) for which
\begin{equation}\label{eq:ord-shell-tail}
\sum_{d(\boldsymbol l)>X^{1/8}}
\frac{P(\boldsymbol l)}{d(\boldsymbol l)}
\le \frac Cq X^{-1/128}.
\end{equation}
\end{lemma}
\begin{proof}
Normalize additive Haar measure \(m_p\) on \(R_p\) by \(m_p(R_p)=1\).
In a component \(R_{\mathfrak p}\), the set of elements of valuation
\(l_{\mathfrak p}\) has measure
\((1-p^{-f_{\mathfrak p}})p^{-f_{\mathfrak p}l_{\mathfrak p}}\).
Consequently the shell \(\boldsymbol l_p\) in \(R_p\) has measure
\begin{equation}\label{eq:ord-shell-measure}
m_p(\text{shell }\boldsymbol l_p)
=\frac1{Z_p\,N_p(\boldsymbol l_p)}.
\end{equation}
Its conditional additive measure is invariant under the transitive
unit action. It is therefore the image of unit Haar probability.
By Equation~\eqref{eq:ord-local-probability},
\(P_p(\boldsymbol l_p)\) is exactly the fraction of this shell lying
in \(\OO_p\). The elements with a zero component have measure zero,
and \(m_p(\OO_p)=q_p^{-1}\). Summing
Equation~\eqref{eq:ord-shell-measure} over the intersections with
\(\OO_p\) gives
\[
\sum_{\boldsymbol l_p}
\frac{P_p(\boldsymbol l_p)}{N_p(\boldsymbol l_p)}
=\frac{Z_p}{q_p}.
\]
Multiplication over \(p\mid q\) proves
Equation~\eqref{eq:ord-shell-mass}.
For the moment, choose \(x\) with normalized additive Haar probability
on \(\OO_p\), and let \(v_{\mathfrak p}=v_{\mathfrak p}(x_{\mathfrak p})\).
The projection \(H_{\mathfrak p}\) of \(\OO_p\) into
\(R_{\mathfrak p}\) contains the embedded copy of \(\Z_p\), because
\(1\in\OO_p\). Its induced probability is normalized Haar on
\(H_{\mathfrak p}\). For an integer \(k\ge1\), the image of this
copy of \(\Z_p\) modulo the ideal of valuation \(k\) has
\(p^{\lceil k/e_{\mathfrak p}\rceil}\) elements. Hence the image of
\(H_{\mathfrak p}\) has at least that many elements, and
\begin{equation}\label{eq:ord-valuation-tail}
\P_{\OO_p}\{v_{\mathfrak p}\ge k\}
\le p^{-\lceil k/e_{\mathfrak p}\rceil}.
\end{equation}
This uses only the size of the additive image of the projection; it
does not require the projections to be independent.
Set
\[
j=\max_{\mathfrak p\mid p}
\left\lceil\frac{v_{\mathfrak p}}{e_{\mathfrak p}}\right\rceil.
\]
If \(j\ge a\ge1\), then some \(v_{\mathfrak p}\) is at least
\(e_{\mathfrak p}(a-1)+1\). Equation~\eqref{eq:ord-valuation-tail}
and a union bound over at most four components give
\(\P_{\OO_p}\{j\ge a\}\le4p^{-a}\). On the other hand,
\[
N_p(x)=p^{\sum_{\mathfrak p\mid p}f_{\mathfrak p}v_{\mathfrak p}}
\le p^{j\sum_{\mathfrak p\mid p}e_{\mathfrak p}f_{\mathfrak p}}
=p^{4j}.
\]
These assertions hold outside the null sets where a component vanishes.
The tail formula for the expectation of \(p^{j/2}\) now yields
\begin{align}
\E_{\OO_p}N_p(x)^{1/8}
&\le \E_{\OO_p}p^{j/2} \notag\\
&=1+\sum_{a\ge1}
(p^{a/2}-p^{(a-1)/2})\P_{\OO_p}\{j\ge a\}
\le 1+4p^{-1/2}.
\label{eq:ord-local-moment}
\end{align}
Applying Equation~\eqref{eq:ord-shell-measure} once more gives the
exact factorization
\[
\sum_{\boldsymbol l}
P(\boldsymbol l)d(\boldsymbol l)^{-1+1/8}
=\frac1q\prod_{p\mid q}
\left(Z_p\,\E_{\OO_p}N_p(x)^{1/8}\right).
\]
Each factor is at most
\((1-p^{-1})^{-4}(1+4p^{-1/2})\). For any fixed \(\eta>0\), this is
at most \(p^\eta\) at all sufficiently large primes. The product of
the factors at the finitely many remaining primes is bounded by a
constant depending only on \(\eta\). Since
\(\prod_{p\mid q}p\le q\), this proves
Equation~\eqref{eq:ord-shell-moment}. Finally, take \(\eta=1/128\)
there and use \(q\le X\):
\[
\sum_{d(\boldsymbol l)>X^{1/8}}\frac{P(\boldsymbol l)}{d(\boldsymbol l)}
\le X^{-1/64}
\sum_{\boldsymbol l}P(\boldsymbol l)d(\boldsymbol l)^{-1+1/8}
\le \frac Cq X^{-1/128}.
\]
This proves Equation~\eqref{eq:ord-shell-tail}.
\end{proof}
\subsection{Ideal counts relative to the residue}
The shell identity supplies the factor \(q^{-1}\). To use it in
Equation~\eqref{eq:ord-unfolding}, the count of the remaining ideal
part must retain the factor \(\kappa\). Recall
\[
a_K(n)=\#\{\mathfrak a\subset R:\mathfrak a\ne0,\ \Nm\mathfrak a=n\},
\qquad n\ge1,
\]
and let \(d_4\) be the fourfold divisor function. Unique factorization
of ideals makes \(a_K\) multiplicative. At every rational prime,
including a ramified one, its local generating series is
\[
\sum_{j\ge0}a_K(p^j)T^j
=\prod_{\mathfrak p\mid p}(1-T^{f_{\mathfrak p}})^{-1}.
\]
Each factor is coefficientwise at most \((1-T)^{-1}\), and there are
at most four factors. Hence \(a_K(n)\le d_4(n)\) for every \(n\);
in particular \(a_K(p)\le4\).
\begin{lemma}[Residue and prime sums]\label{lem:ord-residue}
There are absolute constants \(D_0>1\), \(h_0>0\), \(Y_0>1\), and
\(C\ge1\) with the following properties for every primitive totally
real quartic field \(K\).
\begin{enumerate}
\item If \(D\ge D_0\), then
\begin{equation}\label{eq:ord-residue-comparison}
C^{-1}\kappa\le h\zeta_K(1+h)\le C\kappa
\qquad(0<h\le4/\log D).
\end{equation}
If \(D<D_0\), the same inequalities hold for \(0<h\le h_0\).
\item One has
\begin{equation}\label{eq:ord-residue-lower}
\kappa^{-1}\le C\log(3D).
\end{equation}
\item Whenever \(Y\ge\max\{Y_0,D^{1/4}\}\),
\begin{equation}\label{eq:ord-prime-sum}
\exp\left(\sum_{p\le Y}\frac{a_K(p)}p\right)
\le C\kappa\log Y .
\end{equation}
\end{enumerate}
The sum in Equation~\eqref{eq:ord-prime-sum} is over rational primes.
\end{lemma}
\begin{proof}
We first treat large \(D\), and put \(L=\log D\).
Stark's zero theorem \cite[Lemma~3]{Stark1974} gives at most one
nontrivial zero of \(\zeta_K\) in
\[
\Re s\ge1-\frac1{4L},\qquad |\Im s|\le\frac1{4L};
\]
any such zero is real and simple. His
Lemma~8 \cite[Lemma~8]{Stark1974}, valid also for nonnormal fields,
states in degree four that a real zero
\(\beta\ge1-(96L)^{-1}\) would force a quadratic subfield of \(K\).
Primitivity excludes that subfield. It follows that every nontrivial
zero \(\rho\) satisfies
\begin{equation}\label{eq:ord-zero-disk}
|1-\rho|\ge \frac{c_0}{L},\qquad c_0=\frac1{192}.
\end{equation}
Indeed, a zero in the smaller open disk would lie in Stark's rectangle,
would be real, and would be in the forbidden real interval.
For real \(s>1\), the logarithmic derivative of the completed zeta
function gives
\begin{equation}\label{eq:ord-zero-sum}
\begin{aligned}
S_K(s):=\sum_\rho\Re\frac1{s-\rho}
&=\frac1s+\frac1{s-1}+\frac L2+
\frac{\zeta_K'}{\zeta_K}(s)+G(s),\\
G(s)&=2\left(\frac{\Gamma'(s/2)}{\Gamma(s/2)}-\log\pi\right);
\end{aligned}
\end{equation}
see \cite[Equation~(9)]{Stark1974}. The real zero sum converges and
has nonnegative terms, since \(\Re\rho\le1\). At \(s_0=1+L^{-1}\),
the Euler product gives \(\zeta_K'/\zeta_K(s_0)\le0\). The function
\(G\) is bounded on \([1,2]\), so for sufficiently large \(D\),
\[
0\le S_K(s_0)\ll L.
\]
This estimate persists for all \(1<s\le1+4/L\). To see this term by
term, write \(\rho=\beta+i\gamma\). Since \(\beta\le1\),
\[
s-\beta\le4(s_0-\beta),\qquad
|s-\rho|\ge|1-\rho|.
\]
Equation~\eqref{eq:ord-zero-disk} consequently gives
\[
|s_0-\rho|\le |s-\rho|+\frac3L
\le (1+3/c_0)|s-\rho|.
\]
Using \(\Re(s-\rho)^{-1}=(s-\beta)/|s-\rho|^2\), we obtain
\[
\Re\frac1{s-\rho}
\le4(1+3/c_0)^2\Re\frac1{s_0-\rho}.
\]
Thus \(S_K(s)\ll L\) on the entire interval.
Let \(\mathcal Z_K(s)=(s-1)\zeta_K(s)\). Rearranging
Equation~\eqref{eq:ord-zero-sum} shows that
\[
\frac{\mathcal Z_K'}{\mathcal Z_K}(s)
=S_K(s)-\frac1s-\frac L2-G(s)=O(L)
\qquad(1<s\le1+4/L).
\]
The function \(\mathcal Z_K\) extends to \(s=1\) with value
\(\kappa>0\). Integrating its logarithmic derivative across an
interval of length at most \(4/L\) proves
Equation~\eqref{eq:ord-residue-comparison}, with an absolute constant.
At \(s_0\), the inequality \(\zeta_K(s_0)\ge1\) then gives
\(\kappa\gg L^{-1}\).
There are only finitely many fields with \(D<D_0\), by Hermite's
finiteness theorem \cite[Theorem~8.43]{MilneANT}. For each of them,
\(h\zeta_K(1+h)\) tends to the positive residue \(\kappa\) as
\(h\downarrow0\). Taking a common sufficiently small \(h_0\) over
this finite set proves the remaining comparison, and their positive
residues have a common positive lower bound. Enlarging \(C\) proves
Equation~\eqref{eq:ord-residue-lower} for every field.
Choose \(Y_0\) sufficiently large that \(1/\log Y_0\le h_0\), and
let \(Y\ge\max\{Y_0,D^{1/4}\}\). For \(h=1/\log Y\), the comparison
just proved applies: in the large \(D\) case \(h\le4/\log D\), and
in the bounded \(D\) case \(h\le h_0\). Positivity in the Euler
logarithm gives
\[
\sum_{p\le Y}a_K(p)p^{-1-h}\le\log\zeta_K(1+h).
\]
Moreover, \(a_K(p)\le4\), \(1-p^{-h}\le h\log p\), and the elementary
Chebyshev bound followed by partial summation imply
\[
\sum_{p\le Y}a_K(p)(p^{-1}-p^{-1-h})
\le4h\sum_{p\le Y}\frac{\log p}{p}
\ll h\log Y=1.
\]
It follows that
\[
\sum_{p\le Y}\frac{a_K(p)}p
\le \log\zeta_K(1+h)+O(1)
\le \log(\kappa\log Y)+O(1),
\]
which proves Equation~\eqref{eq:ord-prime-sum}. There is no upper
restriction on \(Y\). This observation is what permits \(D\) to stay
bounded while the order index, and hence \(X\), grows.
\end{proof}
We also need an estimate for ideals in an interval. The following
uniform form of Shiu's bound \cite{Shiu1980} is the one recorded in
\cite[Equation~(4.2)]{PrimeDegreePackets}.
\begin{lemma}\label{lem:ord-shiu}
There are absolute constants \(C,u_0\) such that, for every
nonnegative multiplicative function \(b\) with \(b(n)\le d_4(n)\),
every real \(u\ge u_0\), and every \(u^{1/2}\le v\le u\),
\begin{equation}\label{eq:ord-shiu}
\sum_{u-v<n\le u}b(n)
\le C\frac v{\log u}
\exp\left(\sum_{p\le u}\frac{b(p)}p\right).
