==== TABLE OF CONTENTS ====
The-rational-Hodge-conjecture-for-CM-abelian-varieties//main.tex
The-rational-Hodge-conjecture-for-CM-abelian-varieties//preamble.tex
The-rational-Hodge-conjecture-for-CM-abelian-varieties//sections/01-introduction.tex
The-rational-Hodge-conjecture-for-CM-abelian-varieties//sections/02-tensors.tex
The-rational-Hodge-conjecture-for-CM-abelian-varieties//sections/03-surface.tex
The-rational-Hodge-conjecture-for-CM-abelian-varieties//sections/04-theta.tex
The-rational-Hodge-conjecture-for-CM-abelian-varieties//sections/05a-moduli.tex
The-rational-Hodge-conjecture-for-CM-abelian-varieties//sections/05b-frobenius.tex
The-rational-Hodge-conjecture-for-CM-abelian-varieties//sections/05b-satake-frobenius.tex
The-rational-Hodge-conjecture-for-CM-abelian-varieties//sections/05c-cm-types.tex
The-rational-Hodge-conjecture-for-CM-abelian-varieties//sections/06-assembly.tex
The-rational-Hodge-conjecture-for-CM-abelian-varieties//sections/07-consequences.tex
The-rational-Hodge-conjecture-for-CM-abelian-varieties//references.bib
===========================

==== BEGIN The-rational-Hodge-conjecture-for-CM-abelian-varieties//main.tex ====
\documentclass[11pt]{article}
\input{preamble}

\title{The rational Hodge conjecture for CM abelian varieties}
\author{OpenAI}
\date{September 30, 2026}

\begin{document}
\maketitle

\begin{abstract}
We prove the rational Hodge conjecture for complex abelian varieties
with complex multiplication: every rational Hodge class on such a
variety is a rational linear combination of algebraic cycle classes.
As consequences, we obtain the generalized Hodge conjecture for CM
abelian varieties, the Tate conjecture for abelian varieties over finite
fields, and the Hodge standard conjecture for abelian varieties in
arbitrary characteristic.
\end{abstract}

\input{sections/01-introduction}
\tableofcontents

\input{sections/02-tensors}
\input{sections/03-surface}
\input{sections/04-theta}
\input{sections/05a-moduli}
\input{sections/05b-frobenius}
\input{sections/05b-satake-frobenius}
\input{sections/05c-cm-types}
\input{sections/06-assembly}
\input{sections/07-consequences}

\bibliographystyle{plain}
\phantomsection
\addcontentsline{toc}{section}{References}
\begingroup
\small
\bibliography{references}
\endgroup
\end{document}
==== END The-rational-Hodge-conjecture-for-CM-abelian-varieties//main.tex ====

==== BEGIN The-rational-Hodge-conjecture-for-CM-abelian-varieties//preamble.tex ====
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==== END The-rational-Hodge-conjecture-for-CM-abelian-varieties//preamble.tex ====

==== BEGIN The-rational-Hodge-conjecture-for-CM-abelian-varieties//sections/01-introduction.tex ====
\section{Introduction}\label{sec:introduction}

For a smooth projective complex variety \(X\), the rational Hodge
conjecture asks whether every rational cohomology class of type
\((p,p)\) is a rational linear combination of codimension-\(p\)
algebraic cycle classes.  We prove this assertion for abelian
varieties with complex multiplication, in every dimension and
codimension.

A complex abelian variety \(A\) is of \emph{CM type}, or simply
\emph{CM}, if \(\End(A)\otimes_{\Z}\Q\) contains a commutative
semisimple \(\Q\)-algebra of dimension \(2\dim A\).  We include the
point by convention.  Finite products and powers of CM abelian
varieties are again CM.