\end{equation}
\end{lemma}
\begin{proof}
Apply \cite[Theorem~1.1]{Pollack2020} with no excluded residue classes
and with its parameter \(\beta=1/4\). Its two growth conditions hold
with bounds independent of \(b\):
\[
b(p^j)\le d_4(p^j)=\binom{j+3}{3}\le4^j,\qquad
b(n)\le d_4(n)\ll_\epsilon n^\epsilon
\quad(\epsilon>0).
\]
The theorem's constant and threshold depend only on these growth
bounds and on the fixed parameter. Its strict length condition
\(u^\beta<v\) is satisfied even at \(v=u^{1/2}\) once \(u>1\).
This gives Equation~\eqref{eq:ord-shiu}, including both endpoints.
\end{proof}
\begin{proposition}[Weighted ideal intervals]\label{prop:ord-weighted-intervals}
There is an absolute constant \(C\) such that, for each fixed
\(A\ge1\) and \(c>0\), there is a threshold \(X_0(A,c)\) with the
following property. For every \(K,\OO,\sigma\) under consideration
with \(X\ge X_0(A,c)\), and all real \(x,y\) satisfying
\[
X^{3/4}\le x\le AX,\qquad cx\le y\le x,
\]
one has
\begin{equation}\label{eq:ord-weighted-interval}
\sum_{\substack{\mathfrak a\subset R,\ \mathfrak a\ne0\\
x-y<\Nm\mathfrak a\le x}}P(\mathfrak a)
\le C\,\frac{\kappa y}{q}.
\end{equation}
The constant \(C\) is independent of \(A,c\); the threshold may
depend on them.
\end{proposition}
\begin{proof}
The range is empty for \(c>1\), so suppose \(0<c\le1\).
We adapt the separation of the local norm from the remaining ideal
in \cite[Proposition~4.2]{PrimeDegreePackets}. Put
\[
b(n)=a_K(n)\mathbf 1_{(n,q)=1}.
\]
This is nonnegative and multiplicative, and it is at most \(d_4\).
Every integral ideal splits uniquely into its part above the primes
dividing \(q\), described by one tuple \(\boldsymbol l\), and a part
coprime to \(qR\). The left side of
Equation~\eqref{eq:ord-weighted-interval} is therefore
\begin{equation}\label{eq:ord-ideal-split}
\sum_{\boldsymbol l}P(\boldsymbol l)
\sum_{(x-y)/d(\boldsymbol l)<n\le x/d(\boldsymbol l)}b(n).
\end{equation}
First take \(d=d(\boldsymbol l)\le X^{1/8}\), and put \(u=x/d\),
\(v=y/d\). Then
\[
u\ge X^{5/8}\ge D^{1/4},\qquad v\ge cu.
\]
The second inequality implies \(v\ge u^{1/2}\) when \(X\) is large
in terms of \(c\). The first inequality puts \(u\) in the range of
Equation~\eqref{eq:ord-prime-sum}, once \(X\) is above an absolute
threshold. To remove the primes dividing \(q\) from that Euler bound,
observe that
\begin{equation}\label{eq:ord-euler-removal}
\log Z_q=\sum_{p\mid q}\frac{a_K(p)}p+O(1),\qquad
\sum_{\substack{p\mid q\\p>u}}\frac{a_K(p)}p
\ll\frac{\log(2q)}u\ll1 .
\end{equation}
For the first identity, expand
\(\log Z_p=\sum_{\mathfrak p\mid p}\sum_{j\ge1}p^{-jf_{\mathfrak p}}/j\).
The terms of order \(p^{-1}\) count exactly the primes of residue
degree one, and the remaining terms are \(O(p^{-2})\), uniformly
in the splitting type. Their sum over rational primes is bounded.
For the second estimate, use \(a_K(p)\le4\), at most
\(\log(2q)/\log2\) primes dividing \(q\), and
\(u\ge X^{5/8}\), \(q\le X\).
Consequently Equations~\eqref{eq:ord-prime-sum} and
\eqref{eq:ord-euler-removal} give
\[
\exp\left(\sum_{p\le u}\frac{b(p)}p\right)
\ll \frac{\kappa\log u}{Z_q}.
\]
Lemma~\ref{lem:ord-shiu} and
Equation~\eqref{eq:ord-shell-mass} now bound the contribution of
these tuples to Equation~\eqref{eq:ord-ideal-split} by
\[
C\frac{\kappa y}{Z_q}
\sum_{d(\boldsymbol l)\le X^{1/8}}
\frac{P(\boldsymbol l)}{d(\boldsymbol l)}
\le C\frac{\kappa y}{q}.
\]
It remains to control tuples with \(d>X^{1/8}\). Those with \(d>x\)
contribute nothing. For \(z\ge1\), summing the first three factors
of the divisor convolution gives
\[
\sum_{n\le z}d_4(n)
=\sum_{a_1a_2a_3a_4\le z}1
\le z\left(\sum_{a\le z}\frac1a\right)^3
\le z(1+\log z)^3 .
\]
Use this with \(z=x/d\), and then
Equation~\eqref{eq:ord-shell-tail}. The remaining contribution is
at most
\[
Cx(1+\log x)^3
\sum_{d(\boldsymbol l)>X^{1/8}}
\frac{P(\boldsymbol l)}{d(\boldsymbol l)}
\le C\frac{x}{q}X^{-1/128}(1+\log x)^3 .
\]
For fixed \(A\), once \(X\ge A\) the condition \(x\le AX\) gives
\(\log x\le2\log X\). By
Equation~\eqref{eq:ord-residue-lower} and \(D\le X^2\), the ratio of
this last bound to \(\kappa x/q\) is at most
\[
C X^{-1/128}(\log(3X))^4=o(1).
\]
This bound is independent of the field and the order. Increase the
threshold in terms of \(A,c\) until the ratio is at most \(c\).
The tail is then at most \(\kappa cx/q\le\kappa y/q\). The constant
in the resulting bound has not changed with \(A\) or \(c\), proving
the proposition.
\end{proof}
\subsection{Short vectors and small neighborhoods}
We now apply the weighted interval bound to the two vector counts
announced at the start of the section. If \(f_r=\mathbf 1_{[-r,r]^4}\), direct integration
in Equation~\eqref{eq:ord-fiber} gives
\begin{equation}\label{eq:ord-cube-fiber}
\mathcal V_{f_r}(z)=\frac{16}{6}
\bigl(\log_+(r^4/z)\bigr)^3 .
\end{equation}
Indeed, in a fixed sign component put \(s_i=\log r-u_i\) for
\(1\le i\le3\). The cube conditions become \(s_i\ge0\) and
\(\sum_{i=1}^3s_i\le\log(r^4/z)\), a simplex of the displayed volume
divided by sixteen.
\begin{proposition}[Tightness]\label{prop:ord-tightness}
There is an absolute constant \(C\) such that, for every fixed
\(0<r<1\),
\begin{equation}\label{eq:ord-cusp}
\limsup_{X\longrightarrow\infty}
\mu_*\{\Lambda:\text{some }0\ne v\in\Lambda
\text{ satisfies }\|v\|_\infty<r\}
\le Cr^4 .
\end{equation}
The limit superior ranges over all primitive totally real quartic
fields, all their orders, and all orderings of their real embeddings.
Consequently, for every sequence of these packet measures
\(\mu_i=\mu_{\OO_i,\sigma_i}\) with \(X_i\to\infty\) and every
\(\epsilon>0\), there is a compact \(\mathcal K_\epsilon\subset X_4\) with
\(\mu_i(\mathcal K_\epsilon)\ge1-\epsilon\) for all sufficiently large \(i\).
Such a sequence is tight, and every weak subsequential limit is an
\(A_4\)-invariant probability on \(X_4\).
\end{proposition}
\begin{proof}
The count \(E_{f_r}\) dominates the indicator in
Equation~\eqref{eq:ord-cusp}. Put \(T=r^4X\) and \(T_0=X^{3/4}\).
By Equations~\eqref{eq:ord-unfolding} and \eqref{eq:ord-cube-fiber},
it suffices to bound
\begin{equation}\label{eq:ord-cube-sum}
\frac1{\sqrt D\,\kappa}
\sum_{\Nm\mathfrak a\le T}
P(\mathfrak a)
\bigl(\log(T/\Nm\mathfrak a)\bigr)^3 .
\end{equation}
The sum is empty when \(T<1\). Otherwise its logarithmic factor is
at most \((\log X)^3\), since \(\Nm\mathfrak a\ge1\) and \(r<1\).
Apply Proposition~\ref{prop:ord-weighted-intervals} at
\(x=y=T_0\), with \(A=c=1\). The part with
\(\Nm\mathfrak a\le\min\{T,T_0\}\) is at most
\[
C\frac{\kappa T_0}{q}(\log X)^3.
\]
After division by \(\sqrt D\,\kappa=X\kappa/q\), this contributes
\(O(X^{-1/4}(\log X)^3)=o(1)\).
If \(T>T_0\), put
\[
A_{T_0}(t)=\sum_{T_0<\Nm\mathfrak a\le t}P(\mathfrak a)
\qquad(T_0\le t\le T).
\]
The same interval estimate at \(x=y=t\) gives
\(A_{T_0}(t)\le C\kappa t/q\), since \(T\le X\). Writing each
logarithmic cube as an integral and using Tonelli yields
\begin{align*}
\sum_{T_0<\Nm\mathfrak a\le T}
P(\mathfrak a)\bigl(\log(T/\Nm\mathfrak a)\bigr)^3
&=3\int_{T_0}^T A_{T_0}(t)
\bigl(\log(T/t)\bigr)^2\frac{dt}{t}\\
&\le C\frac{\kappa}{q}
\int_0^T\bigl(\log(T/t)\bigr)^2\,dt
\le C\frac{\kappa T}{q}.
\end{align*}
The last integral is finite after the substitution \(t=Ts\).
This part of Equation~\eqref{eq:ord-cube-sum} is therefore \(O(r^4)\).
If \(T\le T_0\), it is absent. The constants used at \(A=c=1\) are
absolute, so the resulting estimate proves
Equation~\eqref{eq:ord-cusp}. This is the compactness argument of
\cite[Proposition~5.1]{PrimeDegreePackets}, now using the quartic
weighted interval bound.
Mahler's compactness criterion \cite[Section~3, Theorem~2]{Mahler1946}
says that
\[
\mathcal K_r=\{\Lambda\in X_4:\|v\|_\infty\ge r
\text{ for all }0\ne v\in\Lambda\}
\]
is compact. Choose \(r\) so that \(Cr^4<\epsilon/2\).
Equation~\eqref{eq:ord-cusp} then gives
\(\mu_i(\mathcal K_r)\ge1-\epsilon\) for all sufficiently large \(i\).
Each of the finitely many preceding packet measures is supported on a
finite union of compact orbits, by Proposition~\ref{prop:pkt-model}.
Enlarging the compact set to include those supports proves tightness
of the entire sequence. The space \(X_4\) is locally compact,
second countable, and metrizable; the weak compactness theorem for
tight probabilities therefore gives probability subsequential limits.
For \(a\in A_4\) and \(f\in C_c(X_4)\), the equality
\[
\int f(xa)\,d\mu_i(x)=\int f(x)\,d\mu_i(x)
\]
passes to a weak limit, since both test functions are continuous with
compact support. This proves \(A_4\)-invariance.
\end{proof}
\begin{lemma}[Boxes away from the coordinate hyperplanes]\label{lem:ord-boxes}
Let \(V_0\subset(\R^\times)^4\) be compact, and let \(b>0\).
There are constants \(C,r_0>0\), depending only on \(V_0,b\), such
that for every fixed \(0<r<r_0\) there is a threshold
\(X_0(V_0,b,r)\) for which
\begin{equation}\label{eq:ord-vector-box}
\int_{X_4}E_{\mathbf 1_{v+[-br,br]^4}}\,d\mu_*
\le Cr^4\qquad(v\in V_0,\ X\ge X_0(V_0,b,r)).
\end{equation}
The threshold and constants are uniform over \(K,\OO,\sigma\) and
over the centers \(v\in V_0\).
\end{lemma}
\begin{proof}
The assertion is immediate if \(V_0\) is empty. Otherwise choose
\(0<\epsilon\le M\) such that
\(\epsilon\le|v_i|\le M\) for every \(v\in V_0\) and \(i\).
For \(r_0<\epsilon/(2b)\), each box in the statement lies in one
coordinate sign component. Its first three logarithmic widths satisfy
\[
\log\frac{|v_i|+br}{|v_i|-br}\ll_{V_0,b}r .
\]
The product map \(x\mapsto\prod_i|x_i|\) has uniformly bounded first
derivatives on these boxes. If \(t_v=\prod_i|v_i|\), it follows
from Equation~\eqref{eq:ord-fiber} that, for one \(C_1\ge1\),
\begin{equation}\label{eq:ord-box-fiber}
\mathcal V_{\mathbf 1_{v+[-br,br]^4}}(z)
\le C_1r^3\,\mathbf 1_{[t_v-C_1r,t_v+C_1r]}(z)
\qquad(v\in V_0,\ 0<r<r_0).
\end{equation}
Let \(t_->0\) and \(t_+<\infty\) be the minimum and maximum of \(t_v\)
on \(V_0\), and shrink \(r_0\) so that \(6C_1r_0<t_-\).