\Needspace{12\baselineskip}
\begin{theorem}\label{thm:cm}
Let \(A\) be a complex CM abelian variety.  For every integer
\(p\geq0\), the Betti cycle-class map
\[
 \operatorname{cl}_B:\CH^p(A)_{\Q}
 \longrightarrow H^{2p}(A,\Q)\cap H^{p,p}(A)
\]
is surjective, where \(\CH^p(A)_{\Q}=\CH^p(A)\otimes_{\Z}\Q\) and
the intersection is taken in \(H^{2p}(A,\C)\).  In particular, the
rational Hodge conjecture holds in every codimension on every finite
product of complex CM abelian varieties and on every power of such a
product.
\end{theorem}

The theorem also has consequences beyond the algebraicity of individual
CM Hodge classes.  The generalized Hodge conjecture predicts algebraic
supports of the codimension allowed by the Hodge types of a cohomology
substructure.  We obtain it for CM abelian varieties by applying the
theorem to correspondences from auxiliary CM varieties, following
Hazama and Abdulali \cite{Hazama2003CM,Abdulali2005CM}.
Two theorems of Milne then give the Tate conjecture for all abelian
varieties over finite fields, which asks for algebraic representatives
of Galois-invariant \(\ell\)-adic classes, and the Hodge standard
conjecture for abelian varieties in arbitrary characteristic, which
asserts positivity of the intersection form on primitive algebraic
classes \cite{Milne1999Tate,Milne2002Polarizations}.
Section~\ref{cc:section} states these conclusions with their coefficient
and support conventions and explains the separate implications.

\subsection{Hodge classes on CM abelian varieties}

The codimension-one case of the Hodge conjecture is the Lefschetz
\((1,1)\) theorem.  Hodge formulated the broader problem by asking
for an extension of this algebraicity criterion to higher-dimensional
cycles \cite[p.~184]{Hodge1952}.  Cup products of divisor classes therefore give
algebraic Hodge classes in higher degree.  For an abelian variety,
\(H^*(A,\Q)=\Lambda^*H^1(A,\Q)\), but this description of its
cohomology does not make every Hodge class a linear combination of
divisor products.  The problem is to construct cycles for the remaining
classes.

Weil classes provide an important instance of this difficulty.  Suppose
an imaginary quadratic field \(K\) acts on an abelian variety of
dimension \(2n\), with its two embeddings occurring with equal
multiplicity in \(H^{1,0}\).  Then
\(\Lambda_K^{2n}H^1(A,\Q)\) supplies a two-dimensional rational
space of classes of type \((n,n)\).  These determinant classes and
their generalization to CM fields are discussed in
\cite[Proposition~4.4]{Deligne1982}.  Their Hodge type follows from
the multiplicity condition; algebraicity requires a geometric
construction.  Weil classes need not lie in the algebra generated by
divisor classes; this occurs already on abelian fourfolds
\cite[Theorem~0.1]{MoonenZarhin1999}.  Recent work of Markman proves
algebraicity of the Weil classes on polarized abelian sixfolds of
Weil type for an imaginary quadratic field, with polarization
discriminant \(-1\), and deduces the Hodge conjecture for all abelian
fourfolds \cite[Theorem~1.5.1 and Corollary~1.6.1]{Markman2025Secant}.
The sixfold theorem assumes an imaginary quadratic action, without
requiring the full CM condition of Theorem~\ref{thm:cm}.

Deligne proved that every Hodge class on an abelian variety is
\emph{absolute Hodge}: after transport to a conjugate variety through
de Rham cohomology, it remains a Hodge class
\cite[Main Theorem~2.11]{Deligne1982}.  This establishes an arithmetic
compatibility shared by algebraic cycle classes, but does not by itself
produce algebraic cycles.  CM varieties play a central role in that
argument.  Their Hodge groups are tori, and their degree-one cohomology
splits, after extending coefficients, into lines indexed by field
embeddings.  An embedding-line tensor belongs to the scalar extension
of the rational Hodge classes exactly when every scalar conjugate
has equally many holomorphic and antiholomorphic factors.
This is Pohlmann's balance criterion \cite{Pohlmann1968}; see
\cite[\S1.2(c)]{Milne2022} and
\cite[Theorem~3.1]{GaoUllmo2026}.

The CM arguments of Deligne and Andr\'e reduce these Hodge classes to
Weil-type classes on auxiliary CM abelian varieties
\cite{Deligne1982,Andre1992}.  More precisely, the formulation in
\cite[Theorem~1]{Milne2022}, attributed there to Andr\'e, expresses a
CM Hodge class as a sum of pullbacks of split Weil classes.  Here
\emph{split} refers to the Hermitian polarization form having a totally
isotropic subspace of half its dimension.  This reduction identifies
classes whose algebraicity would suffice; it does not supply their
algebraic representatives in all dimensions.

There is also a reduction to relations involving only four labels.
Hazama's arrangement method spans its rational balance kernel by
antipodal pairs and four-label square relations
\cite[Theorem~7.13]{Hazama2003CM}; his Corollary~7.14.1 gives the
corresponding codimension-two algebraicity reduction.  Gao and Ullmo
give equivalent quadratic indicator relations and use them, after
adjoining auxiliary CM factors, to generate Hodge-induced period
relations \cite[Theorems~7.1--7.2]{GaoUllmo2026}.
Their conclusion concerns relations among periods.  Our task is to
construct algebraic representatives for the corresponding four-factor
tensors and to compose those representatives as correspondences.
This distinction matters: the reduction uses auxiliary factors and
polarization contractions, rather than asserting that low-codimension
classes generate the Hodge ring of a fixed power.

\subsection{From balanced tensors to surface periods}

Choose a Galois CM field \(E\) containing the CM fields in
\(H^1(A,\Q)\), and write \(G_E=\Gal(E/\Q)\) and \(c\) for its
complex conjugation.  An odd sign function
\(v:G_E\to\{1,-1\}\), satisfying \(v(c\tau)=-v(\tau)\), specifies
a weight-one Hodge structure \(U(v)\) on the rational vector space
\(E\).  Its embedding line \(U(v)_\lambda\), on which \(a\in E\)
acts as \(\lambda(a)\), has type \((1,0)\) or \((0,1)\) according
as \(v(\lambda)\) is positive or negative.
Section~\ref{sec:tensors} realizes \(U(v)=H^1(B(v),\Q)\) for a CM
abelian variety \(B(v)\).

The elementary relation needed in the proof is
\[
 v_1+v_2=v_3+v_4.
\]
It makes every scalar conjugate of the four-factor line
\[
 U(v_1)_1\otimes U(v_2)_1\otimes U(v_3)_c\otimes U(v_4)_c
 \ \subset H^4\!\left(\prod_{i=1}^4B(v_i),\Qbar\right)
\]
have type \((2,2)\).  We prove that this line lies in the scalar
extension of the rational cycle span.  Section~\ref{sec:tensors}
then assembles arbitrary balanced tensors from these relations:
two-row switches connect lists with the same sum, and algebraic
polarization pairings contract their intermediate factors.  Transfer
back to the original abelian variety and descent of coefficients
give Theorem~\ref{thm:cm}.

The geometric construction of a four-factor class takes place on a
surface.  Suppose there are rational Hodge maps
\(h_i:U(v_i)\to H^1(S,\Q)\), for a smooth projective surface
\(S\), and vectors \(\alpha_i\) on the four selected embedding
lines such that
\[
 \int_S h_1(\alpha_1)\smile h_2(\alpha_2)
              \smile h_3(\alpha_3)\smile h_4(\alpha_4)\ne0.
\]
Degree-one Hodge maps are induced, up to rational multiples, by
morphisms \(S\to B(v_i)\).  The image of \(S\) in their product
therefore gives an algebraic cycle whose pairing with the desired
line is nonzero.  Section~\ref{sec:surface} shows how four algebraic
divisor kernels convert this functional into a vector on that exact
line.  The remaining problem is thus to find four degree-one classes
with both a nonzero surface period and the prescribed rational CM
sources.

\subsection{Theta forms and their CM sources}

Section~\ref{sec:theta} constructs the period on a compact arithmetic
quotient of the complex two-ball.  Its unitary group has signature
\((2,1)\) at one distinguished embedding and is definite at all the
other real places.  After the elementary normalization in
Section~\ref{sec:tensors}, all four lines \(U(v_i)_1\) are holomorphic
at that embedding.  To each type the construction assigns a family
of holomorphic theta one-forms.  Their values span a cotangent fiber,
so two such families supply a wedge that is a nonzero holomorphic
two-form.