For a fixed \(r\) in this range set
\[
x=(t_v+2C_1r)X,\qquad y=4C_1rX.
\]
The possible ideal norms in Equation~\eqref{eq:ord-box-fiber} lie
strictly inside \((x-y,x)\). The enlargement avoids any issue with
integer endpoints. There is a fixed \(A\ge1\), depending only on
\(V_0,b\), such that \(x\le AX\) for all centers and radii under
consideration. For sufficiently large \(X\), uniformly in the
centers, \(x\ge X^{3/4}\). Also \(y\le x\) by the choice of \(r_0\),
and \(y\ge cx\) with \(c=4C_1r/A>0\), after decreasing \(r_0\) if
necessary so that \(c\le1\).
Proposition~\ref{prop:ord-weighted-intervals} applies for every fixed
\(r\), with this \(c\) and the fixed \(A\). Its constant is independent
of \(c\), so the sum of the relevant ideal weights is at most
\(C\kappa rX/q\), with a constant independent of \(r\). Combining
this with Equations~\eqref{eq:ord-unfolding} and
\eqref{eq:ord-box-fiber}, and using \(\sqrt D=X/q\), proves
Equation~\eqref{eq:ord-vector-box}. Only the threshold for \(X\)
depends on the fixed radius.
\end{proof}
For \(r>0\), define the open identity neighborhood
\begin{equation}\label{eq:ord-ball-neighborhood}
B(r)=\{h\in\mathrm{SL}_4(\R):
\max_{i,j}|h_{ij}-\delta_{ij}|<r\}.
\end{equation}
\begin{proposition}[Local bounds in probability limits]\label{prop:ord-balls}
Let \(\mu\) be a weak probability limit of packet measures along
\(X_i\to\infty\). For every compact \(\Omega\subset X_4\), there
are constants \(C_\Omega,r_\Omega>0\) such that
\begin{equation}\label{eq:ord-balls}
\mu(xB(r))\le C_\Omega r^4
\qquad(x\in\Omega,\ 0<r<r_\Omega).
\end{equation}
\end{proposition}
\begin{proof}
The assertion is immediate if \(\Omega\) is empty, so assume otherwise.
We use the passage from vector boxes to ordinary neighborhoods in
\cite[Proposition~5.2]{PrimeDegreePackets}. First, every full lattice
has a vector with no zero coordinate. If \(\Lambda=\Z^4g\), each
coordinate of \((1,N,N^2,N^3)g\) is a nonzero polynomial in \(N\):
its coefficients are a column of the invertible matrix \(g\).
An integer \(N\) outside their finitely many roots gives the required
vector.
For each \(x_0\in\Omega\), choose a continuous local lift \(g(x)\) of
nearby lattices to \(\mathrm{SL}_4(\R)\), and an integer row \(z\)
for which \(zg(x_0)\) has no zero coordinate. On a sufficiently
small neighborhood with compact closure in this chart, \(zg(x)\)
stays bounded and its coordinates stay bounded away from zero.
A finite collection of these neighborhoods covers \(\Omega\).
Consequently we may select, for every \(x\in\Omega\), a vector
\(v_x\in\Lambda_x\) so that all selected vectors lie in one compact
set \(V_0\subset(\R^\times)^4\). The selection is used pointwise and
need not be measurable.
Choose \(M\) with \(|(v_x)_i|\le M\) for all selected vectors, and
fix \(b>4M\). If \(y=xh\) with \(h\in B(r)\), then
\(v_xh\in\Lambda_y\), and
\[
|(v_xh-v_x)_j|
\le \sum_i |(v_x)_i|\,|h_{ij}-\delta_{ij}|<br.
\]
Thus every lattice in \(xB(r)\) has a nonzero vector in
\(v_x+[-br,br]^4\). Lemma~\ref{lem:ord-boxes} gives, for each fixed
small \(r>0\) and all sufficiently large \(i\),
\[
\mu_i(xB(r))
\le \int E_{\mathbf 1_{v_x+[-br,br]^4}}\,d\mu_i
\le C_\Omega r^4
\qquad(x\in\Omega).
\]
The constants are common to all \(x\) and \(r\) in the stated ranges;
the index after which this estimate holds may depend on \(r\).
The set \(xB(r)\) is open in \(X_4\). For each fixed \(x,r\),
Portmanteau's inequality therefore yields
\[
\mu(xB(r))\le\liminf_{i\to\infty}\mu_i(xB(r))
\le C_\Omega r^4.
\]
This proves the bound for every \(r\) with the same constant. It uses
no interchange between a shrinking radius and the packet limit.
\end{proof}
==== END Equidistribution-of-Primitive-Quartic-Torus-Packets-for-Arbitrary-Orders//ordinary.tex ====
==== BEGIN Equidistribution-of-Primitive-Quartic-Torus-Packets-for-Arbitrary-Orders//packets.tex ====
\section{Packets from arbitrary orders}\label{sec:packets}
Fix a primitive totally real quartic field \(K\), an order \(\OO\subset K\),
and an ordering \(\sigma=(\sigma_1,\ldots,\sigma_4)\) of its real embeddings,
as in Theorem~\ref{thm:main}. We describe the packet using finitely many
local lattice choices above each ideal class of the maximal order. This
description lets us average a containment condition at each finite prime
before performing an integral at the real place. The construction is
adapted from the packet model in
\cite[Section~2, Lemma~2.1]{PrimeDegreePackets}; we give the argument with
all sixteen sign patterns in degree four.
Throughout the paper put
\begin{equation}\label{eq:pkt-scale}
R=\OO_K,\qquad D=|\Disc(K)|,\qquad q=[R:\OO],\qquad
X=q\sqrt D,\qquad \kappa=\Res_{s=1}\zeta_K(s).
\end{equation}
In particular, \(|\Disc(\OO)|=q^2D=X^2\). Absolute ideal norms are
extended multiplicatively to fractional \(R\)-ideals. Thus
\begin{equation}\label{eq:pkt-covolume}
\covol(\sigma(I))=\sqrt D\,\Nm I
\end{equation}
for every nonzero fractional \(R\)-ideal \(I\): this is the defining
discriminant identity for \(I=R\), follows by index for integral \(I\),
and then follows for fractional \(I\) after multiplication by an integer.
For a full \(\Z\)-lattice \(J\subset K\), write
\(J_p=J\otimes_\Z\Z_p\), and put
\[
K_p=K\otimes_\Q\Q_p,\qquad R_p=R\otimes_\Z\Z_p,\qquad
\OO_p=\OO\otimes_\Z\Z_p .
\]
Here \(K_p\) is a product of finite extensions of \(\Q_p\), and \(R_p\)
is the product of their integer rings. We use the usual arithmetic of
fractional ideals in the Dedekind domain \(R\); see
\cite[Theorems~3.7, 3.20, and~3.29]{MilneANT}.
\subsection{The finite local choices}
Choose one fractional \(R\)-ideal \(I\) in each ordinary ideal class of
\(R\). Write \(h_K\) for the number of these classes. At every prime
choose a generator
\[
I_p=\xi_p R_p,\qquad \xi_p\in K_p^\times.
\]
Such a generator exists because each component of \(R_p\) is a discrete
valuation ring. Let \(\mathscr L_{\OO}(I)\) be the set of full lattices
\(L\subset I\) whose completions have the form
\begin{equation}\label{eq:pkt-local-choices}
L_p=\xi_p t_p\OO_p,\qquad t_p\in R_p^\times
\quad\text{for every }p.
\end{equation}
Replacing \(\xi_p\) by another generator merely changes \(t_p\) by a
unit, so this set does not depend on the chosen generators.
\begin{lemma}\label{lem:pkt-local-labels}
The set \(\mathscr L_{\OO}(I)\) is finite, and its cardinality is
\begin{equation}\label{eq:pkt-label-count}
N_{\OO}=\prod_{p\mid q}[R_p^\times:\OO_p^\times],
\end{equation}
independently of \(I\). Its members satisfy
\[
[I:L]=q,\qquad RL=I,
\]
and are proper invertible fractional \(\OO\)-ideals. Every ordinary
class of proper invertible fractional \(\OO\)-ideals is represented,
up to multiplication by \(K^\times\), by a member of one of the sets
\(\mathscr L_{\OO}(I)\). Uniform probability on
\(\mathscr L_{\OO}(I)\) is induced by independent normalized Haar
probabilities for \(t_p\in R_p^\times\).
\end{lemma}
\begin{proof}
The additive group \(R/\OO\) has order \(q\), so \(qR\subset\OO\).
For every \(p\) and every unit \(t_p\), it follows that
\[
qI_p=\xi_p t_p qR_p\subset \xi_p t_p\OO_p\subset I_p.
\]
At \(p\nmid q\) the middle lattice is \(I_p\). At the remaining
primes its image is a submodule of the finite module \(I_p/qI_p\).
The primary decomposition of \(I/qI\) identifies its \(p\)-primary
part with \(I_p/qI_p\). Thus a choice of these local submodules
patches to a unique subgroup of \(I/qI\), and its inverse image in
\(I\) is the unique lattice with the prescribed completions.
The stabilizer of \(\OO_p\) under multiplication by \(R_p^\times\)
is exactly \(\OO_p^\times\). Indeed, \(t_p\OO_p=\OO_p\) implies
that both \(t_p\) and \(t_p^{-1}\) belong to \(\OO_p\), by applying
the equality and its inverse to \(1\). The converse is immediate.
The local choices therefore form the finite homogeneous space
\(R_p^\times/\OO_p^\times\). Its invariant probability is uniform
and is the image of normalized unit Haar measure. Patching over the
finitely many primes dividing \(q\) proves
Equation~\eqref{eq:pkt-label-count} and the assertion about
probabilities. The index of the lattice in
Equation~\eqref{eq:pkt-local-choices} is
\([R_p:\OO_p]\), and its \(R_p\)-span is \(I_p\) because \(1\in\OO_p\).
Taking products of the indices and comparing all completions gives
\([I:L]=q\) and \(RL=I\).
We record why the local condition describes exactly invertible ideals
of the order. A full fractional \(\OO\)-ideal \(J\) is invertible
if and only if
\begin{equation}\label{eq:pkt-local-principality}
J_p=\eta_p\OO_p\quad\text{for some }\eta_p\in K_p^\times
\quad\text{at every }p.
\end{equation}
For the forward implication, complete an equality
\(JJ^{-1}=\OO\). The resulting equality over \(\OO_p\) expresses
\(1\) as a finite sum \(\sum_i a_i b_i\), with \(a_i\in J_p\) and
\(b_i\in J_p^{-1}\). At each maximal ideal \(\mathfrak m\) of the
semilocal ring \(\OO_p\), some \(a_i b_i\) is a unit after localization,
and that \(a_i\) generates \((J_p)_{\mathfrak m}\). To obtain one
generator for all the maximal ideals, choose such an \(a_{\mathfrak m}\)
for each \(\mathfrak m\), and use the Chinese remainder theorem
\cite[Theorem~1.14]{MilneANT} to
choose \(c_{\mathfrak m}\in\OO_p\) congruent to \(1\) at
\(\mathfrak m\) and to \(0\) at every other maximal ideal. Then
\(\eta_p=\sum_{\mathfrak m}c_{\mathfrak m}a_{\mathfrak m}\) generates
\((J_p)_{\mathfrak m}\) modulo \(\mathfrak m\) at each \(\mathfrak m\);
Nakayama's lemma \cite[Lemma~1.9]{MilneANT} shows that it generates each localization. Hence it
generates \(J_p\). Since \(J_p\) spans \(K_p\) over \(\Q_p\),
\(\eta_p\) has no zero component and lies in \(K_p^\times\).
Conversely, define
\[
J^{-1}=\{a\in K:aJ\subset\OO\}.
\]
This definition commutes with completion: after choosing \(\Z\)-bases
of \(J\) and \(\OO\), the containment is a finite collection of
integrality conditions on the coordinates of multiplication by \(a\),
and the same conditions over \(\Z_p\) describe the completed lattice.
Under Equation~\eqref{eq:pkt-local-principality} we therefore have
\((J^{-1})_p=\eta_p^{-1}\OO_p\). Consequently
\((JJ^{-1})_p=\OO_p\) for every \(p\), which implies \(JJ^{-1}=\OO\).
This proves the equivalence. It also proves properness: if \(aJ\subset J\)
for an invertible \(J\), multiplication by \(J^{-1}\) gives
\(a\OO\subset\OO\), and hence \(a\in\OO\). Thus its multiplier order
is \(\OO\).
The lattices in Equation~\eqref{eq:pkt-local-choices} are stable under
\(\OO_p\) at every prime, so comparison of completions shows that
\(\OO L\subset L\). They are therefore fractional \(\OO\)-ideals
satisfying Equation~\eqref{eq:pkt-local-principality}. In the other
direction, let \(J\) be an invertible fractional \(\OO\)-ideal. Its \(R\)-span
\(RJ\) is a fractional \(R\)-ideal. Multiplying \(J\) by an element
of \(K^\times\), we may arrange \(RJ=I\) for the chosen representative
of this maximal-order class. If \(J_p=\eta_p\OO_p\), then
\(\eta_p R_p=I_p=\xi_pR_p\), so \(\eta_p=\xi_p t_p\) for a unit
\(t_p\in R_p^\times\). Thus the resulting \(J\) belongs to
\(\mathscr L_{\OO}(I)\), as required.