To compare the two sides of the four-factor relation, we realize each
pair of types by an orthogonal decomposition of the same Hermitian
plane.  The balance relation says that the two decompositions have
the same local sign multiplicities.  Weak approximation produces a
rational second decomposition whose compact-place data are as close
as desired to those of the first.  At the distinguished embedding,
the total period data depend only on the plane and remain unchanged.

A rank-two theta tensor packages the wedges into one total two-form.
We keep its finite theta input and the compact quotient fixed while
varying the compact-place data.  The total form varies continuously,
so its pairing with the initial nonzero form stays close to the
nonzero self-pairing.  At the two rational endpoints it is a finite
sum of wedges of the required one-forms.  Factoring these endpoints
and passing to a common finite cover therefore gives a nonzero mixed
period with two holomorphic and two antiholomorphic factors.

The construction uses Weil's rational theta covariance and the unitary
splittings of the Weil representation
\cite{Weil1964,Kudla1994,BoradeEtAl2025ToriPeriods}.
Kudla and Millson developed differential-form theta kernels in this
unitary setting \cite[Introduction and Theorem~3]{KudlaMillson1990DifferentialTheta}.
The projection turning the rank-two tensor into wedges has a close
predecessor in the pure-cocycle product calculation of Bergeron,
Millson, and M\oe glin \cite[Lemmas~5.7 and~5.11]{BMM2016}.
We derive the required tensor identity in coordinates and use the
fixed-input variation to obtain the mixed period.

These are initially complex one-form classes.  The surface criterion
also requires that they come from specified rational CM Hodge
structures.  Sections~\ref{sec:arithmetic} and~\ref{sec:cm-sources}
establish this arithmetic input.  First we realize the ball quotients
in a projective moduli space and construct algebraic Hecke
endomorphisms of the product of its components' Albanese varieties.
Counting ordinary isogeny
neighbors gives a polynomial relation between these operators and
Frobenius.  We then compute the Hecke parameters of the theta forms.
The Frobenius relation and the degree-one Weil weight force slope
zero or one on the relevant Hecke summands, with the choice determined
by the prescribed CM signs.

The Hecke projectors preserve the Hodge filtration, so weak
admissibility turns these extreme slopes into pure Hodge types.
Applying the calculation
to every scalar conjugate identifies the complete CM type of each
summand.  Scalar descent then expresses every relevant theta class
as a complex linear combination of images of the selected CM line
under rational Hodge maps.  Section~\ref{sec:assembly} puts the
finite families on one projective surface, expands the nonzero
period, and applies the surface criterion.  This completes the
four-factor construction and hence the proof of the main theorem.
==== END The-rational-Hodge-conjecture-for-CM-abelian-varieties//sections/01-introduction.tex ====

==== BEGIN The-rational-Hodge-conjecture-for-CM-abelian-varieties//sections/02-tensors.tex ====
\section{CM tensors and four-factor relations}\label{sec:tensors}

This section reduces Theorem~\ref{thm:cm} to the algebraicity of a
specified family of tensors in the fourth cohomology of products of four
CM abelian varieties.  The reduction changes the CM types of individual
factors while preserving their total Hodge type; its intermediate
factors may therefore be absent from the original variety.  We first
record how to work with individual embedding lines and recover
rationality after extending coefficients.  We then construct the
auxiliary CM varieties and assemble arbitrary balanced tensors from
four-factor relations by algebraic correspondences.

\subsection{Scalar coefficients and degree-one maps}

For a smooth projective complex variety \(X\), write
\[
 \Hdg^{2p}(X)=H^{2p}(X,\Q)\cap H^{p,p}(X),\qquad
 \mathcal Z^{2p}(X)=
 \im\bigl(\CH^p(X)_{\Q}\longrightarrow H^{2p}(X,\Q)\bigr).
\]
Cohomology is Betti cohomology unless another theory is specified.  We
fix \(\Qbar\subset\C\), and write \(T_k=T\otimes_{\Q}k\) for a rational
vector space \(T\).  A class with coefficients in \(\Qbar\) or \(\C\)
is called \emph{algebraic} if it belongs to the corresponding scalar
extension of \(\mathcal Z^{2p}(X)\).  Scalar conjugation therefore
preserves algebraicity on the same complex variety: it fixes each
rational cycle class and changes only its coefficient.  We use the
compatibility of cycle classes with exterior products, cup products,
pullback between smooth varieties, and proper pushforward.

Our Tate convention is \(V(r)^{a,b}=V^{a+r,b+r}\), so \(V(r)\) has
weight \(\operatorname{wt}(V)-2r\).

\begin{lemma}[Scalar descent]\label{lem:trace-descent}
Let \(T\) be a finite-dimensional rational vector space and
\(P\subset T_{\C}\) a complex-linear subspace.  If \(t\in T_{\Qbar}\)
and \(\sigma t\in P\) for every
\(\sigma\in\Gal(\Qbar/\Q)\), then
\[
 t\in (T\cap P)\otimes_{\Q}\Qbar.
\]
The same assertion applies to the subspace of maps of any fixed Hodge
type in a Hom space.
\end{lemma}

\begin{proof}
Choose a finite Galois extension \(K/\Q\) containing the coefficients
of \(t\).  For \(a\in K\), the vector
\[
 T_a=\Tr_{K/\Q}(at)
     =\sum_{\sigma\in\Gal(K/\Q)}\sigma(a)\sigma(t)
\]
is rational.  Each automorphism of \(K\) extends to \(\Qbar\), so every
summand belongs to \(P\); hence \(T_a\in T\cap P\).  If
\(b_1,\ldots,b_r\) is a rational basis of \(K\) and
\(b_1^*,\ldots,b_r^*\) is its trace-dual basis, write
\(t=\sum_jt_jb_j\) with \(t_j\in T\).  Then \(T_{b_j^*}=t_j\), which
proves the assertion.  The condition that a complex-linear map have a
fixed Hodge type is complex-linear, so the same argument applies in a
Hom space.
\end{proof}

We will also use
\begin{equation}\label{eq:cycle-descent}
 \bigl(\mathcal Z^{2p}(X)\otimes_{\Q}\Qbar\bigr)
       \cap H^{2p}(X,\Q)=\mathcal Z^{2p}(X).
\end{equation}
Indeed, extend a rational basis of \(\mathcal Z^{2p}(X)\) to one of
\(H^{2p}(X,\Q)\) and compare coefficients.

\begin{lemma}[Realization in degree one]\label{lem:weight-one-maps}
For complex abelian varieties \(A\) and \(B\), pullback induces an
isomorphism
\[
 \Hom(A,B)\otimes_{\Z}\Q
 \ \simeq\
 \Hom_{\mathrm{Hdg}}\bigl(H^1(B,\Q),H^1(A,\Q)\bigr).
\]
If \(S\) is a smooth connected projective complex variety and
\(h:H^1(B,\Q)\to H^1(S,\Q)\) is a rational Hodge map, then there are
an integer \(m\geq1\) and a morphism \(f:S\to B\) such that
\(f^*=mh\) on \(H^1(B,\Q)\).
\end{lemma}

\begin{proof}
Dualizing a rational Hodge map gives a rational map on homology
commuting with the complex structures.  Multiplication by a positive
integer makes it carry the integral lattice of \(A\) into that of
\(B\).  It consequently induces a holomorphic homomorphism of complex
tori \(A^{\mathrm{an}}\to B^{\mathrm{an}}\), which is algebraic.
Conversely, an algebraic homomorphism induces the indicated Hodge
pullback.  This proves the first assertion; equivalently it is the
dual of the rational form of Riemann's theorem
\cite[Proposition~6.5, Theorems~6.7--6.8, and Corollary~6.9]{MilneShimura2017}.

The Albanese map of \(S\) induces an isomorphism
\(H^1(\Alb S,\Q)\simeq H^1(S,\Q)\).  Apply the first assertion to the
map with target \(H^1(\Alb S,\Q)\), clear a denominator, and compose
the resulting homomorphism \(\Alb S\to B\) with an Albanese map.
\end{proof}

The Lefschetz \((1,1)\) theorem says that
\(\Hdg^2(X)=\mathcal Z^2(X)\).  In conjunction with
Lemma~\ref{lem:trace-descent}, it has the following useful consequence:
if \(t\in H^2(X,\Qbar)\) is of type \((1,1)\) in every scalar
conjugate, then \(t\) is algebraic.  This observation will supply all
two-factor tensors used below.