\end{proof}
\subsection{Real parameters, identifications, and volume}
Let \(U=R^\times\), and let \(U^+\) be its totally positive subgroup.
Put
\[
T_\infty=\{v=(v_1,\ldots,v_4)\in(\R^\times)^4:
\textstyle\prod_i|v_i|=1\}.
\]
We give \(T_\infty\) counting measure on its sixteen sign components
and measure \(du_1\,du_2\,du_3\) on the logarithms
\(u_i=\log|v_i|\), with \(u_4=-u_1-u_2-u_3\). Denote this measure
by \(dv\). The unit theorem
\cite[Theorems~5.1 and~5.9]{MilneANT} says that
\[
\ell(U)=\{(\log|\sigma_i(\epsilon)|)_{i=1}^4:\epsilon\in U\}
\]
is a full lattice in the hyperplane \(\sum_i u_i=0\). Its
covolume for the deleted-coordinate measure is the ordinary regulator
\(R_K\). Its kernel on \(U\) consists of \(1\) and \(-1\), since
these are the roots of unity in a totally real field.
Let \(\mathscr F_U\) be a measurable fundamental domain for the embedded
group \(\sigma(U)\) in \(T_\infty\). Reducing logarithms modulo
\(\ell(U)\) leaves sixteen sign choices, on which the remaining kernel
\(\{1,-1\}\) identifies pairs of opposite signs. It follows that
\(\vol(\mathscr F_U)=8R_K\). The analytic class-number formula in
signature \((4,0)\), with two roots of unity, gives
\begin{equation}\label{eq:pkt-unit-volume}
h_K\vol(\mathscr F_U)=8h_KR_K=\sqrt D\,\kappa;
\end{equation}
see \cite[Chapter~V, Theorem~2.4(a)]{MilneCFT}.
For \(L\in\mathscr L_{\OO}(I)\), set
\[
c_I=(X\Nm I)^{-1/4}.
\]
Equations~\eqref{eq:pkt-covolume} and \([I:L]=q\) show that
\(c_I\sigma(L)v\) has covolume one for every \(v\in T_\infty\).
This notation denotes the corresponding row lattice, and thus a point
of \(X_4\), after choosing a basis of determinant one. This convention
also applies when the diagonal \(v\) has negative determinant.
\begin{proposition}[Packet model and volume]\label{prop:pkt-model}
The maps
\[
(L,v)\longmapsto c_I\sigma(L)v
\qquad
(L\in\mathscr L_{\OO}(I),\ v\in T_\infty)
\]
give a bijection from the disjoint union over the chosen \(I\) of the
quotients by
\begin{equation}\label{eq:pkt-unit-action}
\epsilon:(L,v)\longmapsto
(\epsilon L,\sigma(\epsilon)^{-1}v),\qquad \epsilon\in U,
\end{equation}
onto \(\bigcup_{\mathcal Y\in\mathcal P_{\OO,\sigma}}\mathcal Y\subset X_4\).
Counting measure on the \(L\)'s times \(dv\) descends to the sum of
the Haar measures of those orbits. In particular,
\begin{equation}\label{eq:pkt-total-volume}
\sum_{\mathcal Y\in\mathcal P_{\OO,\sigma}}\vol(\mathcal Y)
=N_{\OO}\sqrt D\,\kappa .
\end{equation}
Writing \(\mu_*=\mu_{\OO,\sigma}\), for every nonnegative Borel
function \(\Phi\) on \(X_4\) one has
\begin{equation}\label{eq:pkt-integration}
\int_{X_4}\Phi\,d\mu_*
=\frac{1}{h_KN_{\OO}\vol(\mathscr F_U)}
\sum_I\sum_{L\in\mathscr L_{\OO}(I)}
\int_{\mathscr F_U}\Phi(c_I\sigma(L)v)\,dv .
\end{equation}
The identity permits infinite values on both sides.
\end{proposition}
\begin{proof}
The action in Equation~\eqref{eq:pkt-unit-action} preserves the
parameter set, since \(\epsilon I=I\) for \(\epsilon\in U\), and
preserves its image lattice. Lemma~\ref{lem:pkt-local-labels} shows
that every ordinary invertible \(\OO\)-ideal class occurs. If two
ideals differ by multiplication by \(\alpha\in K^\times\), their
normalized embedded lattices differ by the diagonal with coordinates
\(\sigma_i(\alpha)/|\Nm_{K/\Q}(\alpha)|^{1/4}\), an element of
\(T_\infty\). Finally \(T_\infty\) is the union of all sixteen
sign translates of \(A_4\). These facts prove surjectivity.
Suppose two displayed normalized lattices coincide. After absorbing
the ratio of their positive scalars, this equality gives a diagonal
map carrying \(\sigma(L)\) onto \(\sigma(L')\). Choose \(0\ne b\in L\)
and let \(b'\in L'\) be its image. None of the four coordinates of
\(\sigma(b)\) is zero. The diagonal entries must therefore be
\(\sigma_i(b')/\sigma_i(b)=\sigma_i(b'/b)\). The map is multiplication
by \(\alpha=b'/b\), and \(L'=\alpha L\). Taking \(R\)-spans gives
\(I'=\alpha I\). The chosen representatives then force \(I'=I\)
and \(\alpha I=I\), so \(\alpha\in U\). The scalar normalizations
are now equal, and the two parameters differ by exactly
Equation~\eqref{eq:pkt-unit-action}. This proves injectivity, including
all identifications involving signs. Equality of covolume-one row
lattices is the same as equality in \(X_4\): any two positively oriented
bases of the same lattice differ by an element of \(\mathrm{SL}_4(\Z)\).
It remains to check that this identification gives the stipulated
orbit weights. Write
\(\OO^{\times,+}=\OO^\times\cap U^+\). The \(U\)-stabilizer of a
label \(L\) is \(\OO^\times\), by its multiplier order. The stabilizer
of \(L\) together with a sign pattern is therefore
\(\OO^{\times,+}\). This subgroup has finite index in \(U\).
For \(q=1\) this follows directly from \(\OO=R\). For \(q>1\),
the kernel of \(U\longrightarrow(R/qR)^\times\) lies in
\(\OO^\times\), since both a unit congruent to \(1\) modulo \(qR\)
and its inverse belong to \(\OO\), and imposing positive signs costs
at most a further factor of sixteen. The same diagonal-identification
argument shows that all periods of the corresponding \(A_4\)-orbit
are precisely \(\ell(\OO^{\times,+})\). They form a full lattice,
so the orbit is compact.
Combining a label with a sign produces \(16N_{\OO}\) finite choices
above each \(I\). Each \(U\)-orbit of these choices has
\([U:\OO^{\times,+}]\) elements and parametrizes one \(A_4\)-orbit.
On that orbit the quotient of logarithmic Lebesgue measure is its
Haar measure, with volume
\[
\covol\bigl(\ell(\OO^{\times,+})\bigr)
=[\ell(U):\ell(\OO^{\times,+})]R_K
=\frac{[U:\OO^{\times,+}]}{2}R_K .
\]
The last equality uses \(\ker(\ell|_U)=\{1,-1\}\) and
\(-1\notin\OO^{\times,+}\). Consequently the number of distinct
orbits times their common volume is
\[
\frac{16h_KN_{\OO}}{[U:\OO^{\times,+}]}
\cdot \frac{[U:\OO^{\times,+}]}2 R_K
=8h_KN_{\OO}R_K.
\]
This proves both the assertion about the descended measure and
Equation~\eqref{eq:pkt-total-volume}, by
Equation~\eqref{eq:pkt-unit-volume}.
One can also take \(\mathscr L_{\OO}(I)\times\mathscr F_U\) as a
fundamental domain for the diagonal action in
Equation~\eqref{eq:pkt-unit-action}: reducing \(v\) into
\(\mathscr F_U\) transports \(L\) by the same unique unit, outside
the usual null boundary set. Counting measure times \(dv\) is
invariant under this action. These domains have total measure
\(h_KN_{\OO}\vol(\mathscr F_U)\), equal to the total volume just
computed. Dividing their integral by that quantity proves
Equation~\eqref{eq:pkt-integration}.
\end{proof}
==== END Equidistribution-of-Primitive-Quartic-Torus-Packets-for-Arbitrary-Orders//packets.tex ====
==== BEGIN Equidistribution-of-Primitive-Quartic-Torus-Packets-for-Arbitrary-Orders//rigidity.tex ====
\section{The homogeneous obstruction in degree four}\label{sec:rigidity}
For a sequence with \(X\to\infty\), Proposition~\ref{prop:ord-tightness}
shows that every weak subsequential limit of the packet measures is an
\(A_4\)-invariant probability. Proposition~\ref{prop:ord-balls}
supplies the compact ball estimate in Equation~\eqref{eq:ord-balls}.
We first turn that estimate
into positive entropy in almost every \(A_4\)-ergodic component.
Measure classification then makes those components homogeneous.
In degree four, the proper homogeneous components that remain have two
coordinate blocks of dimension two. We will show that each of them
carries an exterior-square lattice vector in one of three fixed planes.
The arithmetic part of the proof will exclude these three events.
\subsection{Entropy from the compact ball estimate}
Fix distinct real numbers \(\lambda_1,\ldots,\lambda_4\) with sum zero,
and set
\[
a(t)=\diag(e^{\lambda_1t},\ldots,e^{\lambda_4t}),
\qquad
\lambda_a=\min_{i\ne j}|\lambda_i-\lambda_j|>0.
\]
For an \(a(\R)\)-invariant probability \(\xi\), let \(h_\xi(a(1))\)
denote the measure-theoretic entropy of right multiplication by \(a(1)\).
The argument below is the degree-four specialization of the
ordinary-ball argument in
\cite[Proposition~6.1 and Lemma~6.2]{PrimeDegreePackets}, which in turn
follows the strategy of \cite[Section~2.7.1]{ELMV2011}.
We include the argument because its statement for every dominated
invariant probability is needed when passing to ergodic components.
Here is the entropy input. For an open identity neighborhood \(B\)
in \(\mathrm{SL}_4(\R)\), write
\[
B^{(s,t)}=a(-s)Ba(s)\cap a(-t)Ba(t).
\]
The following is the specialization to \(X_4\) of the
Einsiedler--Lindenstrauss--Michel--Venkatesh tube criterion
\cite[Corollary~3.3 and Equation~(3.4)]{ELMV2009}. The flow \(a(t)\)
is \(\R\)-diagonalizable, as required there.
\begin{theorem}[Entropy criterion for two-sided tubes]
\label{thm:rig-tube-criterion}
Let \(\xi_i\) and \(\xi\) be \(a(\R)\)-invariant Borel probabilities
on \(X_4\), with \(\xi_i\) converging weak-* to \(\xi\).
Suppose that \(t_i>0\) tend to infinity and that \(\eta>0\).
Assume that for every compact \(\Omega\subset X_4\) there are an open
identity neighborhood \(B\subset\mathrm{SL}_4(\R)\) and a constant
\(C_\Omega\) such that
\[
\xi_i\bigl(xB^{(-t_i,t_i)}\bigr)
\le C_\Omega e^{-2\eta t_i}
\qquad(x\in\Omega,\ i\ge1).
\]
Then \(h_\xi(a(1))\ge\eta\).
\end{theorem}
In this criterion \(B\) may depend on \(\Omega\), and the
probabilities need not be ergodic under the flow. The requirement
that the weak-* limit is a probability is part of the theorem.
\begin{proposition}[Entropy for dominated probabilities]
\label{prop:rig-dominated-entropy}
Let \(\mu\) be an \(A_4\)-invariant probability on \(X_4\) satisfying
the compact ball bound in Equation~\eqref{eq:ord-balls} for every compact set of
centers. If \(\xi\) is an \(A_4\)-invariant probability with
\(\xi\le c\mu\) for some finite \(c\), then
\[
h_\xi(a(1))\ge \lambda_a/3.
\]
\end{proposition}
\begin{proof}
Use the entrywise neighborhoods \(B(r)\) defined in
Equation~\eqref{eq:ord-ball-neighborhood}. Fix \(0<b<1/4\) and put
\[
T_t=a(t)B(b)a(-t)\cap a(-t)B(b)a(t),
\qquad r=e^{-\lambda_a t}\quad(t\ge0).
\]
For \(h\in T_t\), the two conjugations give
\begin{equation}\label{eq:rig-tube-entries}
|h_{ii}-1|<b,\qquad
|h_{ij}|<b e^{-|\lambda_i-\lambda_j|t}\le br
\quad(i\ne j).
\end{equation}
In the determinant expansion, every term except the product of the
diagonal entries contains at least two off-diagonal entries.
Since \(\det h=1\), Equation~\eqref{eq:rig-tube-entries} implies
\(\prod_i h_{ii}=1+O(r^2)\). The diagonal matrix
\[
d(h)=\diag\bigl(h_{11},h_{22},h_{33},
(h_{11}h_{22}h_{33})^{-1}\bigr)
\]
therefore belongs to a fixed compact subset
\(\mathcal D_\infty\subset A_4\), and \(d(h)^{-1}h\in B(Cr)\).