\subsection{Rank-one CM structures}

For a polarizable rational Hodge structure \(H\) of weight one, let
\(J(H)\) be the smallest rational algebraic subgroup of \(\GL(H)\)
containing the Hodge circle.  The circle acts by \(z\) on \(H^{1,0}\)
and by \(\bar z\) on \(H^{0,1}\), for \(|z|=1\).  It preserves an
alternating polarization.  Its connectedness shows that it lies in the
rational identity component of \(J(H)\); minimality therefore makes
\(J(H)\) connected.

\begin{lemma}\label{lem:toric-cm}
For a complex abelian variety \(A\), the following conditions are
equivalent:
\begin{enumerate}[label=\textup{(\roman*)}]
\item \(J(H^1(A,\Q))\) is a torus.
\item \(H^1(A,\Q)\) is a direct sum of rational Hodge structures,
      each of rank one over a CM field acting by Hodge endomorphisms.
\item \(\End(A)\otimes_{\Z}\Q\) contains a commutative semisimple
      algebra of rational dimension \(2\dim A\).
\end{enumerate}
\end{lemma}

\begin{proof}
The zero-dimensional case is immediate.  Put \(H=H^1(A,\Q)\), and
first assume that \(J(H)\) is a torus.  Its associative envelope
\(E_0\subset\End_{\Q}(H)\) is a commutative semisimple algebra: after
splitting the torus, its elements act diagonally by scalars on the
weight spaces, so the envelope is a subalgebra of a product of fields
and is reduced.  This description descends to \(\Q\).

Let \(*\) be the polarization adjoint.  It preserves \(E_0\), since
it sends every symplectic torus element to its inverse.  Every element
of \(E_0\) commutes with the complex structure \(I\) supplied by the
Hodge circle.  Consequently \(*\) is also the adjoint for the positive
definite real form obtained by inserting \(I\) in the polarization.
In particular,
\[
 \Tr_{H_{\R}}(a^*a)>0\qquad(0\ne a\in E_0\otimes_{\Q}\R).
\]
The involution cannot exchange two primitive factors of the real
finite étale algebra \(E_0\otimes\R\): for the idempotent \(e\) of one
factor, this would give \(e^*e=0\).  On a complex factor it cannot be
the identity, since for its idempotent \(e\) one would have
\((ie)^*(ie)=-e\), of negative trace.  Thus it acts as complex
conjugation on each complex factor.  There are no real factors.  On
the nonzero module of such a factor the circle would act by real
scalars, and a continuous homomorphism from the circle to
\(\R^\times\) is trivial; this contradicts the weight-one characters.

It follows that each rational field factor \(K\) of \(E_0\) is CM.
Indeed, it has only complex archimedean factors, and its involution
satisfies \(\tau(a^*)=\overline{\tau(a)}\) at every embedding.  Its
fixed field is therefore totally real and has index two.  On the
corresponding \(K\)-module the circle acts by \(K\otimes\R\)-scalars.
Splitting that module into \(K\)-lines thus gives rational Hodge
substructures.  This proves (ii), without discarding multiplicities.

The separate scalar actions on the field lines in (ii) form a
commutative semisimple Hodge-endomorphism algebra of dimension
\(\dim_{\Q}H\).  Lemma~\ref{lem:weight-one-maps} realizes it in
\(\End(A)\otimes\Q\), proving (iii).  Conversely, write an algebra as
in (iii) as \(\prod_jK_j\).  Its faithful action on \(H\) has a
positive rank \(m_j\) over each \(K_j\), and
\[
 \dim_{\Q}H=\sum_jm_j[K_j:\Q]=\sum_j[K_j:\Q].
\]
Hence every \(m_j=1\).  The centralizer of this action in \(\GL(H)\)
is the torus \(\prod_j\Res_{K_j/\Q}\mathbf G_m\).  It contains the
Hodge circle because the algebra acts by Hodge endomorphisms, proving
(i).
\end{proof}

These equivalences give the rank-one description of the CM varieties
defined in the introduction.
A finite product of CM abelian varieties is CM: the circle on the
direct sum of their degree-one cohomologies lies in the product of
their tori, and its connected rational envelope is a subtorus.