The constant \(C\) here and below may depend on \(b\) and the fixed flow.
Indeed, the first three diagonal entries of \(d(h)^{-1}h\) are one,
the fourth differs from one by \(O(r^2)\), and all its off-diagonal
entries are \(O(r)\).
The logarithmic coordinates on \(A_4\) identify
\(\mathcal D_\infty\) with a bounded subset of \(\R^3\).
A mesh of spacing comparable to \(r\), followed by a fixed enlargement
to account for matrix multiplication, gives
\begin{equation}\label{eq:rig-tube-cover}
T_t\subset\bigcup_{j=1}^{N_t}d_jB(Cr),
\qquad d_j\in\mathcal D_\infty,\qquad N_t\le Cr^{-3}.
\end{equation}
For a compact \(\Omega\subset X_4\), the centers \(xd_j\) lie in the
compact set \(\Omega\mathcal D_\infty\). Domination and
Equation~\eqref{eq:ord-balls} now imply
\begin{equation}\label{eq:rig-tube-mass}
\xi(xT_t)\le C_{\Omega,c}r^{-3}r^4
=C_{\Omega,c}e^{-\lambda_a t}
\qquad(x\in\Omega)
\end{equation}
for all sufficiently large \(t\).
Apply Theorem~\ref{thm:rig-tube-criterion} to the constant sequence
\(\xi_i=\xi\), with \(t_i=i\), \(B=B(b)\), and
\(\eta=\lambda_a/3\). The identity \(B(b)^{(-t,t)}=T_t\) and
\(2\lambda_a/3<\lambda_a\) verify its estimate for large \(i\).
For the finitely many earlier \(i\), enlarge \(C_{\Omega,c}\), using
that \(\xi\) is a probability. The criterion gives the assertion.
\end{proof}
\begin{corollary}[Positive entropy in the ergodic components]
\label{cor:rig-positive-components}
Let \(\mu\) be an \(A_4\)-invariant probability on \(X_4\) satisfying
Equation~\eqref{eq:ord-balls} on every compact set of centers. Almost every
component in its \(A_4\)-ergodic decomposition has positive
\(a(1)\)-entropy. In particular, this holds for every probability
limit in Proposition~\ref{prop:ord-balls}.
\end{corollary}
\begin{proof}
The group \(A_4\simeq\R^3\) is locally compact and second countable,
and acts continuously on the standard Borel space \(X_4\).
The ergodic decomposition theorem
\cite[Theorem~5.2]{GreschonigSchmidt2000}, specialized to invariant
probabilities, gives a measurable decomposition
\[
\mu=\int_Z\nu_z\,d\tau(z)
\]
into \(A_4\)-invariant, \(A_4\)-ergodic probabilities over a factor
fixed by \(A_4\). Put \(T(x)=xa(1)\).
We record why a restriction to the zero-entropy components is
measurable. Choose increasing finite Borel partitions
\(\mathcal P_k\) generating the Borel \(\sigma\)-algebra of \(X_4\).
For any \(T\)-invariant probability \(\nu\), the
Kolmogorov--Sinai approximation theorem and subadditivity give
\[
h_\nu(T)=\sup_k\inf_{N\ge1}\frac1N
H_\nu\left(\bigvee_{j=0}^{N-1}T^{-j}\mathcal P_k\right),
\]
where \(H_\nu\) is the Shannon entropy of a finite partition.
Each finite-partition entropy is a Borel function of \(\nu\), so
\(z\mapsto h_{\nu_z}(T)\) is measurable.
For each probability under consideration, complete the standard Borel
probability space; it is then a normalized Lebesgue space.
The entropy integral formula for an invariant component factor
\cite[Theorem~9.8]{Rokhlin1967} gives
\begin{equation}\label{eq:rig-entropy-decomposition}
h_\mu(T)=\int_Z h_{\nu_z}(T)\,d\tau(z),
\end{equation}
and the same formula after restricting and normalizing \(\tau\) on
any measurable subset of positive measure. This formula applies
because the component factor is fixed by \(T\). It does not require
the individual \(\nu_z\) to be \(T\)-ergodic.
If \(Z_0=\{z:h_{\nu_z}(T)=0\}\) had measure \(w>0\), then
\[
\xi=\frac1w\int_{Z_0}\nu_z\,d\tau(z)
\]
would be an \(A_4\)-invariant probability with \(\xi\le w^{-1}\mu\).
The restricted form of Equation~\eqref{eq:rig-entropy-decomposition}
would give \(h_\xi(T)=0\), contrary to
Proposition~\ref{prop:rig-dominated-entropy}.
\end{proof}
\subsection{Finite-volume subgroups containing the diagonal group}
We next use the measure-classification theorem of
Einsiedler--Katok--Lindenstrauss
\cite[Theorem~1.3; definition in Conjecture~1.1]{EKL2006}.
We state the exact specialization that supplies the homogeneous
measures needed here.
\begin{theorem}[Positive-entropy measure classification]
\label{thm:rig-ekl-classification}
For \(n\ge3\), let \(A_n\) be the group of positive diagonal matrices
in \(\mathrm{SL}_n(\R)\). Let \(\nu\) be an \(A_n\)-invariant and
\(A_n\)-ergodic probability on
\(\mathrm{SL}_n(\R)/\mathrm{SL}_n(\Z)\), with the left action.
If \(h_\nu(a_0(1))>0\) for some one-parameter subgroup
\(a_0:\R\to A_n\), then there are a
closed connected subgroup \(L\subset\mathrm{SL}_n(\R)\) containing
\(A_n\) and a point \(x\) such that \(Lx\) is closed and \(\nu\) is
the \(L\)-invariant probability on \(Lx\).
\end{theorem}
No ergodicity for the chosen one-parameter subgroup is assumed.
For \(n=4\), inversion
\(\mathrm{SL}_4(\Z)g\mapsto g^{-1}\mathrm{SL}_4(\Z)\)
turns right multiplication by \(a(t)\) into left multiplication by
\(a(-t)\), and entropy is unchanged upon inverting an invertible
transformation. Thus, on our right quotient, the theorem says that
a positive-entropy \(A_4\)-ergodic probability is invariant
probability on an orbit \(\Lambda J\), where \(J\) is closed and
connected, contains \(A_4\), and the stabilizer of \(\Lambda\) is a
lattice in \(J\).
For a partition \(\mathcal I=\{I_1,\ldots,I_s\}\) of
\(\{1,2,3,4\}\) into nonempty subsets, define
\begin{equation}\label{eq:rig-block-group}
J_{\mathcal I}
=\left\{g\in\mathrm{SL}_4(\R):
\begin{array}{l}
g_{ij}=0\text{ if \(i,j\) belong to different \(I_v\)},\\
\det(g|_{\R^{I_v}})>0\text{ for every }v
\end{array}\right\}.
\end{equation}
This is the identity component of the corresponding block-diagonal
group. Lindenstrauss and Weiss identified the equal-block phenomenon
for orbit closures containing a compact diagonal orbit
\cite[Theorem~1.3]{LindenstraussWeiss2001}; their result also enters the
proof of \cite[Corollary~1.4]{EKL2006}. The next lemma gives a direct
finite-volume argument and records the rational structure of the blocks.
\begin{lemma}[Equal blocks and a rational commutant]
\label{lem:rig-equal-blocks}
Let \(J\) be a closed connected subgroup of \(\mathrm{SL}_4(\R)\)
containing \(A_4\), and let \(\Lambda\in X_4\). Suppose that the
stabilizer \(\Gamma_\Lambda=\{j\in J:\Lambda j=\Lambda\}\) is a
lattice in \(J\).
Then \(J=J_{\mathcal I}\) for a partition \(\mathcal I\) whose blocks
all have the same size.
More precisely, put \(W_\Q=\Q\Lambda\) and
\(W_\R=W_\Q\otimes_\Q\R=\R^4\). There is a totally real number field
\(E\subset\End_\Q(W_\Q)\), of degree \(s=|\mathcal I|\), whose scalar
extension to \(\R\) is exactly the commutant of \(J\) in
\(\End_\R(W_\R)\). The coordinate blocks \(\R^{I_v}\) are the real
eigenspaces of \(E\) on \(W_\R\).
\end{lemma}
\begin{proof}
Let \(\mathfrak j\) be the Lie algebra of \(J\).
It contains the trace-zero diagonal algebra and is invariant under
diagonal conjugation. The other weight spaces in
\(\mathfrak{sl}_4(\R)\) are the distinct one-dimensional spaces
\(\R E_{ij}\), \(i\ne j\), where \(E_{ij}\) is a matrix unit.
An invariant subspace is a sum of these weight spaces, so
\(\mathfrak j\) consists of the diagonal algebra and a selection of
the root spaces.
The group \(J\) is unimodular.
One way to see this standard fact is to use the quotient
criterion for Haar measures: existence of an invariant quotient
measure and triviality of the modular character of the discrete
group \(\Gamma_\Lambda\) imply that the modular character of \(J\)
is trivial on \(\Gamma_\Lambda\). It therefore descends to
\(\Gamma_\Lambda\backslash J\). Its pushforward of invariant
probability would be invariant under translations by its image.
A nontrivial positive dilation admits no invariant probability on
\(\R_{>0}\), as its disjoint dilates of a fundamental interval all
have the same mass. Hence the modular character is trivial on \(J\).
Draw a directed edge \(i\to j\) when \(E_{ij}\in\mathfrak j\).
The trace of the adjoint action of
\(\diag(u_1,\ldots,u_4)\), with \(\sum_i u_i=0\), on
\(\mathfrak j\) is the sum of the selected roots \(u_i-u_j\).
Unimodularity makes this sum zero. Its coefficient at \(u_i\) is
the outgoing degree of \(i\) minus its incoming degree. These
coefficients sum to zero, and a linear form vanishing on
\(\sum_i u_i=0\) has all coefficients equal. Thus every vertex
has equal outgoing and incoming degrees.
Every edge now lies within a strongly connected component.
Otherwise the finite directed acyclic graph of components would
have a source with an outgoing edge; summing degree balance over
that source would give a positive outgoing degree and zero incoming
degree. For distinct indices, the bracket identity
\([E_{ij},E_{jk}]=E_{ik}\) makes the edge relation transitive.
Each strongly connected component is consequently a complete
directed graph. These components form a partition \(\mathcal I\),
and \(\mathfrak j\) is the Lie algebra of \(J_{\mathcal I}\).
Connected Lie subgroups with the same Lie algebra agree, giving
\(J=J_{\mathcal I}\).
We use \(\Gamma_\Lambda\) to recover a rational structure on the
block scalars. Its elements preserve \(\Lambda\), hence act
rationally on \(W_\Q\). Let
\[
\mathcal C_\Q
=\{M\in\End_\Q(W_\Q):M\gamma=\gamma M
\text{ for every }\gamma\in\Gamma_\Lambda\}.
\]
In a lattice basis the commutation equations have integer
coefficients. A finite subcollection of these linear equations
defines the same solution space, by finite dimensionality.
It follows that \(\mathcal C_\Q\otimes_\Q\R\) is the real
commutant of \(\Gamma_\Lambda\).
We claim that this is also the real commutant of \(J\).
Let \(M\) be a real endomorphism commuting with \(\Gamma_\Lambda\).
The continuous map
\[
\Gamma_\Lambda\backslash J\longrightarrow\End_\R(W_\R),
\qquad \Gamma_\Lambda j\longmapsto j^{-1}Mj
\]
is well-defined. Its pushforward of invariant probability is
invariant under conjugation by \(A_4\). Each off-diagonal
coordinate is multiplied by a nontrivial positive dilation under
some element of \(A_4\). A probability on \(\R\) invariant under
such a dilation is supported at zero: the annuli
\(\{x:r^k\le |x|<r^{k+1}\}\), \(k\in\Z\), for a fixed \(r>1\),
are disjoint dilates covering \(\R\setminus\{0\}\).
Thus every off-diagonal coordinate of \(j^{-1}Mj\) is zero almost
everywhere. Invariant probability on the transitive space
\(\Gamma_\Lambda\backslash J\) has full support; continuity makes
\(j^{-1}Mj\) diagonal for every \(j\in J\).
In particular \(M\) is diagonal. Conjugating by
\(1+tE_{ij}\) for \(i,j\) in the same block forces its \(i\)-th
and \(j\)-th diagonal entries to agree. Therefore \(M\) is scalar
on each block, which is exactly the condition that it commute with
\(J\). The reverse inclusion is immediate, proving the claim.
It follows that \(\mathcal C_\Q\otimes_\Q\R\simeq\R^s\).
In particular, \(\mathcal C_\Q\) is commutative and has no
nonzero nilpotents. The structure theorem for finite reduced
commutative algebras over \(\Q\) expresses it as a product of
number fields, all totally real because its real scalar extension
has no complex factor.
This product has only one factor. If
\(e\in\mathcal C_\Q\) were an idempotent other than \(0\) and \(1\),
then \(W=e(W_\R)\) would be a nonempty proper union of coordinate
blocks, rational relative to \(\Lambda\). Every
\(\gamma\in\Gamma_\Lambda\) preserves the full lattice
\(\Lambda\cap W\) in \(W\), so
\(\lvert\det(\gamma|_W)\rvert=1\).
The continuous character
\[
\chi:J\longrightarrow\R_{>0},\qquad
\chi(j)=\lvert\det(j|_W)\rvert
\]
is therefore trivial on \(\Gamma_\Lambda\).
It is surjective: if \(k=\dim W\), a diagonal matrix may scale the
coordinates in \(W\) by \(e^{(4-k)t}\) and those in its complement
by \(e^{-kt}\). Its total determinant is one and its determinant
on \(W\) is \(e^{k(4-k)t}\).
The character \(\chi\) would consequently push the invariant
probability on \(\Gamma_\Lambda\backslash J\) to a probability on
\(\R_{>0}\) invariant under all multiplicative translations.
The disjoint-interval argument above rules this out.
Thus \(\mathcal C_\Q\) is a field, which we denote by \(E\).
Since \(E\otimes_\Q\R\simeq\R^s\), its degree is \(s\).
The space \(W_\Q\) is a vector space over \(E\), and
\[
W_\Q\otimes_\Q\R
\simeq\bigoplus_{\tau:E\hookrightarrow\R}
W_\Q\otimes_{E,\tau}\R.
\]
Each summand has dimension \(\dim_E W_\Q=4/s\).
These summands are the eigenspaces of the block-scalar algebra,
namely the coordinate blocks. The blocks therefore have equal size.
\end{proof}
For a positive-entropy homogeneous probability the lemma leaves
\(J=\mathrm{SL}_4(\R)\) or two blocks of size two.
Indeed, the only other equal-block possibility is four singleton
blocks, giving \(J=A_4\). A lattice in \(A_4\simeq\R^3\) has
compact quotient, and translations on this torus have zero entropy:
for a flat metric they are isometries, so the iterated Bowen metrics
equal the original metric and the topological entropy is zero;
the variational principle bounds measure entropy by topological entropy.
For a possible two-block limit orbit \(\Lambda J\), the quadratic field
\(E\subset\End_\Q(\Q\Lambda)\) has real scalar extension equal to
the commutant of \(J\) in \(\End_\R(\R^4)\).
The construction supplies no embedding of \(E\) into the packet fields
\(K_i\), so their primitivity does not by itself exclude this orbit.
We will exclude it by proving that packet limits give zero mass to the
exterior obstruction below.
\subsection{The three exterior obstruction events}
Let \(e_1,\ldots,e_4\) be the coordinate basis of \(\R^4\).
For \(w\in\bigwedge^2\R^4\), write \(w_{ij}\), \(i<j\), for its
coefficients in the basis \(e_i\wedge e_j\), and define
\begin{equation}\label{eq:rig-Q}
Q(w)=w_{12}w_{34}-w_{13}w_{24}+w_{14}w_{23}.
\end{equation}
The identity
\[
w\wedge w=2Q(w)\,e_1\wedge e_2\wedge e_3\wedge e_4
\]
shows that
\(Q((\bigwedge^2g)w)=\det(g)Q(w)\) for \(g\in\mathrm{GL}_4(\R)\).
Expressing a unimodular lattice in a determinant-one basis gives
\begin{equation}\label{eq:rig-Q-integral}
Q\bigl(\bigwedge\nolimits^2\Lambda\bigr)\subset\Z
\qquad(\Lambda\in X_4).
\end{equation}
This also covers the sign translates in the packet definition:
a coordinate reflection of determinant \(-1\) changes only the
sign of \(Q\).
Let
\[
\mathcal B=\{12|34,\ 13|24,\ 14|23\}.
\]
For \(b=ij|kl\in\mathcal B\), with the indices within each pair
increasing, put
\begin{equation}\label{eq:rig-planes}
V_b=\operatorname{span}_\R\{e_i\wedge e_j,e_k\wedge e_l\}.
\end{equation}
The restriction of \(Q\) to \(V_b\) is, up to sign, the product of
its two displayed coordinates. Define
\begin{equation}\label{eq:rig-events}
\mathcal E_b
=\left\{\Lambda\in X_4:
\text{there exists }w\in\bigwedge\nolimits^2\Lambda\cap V_b
\text{ with }Q(w)\ne0\right\}.
\end{equation}
Each \(\mathcal E_b\) is \(A_4\)-invariant and Borel.
For Borel measurability, take a countable cover of \(X_4\) by local
charts admitting continuous lattice bases. In a chart, the six
integer coefficients of an exterior vector label it continuously.
For each label, membership in \(V_b\) is a set of continuous
coordinate equalities, and \(Q(w)\ne0\) is an open condition.
Taking the countable union over labels and charts proves the claim.
Invariance follows because \(A_4\) preserves \(V_b\) and \(Q\).
\begin{proposition}[The quartic homogeneous obstruction]
\label{prop:rig-obstruction}
Let \(\nu\) be an \(A_4\)-invariant, \(A_4\)-ergodic probability
on \(X_4\). Suppose that \(h_\nu(a_0(1))>0\) for some
one-parameter subgroup \(a_0:\R\to A_4\).
Then either \(\nu=m_4\), or there exists
\(b\in\mathcal B\) such that \(\nu(\mathcal E_b)=1\).
In the latter case every lattice \(\Lambda'\) in the homogeneous orbit carrying
\(\nu\) has a vector in \(\bigwedge^2\Lambda'\cap V_b\) with both
coordinates nonzero and with nonzero integer \(Q\)-value.
\end{proposition}
\begin{proof}
Theorem~\ref{thm:rig-ekl-classification}, in the right-quotient
form explained above, represents \(\nu\) as invariant probability
on \(\Lambda J\) for a connected closed subgroup \(J\) containing
\(A_4\). By Lemma~\ref{lem:rig-equal-blocks} and the
positive-entropy assumption, either \(J=\mathrm{SL}_4(\R)\), which
gives \(\nu=m_4\), or \(J\) has two coordinate blocks of size two.
In the second case write the blocks as \(\{i,j\}\) and \(\{k,l\}\),
and put \(b=ij|kl\). The commutant field \(E\) in
Lemma~\ref{lem:rig-equal-blocks} is quadratic. Choose a nonzero
\(\alpha\in E\) with \(\Tr_{E/\Q}(\alpha)=0\).
Its two scalar values on the blocks are \(c\) and \(-c\), with
\(c\ne0\). The rational endomorphism \(\alpha\) of
\(W_\Q=\Q\Lambda\) induces a rational endomorphism
\[
D_\alpha:\bigwedge\nolimits^2 W_\Q\longrightarrow
\bigwedge\nolimits^2 W_\Q,\qquad
D_\alpha(v\wedge w)=\alpha(v)\wedge w+v\wedge\alpha(w).
\]
On the exterior square of the first block it acts by \(2c\), and
on that of the second by \(-2c\). It vanishes on the cross-block
summand. Its real image is therefore exactly
\[
\bigwedge\nolimits^2\R^{\{i,j\}}
\oplus \bigwedge\nolimits^2\R^{\{k,l\}}=V_b.
\]
Because \(D_\alpha\) is rational, \(V_b\) is a rational subspace
relative to \(\bigwedge^2 W_\Q\).
It follows that \(V_b\cap\bigwedge^2\Lambda\) is a rank-two lattice
in \(V_b\). It contains a vector off both coordinate axes: take
two independent lattice vectors; if neither is already off the
axes, their sum is. For this vector \(Q(w)\ne0\), and
Equation~\eqref{eq:rig-Q-integral} makes its value a nonzero integer.
Every element of \(J\) preserves each of the two exterior lines,
multiplying its coefficient by the nonzero determinant of the
corresponding block. The translated vector therefore has both
coordinates nonzero at every lattice of \(\Lambda J\).
This proves \(\nu(\mathcal E_b)=1\).
\end{proof}
Corollary~\ref{cor:rig-positive-components} and
Proposition~\ref{prop:rig-obstruction} reduce identification of a
packet probability limit \(\mu\) to proving
\(\mu(\mathcal E_b)=0\) for all three \(b\in\mathcal B\).
Equation~\eqref{eq:rig-Q-integral} lets us test those events one
nonzero integer \(Q\)-value at a time. The next sections establish
the exterior-vector estimate used for this exclusion.
==== END Equidistribution-of-Primitive-Quartic-Torus-Packets-for-Arbitrary-Orders//rigidity.tex ====
==== BEGIN Equidistribution-of-Primitive-Quartic-Torus-Packets-for-Arbitrary-Orders//sieve.tex ====
\section{Averaging the unramified factors}\label{sec:sieve}
Proposition~\ref{prop:additive-equidistribution} distributes the
nonnegative mass $\phi(z)\prod_{p\in S}f_p^*(z)$ among residue classes at
primes outside $S$. To prove Theorem~\ref{thm:exterior-count}, we must
insert the remaining
factors $b_p(z)$ while retaining their local means. Some of those means are
less than one, so the argument must also keep track of the probability that
a prime does not divide any component of $z$. We first obtain a weighted
sieve estimate with this feature and then combine its local means with the
Euler factors in Proposition~\ref{prop:ext-multiplicity-majorant}.
\subsection{Residue functions and the local bounds}
Fix $m\in\Z\setminus\{0\}$ and a bounded region of the real trace-$m$
plane. Throughout the sieve argument, $\phi$ is one of the nonnegative
smooth box functions in Proposition~\ref{prop:additive-equidistribution},
with its planar support in this region. Its two scales lie between
$X^{-\tau}$ and
fixed constants. Define a measure on $F$ by
\begin{equation}\label{eq:sieve-measure}
\nu_\phi
=\sum_{\substack{z\in F,\ \Tr z=m\\z\text{ integral outside }S}}
\phi(z)\prod_{p\in S}f_p^*(z)\,\delta_z,
\qquad
V_\phi=A_\phi\prod_{p\in S}d_p(m).
\end{equation}
Here $\delta_z$ is unit point mass at $z$. All its coefficients are
nonnegative, and Proposition~\ref{prop:additive-equidistribution} with
modulus one gives a finite total mass.
Lemma~\ref{lem:fourier-scalar-density} gives
\begin{equation}\label{eq:sieve-volume}
V_\phi\asymp A_\phi
\end{equation}
uniformly in $S$: for large primes the logarithms of $d_p(m)=1+O(p^{-2})$
have an absolutely convergent sum, and the finitely many small-prime
densities have uniform positive upper and lower bounds.
For a positive integer $l$ prime to every prime of $S$, the integral
trace-$m$ coset has $l^2$ residue classes modulo $l$ in its two integral
coordinates. Write $\E_l W$ for the uniform mean of a function $W$ on
these classes, and also write $W(z)$ for its pullback to points integral
outside $S$.
\begin{lemma}[Nonnegative residue functions]\label{lem:sieve-residue-function}
With $\phi$ and $V_\phi$ as in Equation~\eqref{eq:sieve-measure}, let
$1\le l\le X^\theta$ be an integer prime to every prime of $S$.
For every nonnegative
function $W$ on the residue classes of the integral trace-$m$ coset
modulo $l$,
\begin{equation}\label{eq:sieve-residue-function}
\int W\,d\nu_\phi
=V_\phi\E_l W+O\bigl(V_\phi X^{-\theta}\E_l W\bigr).
\end{equation}
The implied constant is uniform in the fields, orders, embedding
orderings, permitted box locations and scales, $l$, and $W$. It may
depend on $m$, the fixed bounded region, and the fixed smooth box function.
\end{lemma}
\begin{proof}
For each residue class $a$, Equation~\eqref{eq:add-equidistribution} gives
\[
\nu_\phi(a)=V_\phi l^{-2}(1+\epsilon_a),
\qquad |\epsilon_a|\le C_{\mathrm{add}}X^{-\theta},
\]
with the same $C_{\mathrm{add}}$ for every $a$ and $l$ in the stated range.
Multiply by
$W(a)$ and sum. Since $W(a)\ge0$, the absolute value of the summed error
is at most
$C_{\mathrm{add}}V_\phi X^{-\theta}l^{-2}\sum_a W(a)
=C_{\mathrm{add}}V_\phi X^{-\theta}\E_l W$.
This also covers $\E_l W=0$. In particular the error retains the local
mean even when that mean is very small.
\end{proof}
We recall the information about the functions to be inserted. For
$p\notin S$, let $\E_p$ and $\P_p$ denote expectation and probability for
normalized additive Haar measure on
$\{z\in R_{F,p}:\Tr z=m\}$. On $R_{F,p}\cap F_p^\times$, recall that
$n_p(z)=\max_{\mathfrak q\mid p}v_{\mathfrak q}(z)$ is the largest
component valuation, with $v_{\mathfrak q}(p)=1$; see
Equation~\eqref{eq:ext-unramified-factor}. On the null set where a
component vanishes, keep the convention $n_p=\infty$, $b_p=0$.