Choose a Galois CM field \(E\subset\Qbar\), and put
\[
 F=E^+,\qquad G_E=\Gal(E/\Q),\qquad s=[F:\Q].
\]
Its complex conjugation \(c\in G_E\) is central.  One such \(E\) can
contain any finite collection of CM fields and all their conjugates.
To see this, take the compositum of all those conjugates.  For a CM
field \(K\), every embedding \(\rho:K\to\C\) satisfies
\(\rho c_K=c_{\C}\rho\).  These identities show that complex
conjugation on the Galois compositum commutes with its Galois group.
Every conjugate of an element fixed by it is real, so the fixed field
is totally real.  The compositum is therefore a Galois CM field.

An \emph{odd sign function} on \(G_E\) is a function
\(v:G_E\to\{1,-1\}\) satisfying \(v(c\tau)=-v(\tau)\).  Give the
rational vector space \(E\) a weight-one Hodge structure \(U(v)\) by
declaring its embedding line
\[
 U(v)_\lambda=
 \{u\in U(v)_{\Qbar}: au=\lambda(a)u\text{ for every }a\in E\}
 \qquad(\lambda\in G_E)
\]
to have type \((1,0)\) if \(v(\lambda)=1\), and type \((0,1)\)
otherwise.  Oddness makes these declarations compatible with the real
structure.

This Hodge structure is polarizable.  Choose \(0\ne\delta\in E\) with
\(c(\delta)=-\delta\).  Multiplying \(\delta\) by an element of
\(F^\times\), weak approximation gives an element \(\xi\in E^\times\)
such that \(c(\xi)=-\xi\) and
\[
 \operatorname{sgn}\bigl(\operatorname{Im}\tau(\xi)\bigr)=-v(\tau)
\]
at one representative \(\tau\) of each conjugate pair.  The form
\[
 \psi_v(x,y)=\Tr_{E/\Q}\bigl(\xi x c(y)\bigr)
\]
is alternating and nondegenerate.  It pairs only conjugate embedding
lines.  On a vector \(u\) in the sum of the \((1,0)\) lines,
\(i\psi_v(u,\bar u)\) is a sum of positive multiples of the squared
absolute values of its embedding coordinates; hence it is positive
for \(u\ne0\).  This is the weight-one polarization condition.
After scaling, the form is integral on a lattice.  The Riemann
criterion, applied to the dual homological Hodge structure, gives a
CM abelian variety \(B(v)\) of dimension \(s\) with
\begin{equation}\label{eq:cm-types}
 U(v)=H^1(B(v),\Q).
\end{equation}
This construction permits every odd sign function, including
imprimitive CM types; it does not require a principal polarization.
The polarization and Riemann-criterion argument also applies to any
CM field, without a Galois hypothesis, and any CM type, that is, any
choice of one embedding from each conjugate pair.  The Galois field
\(E\) provides a common indexing set for the types used below.

We will need the exact action of scalar conjugation on labels.  If
\(\sigma\in\Gal(\Qbar/\Q)\) restricts to \(g\in G_E\) and
\(u\in U(v)_\lambda\), rationality of the \(E\)-action gives
\[
 a\,\sigma(u)=\sigma(au)
             =\sigma(\lambda(a))\sigma(u)
             =(g\lambda)(a)\sigma(u).
\]
Consequently
\begin{equation}\label{eq:cm-label-conjugation}
 \sigma U(v)_\lambda=U(v)_{g\lambda}.
\end{equation}
In particular the complete type signs of \(U(v)_1\) and \(U(v)_c\)
after \(g\) are \(v(g)\) and \(-v(g)\), respectively.  Here and below
the commutation of \(c\) with \(g\) takes place in \(G_E\); no
centrality assertion is made in \(\Gal(\Qbar/\Q)\).