Lemma~\ref{lem:ext-unramified-data} permits one fixed integer $C\ge1$
such that, on the nonvanishing locus,
\begin{align}
&b_p\ge0,\qquad b_p=1\ \text{if }n_p=0,\qquad
b_p\le C(1+n_p)^C,\label{eq:sieve-local-weight}\\
&\P_p(n_p\ge j)\le Cp^{-j}\quad(j\ge1),\qquad
\vartheta_p:=\P_p(n_p\ge1)\le C/p,\quad
\vartheta_p\le\tfrac12. \label{eq:sieve-local-tail}
\end{align}
The event $\{n_p\ge j\}$ and the function
$b_p\mathbf 1_{\{n_p=j\}}$ are determined by the two integral coordinates
modulo $p^{C(j+1)}$ for every integer $j\ge1$. They have extensions
defined by congruence classes on the whole local coset, agreeing with their values at every
global point. A prime with $n_p(z)\ge1$ will be called active at $z$.
In particular,
\begin{equation}\label{eq:sieve-mean-lower}
\E_p b_p\ge1-\vartheta_p\ge\tfrac12.
\end{equation}
Lemma~\ref{lem:ext-support} places the support of $\nu_\phi$ in the
trace-$m$ coset of $(C_F^\sharp)^\vee$. On this support all $n_p(z)$,
$p\notin S$, are finite. The lemma's norm and depth conclusions bound
the product of the depth factors by $O(X^{84})$ in our fixed bounded real region. Thus,
for a fixed $B>84$ and all sufficiently large $X$,
\begin{equation}\label{eq:sieve-depth-product}
\prod_{p\notin S}p^{n_p(z)}\le X^B
\qquad\bigl(\nu_\phi(\{z\})>0\bigr).
\end{equation}
Two consequences will be used repeatedly. There is a fixed $A_0>1$ such
that $b_p(z)\le A_0^{n_p(z)}$. This follows from
Equation~\eqref{eq:sieve-local-weight} for $n_p\ge1$, since
$\sup_{j\ge1}(C(1+j)^C)^{1/j}<\infty$, and from $b_p=1$ at depth zero.
Also, with
\[
\mathcal W(z)=\prod_{p\notin S}b_p(z),
\]
the product is finite on the support of $\nu_\phi$ and satisfies
\begin{equation}\label{eq:sieve-pointwise-weight}
\mathcal W(z)\ll_\epsilon X^\epsilon
\qquad(\epsilon>0).
\end{equation}
Indeed, for any $t>0$, the inequality
$C(1+j)^C\le p^{tj}$ holds for all $j\ge1$ once $p$ is sufficiently large.
For each of the remaining finitely many primes it holds with a constant
factor independent of $j$. Multiplying these inequalities and using
Equation~\eqref{eq:sieve-depth-product}, with $t=\epsilon/B$, proves
Equation~\eqref{eq:sieve-pointwise-weight}. All constants here are uniform
in the arithmetic data and in the permitted boxes.
\begin{proposition}[Weighted averaging]\label{prop:sieve-averaging}
For the measure $\nu_\phi$ in Equation~\eqref{eq:sieve-measure}, there is
a fixed $\delta>0$ such that
\begin{equation}\label{eq:sieve-average}
\int\mathcal W(z)\,d\nu_\phi(z)
\ll V_\phi\prod_{\substack{p\le X^\delta\\p\notin S}}\E_p b_p.
\end{equation}
The choice of $\delta$ and the implied constant are independent of the
field, order, embedding ordering, and the permitted box location and
scales. They may depend on $m$, the fixed bounded real region, and the
fixed smooth box function.
\end{proposition}
The proof first counts bounded products of prescribed positive depths,
while requiring inactivity at any other chosen small primes. The
inactivity condition preserves the factors in
Equation~\eqref{eq:sieve-mean-lower}. We then bound the contribution of two
exceptional sets of points and truncate the remaining depth products within
the modulus range of
Lemma~\ref{lem:sieve-residue-function}.
\subsection{Prescribed depths and inactive primes}
Write
$\mathcal P_y=\{p\le y:p\notin S\}$. Chebyshev's estimate
$\sum_{p\le y}(\log p)/p\ll\log y$ and
Equation~\eqref{eq:sieve-local-tail} give a fixed $C_*\ge1$ such that
\[
\sum_{p\in\mathcal P_y}\vartheta_p\log p\le C_*\log y
\qquad(y\ge2).
\]
Choose fixed positive parameters satisfying
\begin{equation}\label{eq:sieve-parameters}
20(C+1)\eta<\theta,\qquad
0<\delta<\min\{\eta/4,\eta/(4C_*)\},
\qquad T_0=X^\eta,\quad Y=X^\delta.
\end{equation}
Here $T_0$ truncates the squarefree Selberg divisors and is the stopping
threshold for depth products, whose crossing prefixes may reach $T_0^2$;
$Y$ bounds the small primes outside $S$ handled by the sieve.
For every $\mathcal I\subset\mathcal P_Y$, Mertens' estimate and
$-\log(1-\vartheta_p)\le2\vartheta_p\le2C/p$ give, for a fixed $C_1$,
\begin{equation}\label{eq:sieve-inactivity-lower}
\prod_{p\in\mathcal I}(1-\vartheta_p)
\gg(\log X)^{-C_1}.
\end{equation}
The bound is uniform in the subset $\mathcal I$ and in $S$.
\begin{lemma}[Prescribed depths and inactive primes]\label{lem:sieve-assignment}
Let $\mathcal A,\mathcal I$ be disjoint subsets of $\mathcal P_Y$, and
assign an integer $j_p\ge1$ to each $p\in\mathcal A$. Suppose that
$\prod_{p\in\mathcal A}p^{j_p}\le T_0^2$. Then
\begin{equation}\label{eq:sieve-assignment}
\int
\prod_{p\in\mathcal A}
b_p\mathbf 1_{\{n_p=j_p\}}
\prod_{p\in\mathcal I}\mathbf 1_{\{n_p=0\}}\,
d\nu_\phi
\ll V_\phi
\prod_{p\in\mathcal A}\E_p
\bigl(b_p\mathbf 1_{\{n_p=j_p\}}\bigr)
\prod_{p\in\mathcal I}(1-\vartheta_p).
\end{equation}
The constant is uniform in the two subsets and all their assigned depths.
\end{lemma}
\begin{proof}
We use the Selberg upper-bound square
\cite{Selberg1947}; see also \cite[Chapter~21]{Koukoulopoulos2019}.
Its calculation is included because the relative error in
Lemma~\ref{lem:sieve-residue-function} must preserve the assigned means.
If $\vartheta_p=0$, the activity event at $p$ is a union of residue classes
of Haar measure zero, hence is empty on global points. Such a prime can
be removed from $\mathcal I$. For the remaining primes put
\[
\vartheta(d)=\prod_{p\mid d}\vartheta_p,\qquad
h(d)=\prod_{p\mid d}\frac{\vartheta_p}{1-\vartheta_p}
\]
for squarefree $d$ supported on $\mathcal I$. Both functions take value
one at $d=1$. Let $\mathcal D$ be the set of those integers with
$d\le T_0$, and put $\mathcal H=\sum_{d\in\mathcal D}h(d)$.
For coefficients $\lambda_d$ supported on $\mathcal D$ with
$\lambda_1=1$, the square
\[
\left(\sum_{d\in\mathcal D}\lambda_d
\prod_{p\mid d}\mathbf 1_{\{n_p\ge1\}}\right)^2
\]
majorizes the indicator that every prime of $\mathcal I$ is inactive.
Its expectation in the independent product of the local probability
spaces is
\begin{equation}\label{eq:sieve-quadratic}
\sum_{d,e\in\mathcal D}\lambda_d\lambda_e\vartheta([d,e])
=\sum_{l\in\mathcal D}\frac1{h(l)}
\left(\sum_{\substack{d\in\mathcal D\\l\mid d}}
\lambda_d\vartheta(d)\right)^2.
\end{equation}
Here $[d,e]$ denotes the least common multiple. The identity follows
by expanding the right side and using
$\vartheta([d,e])
=\vartheta(d)\vartheta(e)\sum_{l\mid(d,e)}h(l)^{-1}$.
The set $\mathcal D$ is closed under divisors. Finite M\"obius inversion
therefore permits the choice
\[
\sum_{\substack{d\in\mathcal D\\l\mid d}}\lambda_d\vartheta(d)
=\frac{\mu_{\mathrm{Mob}}(l)h(l)}{\mathcal H},
\qquad l\in\mathcal D,
\]
where $\mu_{\mathrm{Mob}}$ is the M\"obius function. Explicitly,
\[
\lambda_d=
\frac{\mu_{\mathrm{Mob}}(d)h(d)}
{\vartheta(d)\mathcal H}
\sum_{\substack{r\le T_0/d\\(r,d)=1\\dr\in\mathcal D}}h(r).
\]
Consequently $\lambda_1=1$, the value of
Equation~\eqref{eq:sieve-quadratic} is $\mathcal H^{-1}$, and
\begin{equation}\label{eq:sieve-coefficients}
|\lambda_d|
\le\frac{h(d)}{\vartheta(d)}
=\prod_{p\mid d}(1-\vartheta_p)^{-1}
\le2^{\omega(d)}\le T_0.
\end{equation}
To estimate $\mathcal H$, give every squarefree integer supported on
$\mathcal I$ probability proportional to $h(d)$, without the bound
$d\le T_0$. In this finite product distribution, prime $p$ is selected
with probability $\vartheta_p$. Thus
\[
\E\log d=\sum_{p\in\mathcal I}\vartheta_p\log p
\le C_*\log Y<\tfrac14\log T_0.
\]
Markov's inequality shows that at least half the mass lies at $d\le T_0$.
It follows that
\begin{equation}\label{eq:sieve-selberg-main}
\mathcal H\ge\tfrac12\prod_{p\in\mathcal I}(1+h(p)),
\qquad
\mathcal H^{-1}\le2\prod_{p\in\mathcal I}(1-\vartheta_p).
\end{equation}
It remains to average this square against $\nu_\phi$ together with the
assigned weights. Write
\[
A_{\mathcal A}=
\prod_{p\in\mathcal A}\E_p
\bigl(b_p\mathbf 1_{\{n_p=j_p\}}\bigr).
\]
The assigned weight is defined modulo
\[
\prod_{p\in\mathcal A}p^{C(j_p+1)}
\le\left(\prod_{p\in\mathcal A}p^{j_p}\right)^{2C}
\le T_0^{4C}.
\]
For a pair $d,e\in\mathcal D$, its activity indicators require only the
additional modulus $[d,e]^{2C}\le T_0^{4C}$. The two prime sets are
disjoint, so the total modulus is at most
\begin{equation}\label{eq:sieve-modulus-budget}
T_0^{8C}\le X^\theta
\end{equation}
and is prime to every prime of $S$. The function before multiplication
by $\lambda_d\lambda_e$ is nonnegative, and its residue mean is
$A_{\mathcal A}\vartheta([d,e])$ by the Chinese remainder theorem.
Lemma~\ref{lem:sieve-residue-function} therefore gives the corresponding
main term $V_\phi A_{\mathcal A}\vartheta([d,e])$ and an error of absolute
value at most
$C V_\phi X^{-\theta}A_{\mathcal A}\vartheta([d,e])$.
There are at most $T_0^2$ pairs and, by
Equation~\eqref{eq:sieve-coefficients}, each coefficient product has
absolute value at most $T_0^2$. Summing the errors absolutely costs at
most
\begin{equation}\label{eq:sieve-square-error}
C V_\phi A_{\mathcal A}X^{-\theta}T_0^4.
\end{equation}
This estimate keeps $A_{\mathcal A}$ even if it is very small.
The main term is $V_\phi A_{\mathcal A}\mathcal H^{-1}$ by
Equation~\eqref{eq:sieve-quadratic}. Finally,
$X^{-\theta}T_0^4=o((\log X)^{-C_1})$ by
Equation~\eqref{eq:sieve-parameters}. Equations
\eqref{eq:sieve-inactivity-lower} and
\eqref{eq:sieve-selberg-main} absorb
Equation~\eqref{eq:sieve-square-error} and prove the lemma.
\end{proof}
\subsection{Truncating and summing the depths}
\begin{proof}[Proof of Proposition~\ref{prop:sieve-averaging}]
Put
$P_Y=\prod_{p\in\mathcal P_Y}\E_p b_p$. Equations
\eqref{eq:sieve-mean-lower} and
\eqref{eq:sieve-inactivity-lower} imply
\begin{equation}\label{eq:sieve-product-lower}
P_Y\gg(\log X)^{-C_1}.
\end{equation}
First observe that primes exceeding $Y$ cost only a fixed factor.
Equations~\eqref{eq:sieve-depth-product} and
\eqref{eq:sieve-pointwise-weight}, together with $b_p\le A_0^{n_p}$, give
\begin{equation}\label{eq:sieve-large-primes}
\prod_{\substack{p>Y\\p\notin S}}b_p(z)
\le A_0^{\sum_{p>Y,\ p\notin S}n_p(z)}
\le \exp(B\log A_0/\delta).
\end{equation}
Consider the points for which
$\prod_{p\in\mathcal P_Y}p^{n_p(z)}\le T_0$.
Assign all their active primes in $\mathcal P_Y$ and require inactivity
at the other primes in $\mathcal P_Y$. These assignments satisfy
Lemma~\ref{lem:sieve-assignment}. After applying it, we may enlarge the
sum of its nonnegative upper bounds by allowing every subset and every
positive depth. The resulting product is
\[
V_\phi\prod_{p\in\mathcal P_Y}
\left(1-\vartheta_p+
\sum_{j\ge1}\E_p(b_p\mathbf 1_{\{n_p=j\}})\right)
=V_\phi P_Y.
\]
Together with Equation~\eqref{eq:sieve-large-primes}, this bounds the
contribution from these points.
For a remaining point, the product through $\mathcal P_Y$ exceeds $T_0$.
Order its active primes in $\mathcal P_Y$ increasingly and stop at the
first prefix whose product exceeds $T_0$. If every individual factor
$p^{n_p(z)}$ for $p\in\mathcal P_Y$ is at most $T_0$, the prefix has product in
$(T_0,T_0^2]$, since its preceding product and its last factor are both
at most $T_0$. If also
$\prod_{p\in\mathcal P_{y_0}}p^{n_p(z)}\le T_0$, for a fixed cutoff
$y_0>4$ to be chosen below independently of $X$, its last prime exceeds
$y_0$. The next two estimates allow these restrictions at negligible
weighted cost.
First, if $p^{n_p(z)}>T_0$ for $p\in\mathcal P_Y$, then
$n_p(z)\ge J_p$, where
$J_p=\lfloor\log T_0/\log p\rfloor+1$. The threshold is determined modulo
\[
p^{C(J_p+1)}\le T_0^C Y^{2C}\le X^\theta.
\]
Lemma~\ref{lem:sieve-residue-function} and
Equation~\eqref{eq:sieve-local-tail} bound its $\nu_\phi$-measure by
$O(V_\phi p^{-J_p})=O(V_\phi/T_0)$. Summing over $p\in\mathcal P_Y$
gives $O(V_\phi Y/T_0)$. By Equation~\eqref{eq:sieve-pointwise-weight},
the contribution after multiplication by $\mathcal W$ is
$O_\epsilon(V_\phi X^{\epsilon+\delta-\eta})$.
Choose $\epsilon<\eta-\delta$. This is smaller than $V_\phi P_Y$ by a
power of $X$, in view of Equation~\eqref{eq:sieve-product-lower}.
We also need the analogous removal for a fixed nonempty finite set
$\mathcal Q$ of primes, even though some of them may belong to the varying
set $S$. Put $N=|\mathcal Q|$. If
$\prod_{p\in\mathcal Q\setminus S}p^{n_p(z)}>T_0$, some prime in
$\mathcal Q\setminus S$ has $p^{n_p(z)}>T_0^{1/N}$. Put
$J_{p,\mathcal Q}=\lfloor\log T_0/(N\log p)\rfloor+1$. The event
$n_p\ge J_{p,\mathcal Q}$ is defined modulo at most
$T_0^{C/N}p^{2C}\le X^\theta$ for large $X$, and its local probability
is at most $Cp^{-J_{p,\mathcal Q}}\le C T_0^{-1/N}$.
Lemma~\ref{lem:sieve-residue-function} therefore bounds the measure of
the union by
$O_{\mathcal Q}(V_\phi T_0^{-1/N})$. After multiplication by
Equation~\eqref{eq:sieve-pointwise-weight} with $\epsilon<\eta/N$, its
weighted contribution again saves a power of $X$ relative to $V_\phi$.
If $\mathcal Q\setminus S$ is empty the event is empty. Thus this
removal is uniform in the possible intersections of $\mathcal Q$ with $S$.
Choose a fixed $M_0>0$ and then a fixed $y_0>4$ such that
\begin{equation}\label{eq:sieve-crossing-parameters}
\eta M_0>2B\log A_0+1,\qquad M_0/\log y_0<\tfrac14.
\end{equation}
Apply the finite-set removal with $\mathcal Q=\{p:p\le y_0\}$.
These choices precede the choice of the small $\epsilon$ used in that
removal, and all are independent of $X$.
The retained first-crossing prefixes therefore have product in
$(T_0,T_0^2]$ and last prime greater than $y_0$. Group these assignments
by the interval containing their last prime:
\begin{equation}\label{eq:sieve-buckets}
\sqrt y<p\le y,\qquad y=Y^{1/2^i},\quad i=0,1,2,\ldots.
\end{equation}
Only groups with $y>y_0$ occur. Every unassigned active prime is greater
than the last assigned prime and hence greater than $\sqrt y$. The total
weight of those primes, including primes beyond $Y$, is therefore at most
\begin{equation}\label{eq:sieve-unassigned}
\exp\left(\frac{2B\log A_0}{\log y}\log X\right).
\end{equation}
For each assignment in the group, require inactivity at its unassigned
primes in $\mathcal P_{\sqrt y}$ and apply
Lemma~\ref{lem:sieve-assignment}. We use Rankin's power majorant
\cite{Rankin1938}: since the assigned product exceeds $T_0$, its upper
bound can be multiplied by
\[
T_0^{-s}\prod_{\text{assigned }p}p^{s j_p}\ge1,
\qquad s=M_0/\log y.
\]
Now enlarge the sum of these nonnegative upper bounds to all subsets and
all positive depths through $y$. This enlarges only a numerical sum after
the lemma has been applied to the genuine prefixes; it invokes no residue
estimate for the newly admitted products above $T_0^2$.
For $p\in\mathcal P_{\sqrt y}$, the option of no assignment has weight
$1-\vartheta_p$ and the assigned options have total weight
$\sum_{j\ge1}\E_p(b_p\mathbf 1_{\{n_p=j\}})p^{sj}$. Their sum is
$\E_p(b_p p^{s n_p})$. For $\sqrt y<p\le y$ outside $S$, inactivity was
not imposed, so the option of no assignment has weight one. The total
there is $\E_p(b_p p^{s n_p})+\vartheta_p$. Thus the group's contribution
is at most
\begin{equation}\label{eq:sieve-bucket-bound}
C V_\phi T_0^{-s}
\exp\left(\frac{2B\log A_0}{\log y}\log X\right)
\prod_{p\in\mathcal P_{\sqrt y}}\E_p(b_p p^{s n_p})
\prod_{\substack{\sqrt y<p\le y\\p\notin S}}
\bigl(\E_p(b_p p^{s n_p})+\vartheta_p\bigr).
\end{equation}
We estimate these local moments uniformly for $y>y_0$. By
Equation~\eqref{eq:sieve-crossing-parameters}, $s<1/4$. The total
contribution of depths $j\ge2$ to $\E_p(b_p p^{s n_p})$ is at most
\[
C^2\sum_{j\ge2}(1+j)^C p^{-(1-s)j}\ll_C p^{-3/2}.
\]
At depth one, the increase over $\E_p b_p$ is bounded by
\[
\frac{C'}p(p^s-1)
\ll_{M_0}\frac{\log p}{p\log y}
\qquad(p\le y),
\]
because $0\le s\log p\le M_0$. Chebyshev's estimate shows that these
increases have bounded sum over $p\le y$. Mertens' estimate also gives
$\sum_{\substack{\sqrt y<p\le y\\p\notin S}}\vartheta_p\ll1$. Since
$\E_p b_p\ge1/2$, comparison of the products in
Equation~\eqref{eq:sieve-bucket-bound} with their base means gives
\[
\prod_{p\in\mathcal P_{\sqrt y}}\E_p(b_p p^{s n_p})
\prod_{\substack{\sqrt y<p\le y\\p\notin S}}
\bigl(\E_p(b_p p^{s n_p})+\vartheta_p\bigr)
\ll_{M_0}\prod_{p\in\mathcal P_y}\E_p b_p.
\]
For a fixed $C_2$, a final use of
Equation~\eqref{eq:sieve-mean-lower} and Mertens' estimate yields
\[
\prod_{p\in\mathcal P_y}\E_p b_p
\le P_Y\prod_{\substack{y<p\le Y\\p\notin S}}
(1-\vartheta_p)^{-1}
\ll P_Y\left(\frac{\log Y}{\log y}\right)^{C_2}.
\]
By Equation~\eqref{eq:sieve-crossing-parameters}, the bound for the group
in Equation~\eqref{eq:sieve-bucket-bound} is consequently at most
\[
C_{M_0}V_\phi P_Y
\exp\left(-\frac{\log X}{\log y}\right)
\left(\frac{\log Y}{\log y}\right)^{C_2}.
\]
For the values of $y$ in Equation~\eqref{eq:sieve-buckets}, its sum is
bounded by
\[
C_{M_0}V_\phi P_Y
\sum_{i\ge0}\exp(-2^i/\delta)\,2^{iC_2}
\ll V_\phi P_Y.
\]
Together with the small-product part and the two discarded sets, this
proves Equation~\eqref{eq:sieve-average}.
\end{proof}
\subsection{The planar exterior count}
\begin{proof}[Proof of Theorem~\ref{thm:exterior-count}]
Fix the nonzero trace and bounded region in that theorem. Let
$\mathcal R$ be any of its rectangles, in the specified two partition
coordinates. Choose an allowed smooth function $\phi$ that majorizes
$\mathbf 1_{\mathcal R}$, with its two scales equal to the side lengths
of $\mathcal R$. Its support lies in a fixed enlargement of the bounded
region and
\[
A_\phi\asymp\operatorname{area}(\mathcal R).
\]
The constants are uniform over the positions, side lengths, and aspect
ratios of the rectangles. Proposition~\ref{prop:ext-multiplicity-majorant}
and Proposition~\ref{prop:sieve-averaging} give
\begin{equation}\label{eq:sieve-before-euler}
\sum_{\substack{\Tr z=m\\z\in\mathcal R}}M(z)
\ll \sqrt D\, A_\phi
\prod_{p\in S}(1-p^{-1})Z_p
\prod_{p\in\mathcal P_Y}\E_p b_p,
\qquad
Z_p=\prod_{\substack{\mathfrak p\mid p\\\mathfrak p\subset R}}
(1-\Nm(\mathfrak p)^{-1})^{-1}.
\end{equation}
Here $\prod_{p\in S}|D|_p^{-1/2}=\sqrt D$, because $S$ contains every
prime divisor of $D$, and Equation~\eqref{eq:sieve-volume} absorbed the
trace densities. The restriction to integral points outside $S$ in the
multiplicity proposition agrees with the support of $\nu_\phi$.
The exact unramified mean in Lemma~\ref{lem:ext-unramified-data} is
\[
\E_p b_p=(1-p^{-1})(1-p^{-3})Z_p\qquad(p\notin S).
\]
Since $1-p^{-3}\le1$, the two Euler products on the right of
Equation~\eqref{eq:sieve-before-euler} are at most the product of
$(1-p^{-1})Z_p$ over $p\in S\cup\mathcal P_Y$.
Ideal factorization in the degree-four maximal order gives, uniformly
also at ramified primes,
\[
\log Z_p=\frac{a_K(p)}p+O(p^{-2}).
\]
Thus
\begin{equation}\label{eq:sieve-euler}
\prod_{p\in S}(1-p^{-1})Z_p
\prod_{p\in\mathcal P_Y}\E_p b_p
\ll
\exp\left(\sum_{p\in S\ \mathrm{or}\ p\le Y}
\frac{a_K(p)-1}{p}\right).
\end{equation}
For fixed $m$, the radical of $S$ is polynomial in $X$. Choose a fixed
$c\ge1/2$ large enough that, for all sufficiently large $X$, every prime
in $S$ and every prime at most $Y$ is at most $Z=X^c$. Since
$a_K(p)\ge0$, the exponent in Equation~\eqref{eq:sieve-euler} is at most
\[
\sum_{p\le Z}\frac{a_K(p)}p-\sum_{p\le Y}\frac1p.
\]
We have $Z\ge D^{1/4}$ because $X=q\sqrt D\ge\sqrt D$.
Equation~\eqref{eq:ord-prime-sum} and Mertens' estimate now give
\[
\exp\left(\sum_{p\in S\ \mathrm{or}\ p\le Y}
\frac{a_K(p)-1}{p}\right)
\ll \frac{\kappa\log Z}{\log Y}
=\frac c\delta\,\kappa.
\]
This also applies when $D$ remains bounded and $q$ tends to infinity,
by the stated uniform range of Lemma~\ref{lem:ord-residue}.
Substitution in Equation~\eqref{eq:sieve-before-euler} proves
\[
\sum_{\substack{\Tr z=m\\z\in\mathcal R}}M(z)
\ll\kappa\sqrt D\,\operatorname{area}(\mathcal R).
\]
The only box-dependent quantity in the proof is $A_\phi$, which is
comparable to the area with a fixed constant. All other estimates are
uniform over the box translations and the two scales between
$X^{-\tau}$ and the fixed upper bounds. The local multiplicity majorant
has the same constant for every chosen ideal representative system, and
its right side is independent of that choice. Thus the bound is uniform
in the representative system as well. This proves the precise
uniformity asserted in Theorem~\ref{thm:exterior-count}.
\end{proof}
==== END Equidistribution-of-Primitive-Quartic-Torus-Packets-for-Arbitrary-Orders//sieve.tex ====
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