The same convention applies to a line in a rank-one module over a CM
subfield of \(E\).  Choose \(\lambda\in G_E\) extending the embedding
that labels the line.  Its \emph{complete type function} takes the value
\(+1\) at \(g\in G_E\) if the line labeled by the restriction of
\(g\lambda\) has type \((1,0)\), and \(-1\) if it has type \((0,1)\).
It is odd.  Since the embedding-line decomposition is defined over
\(E\), this is the type of the scalar-conjugate line for every
automorphism of \(\Qbar\) restricting to \(g\).

\begin{lemma}[Transfer of an embedding line]\label{lem:cm-line-transfer}
Let \(H\) and \(H'\) be direct sums of rank-one weight-one Hodge
modules over CM fields contained in \(E\).  Choose an embedding line
\(L\subset H_{\Qbar}\) and an embedding line \(M\subset H'_{\Qbar}\)
with the same complete type function.  Extend any
\(\Qbar\)-linear map \(T:L\to M\) by zero on all other source
embedding lines.  Then
\[
 T\in\Hom_{\mathrm{Hdg}}(H,H')\otimes_{\Q}\Qbar.
\]
When \(H\) and \(H'\) are degree-one cohomologies of abelian varieties,
this map is therefore an algebraic-coefficient linear combination of
pullbacks of abelian morphisms, up to rational denominators.
\end{lemma}

\begin{proof}
Scalar conjugation transports the entire source and target
decompositions.  For each \(\sigma\), the only possibly nonzero
component of \(\sigma T\) maps \(\sigma L\) to \(\sigma M\), whose
types agree by hypothesis.  All other components are zero.  Thus
\(\sigma T\) is a Hodge map on the whole source.  Apply
Lemma~\ref{lem:trace-descent} in \(\Hom_{\Q}(H,H')\), followed, when
applicable, by Lemma~\ref{lem:weight-one-maps}.
\end{proof}

The map \(T\) in this lemma need not be invariant under a stabilizer
of either line.  This matters when a field is a proper subfield of
\(E\), or when a rational field module occurs more than once.  The
proof treats every scalar conjugate of the zero-extended map and
retains the repeated rational summands separately.

\subsection{Reduction to four factors}

The four-factor assertion that remains to be proved is the following.

\begin{theorem}[Four-factor switch]\label{thm:switch}
Let \(E\) be a Galois CM field and let
\(v_1,v_2,v_3,v_4:G_E\to\{1,-1\}\) be odd sign functions satisfying
\(v_1+v_2=v_3+v_4\).  Under the Künneth inclusion, the following line is algebraic:
\begin{equation}\label{eq:switch-line}
 U(v_1)_1\otimes U(v_2)_1\otimes U(v_3)_c\otimes U(v_4)_c
 \ \subset\
 H^4\!\left(\prod_{i=1}^4B(v_i),\Qbar\right)
\end{equation}
\end{theorem}

This line is already of Hodge type \((2,2)\) in every scalar
conjugate.  Indeed, after an automorphism restricting to \(g\in G_E\),
its four type signs have sum
\begin{equation}\label{eq:switch-conjugate-balance}
 v_1(g)+v_2(g)-v_3(g)-v_4(g)=0.
\end{equation}
The theorem asks for algebraicity, which does not follow from this
balance alone.  We leave its proof to the subsequent construction;
the present section proves that it is sufficient.

Hazama's coordinate-arrangement calculation rationally spans the
kernel of its Hodge matrix by antipodal pairs and elementary
four-label relations \cite[Theorem~7.13]{Hazama2003CM}.  His
Corollary~7.14.1 states a corresponding codimension-two algebraicity
reduction.  We prove the particular tensor reduction needed here
directly, including the transfers to auxiliary abelian factors and
the algebraic contractions that assemble the relations.

\begin{proposition}\label{prop:switch-reduction}
If Theorem~\ref{thm:switch} holds for every Galois CM field and every
quadruple of odd sign functions satisfying its relation, then
Theorem~\ref{thm:cm} holds.
\end{proposition}

\begin{proof}
Let \(A\) be a CM abelian variety and let \(h\in\Hdg^{2p}(A)\).
The cases \(p=0\) and \(p=1\) follow from the unit class and the
Lefschetz \((1,1)\) theorem, respectively.  For higher codimension,
we first replace a Hodge monomial by a tensor on auxiliary CM
factors.  We then connect its two lists of types by four-factor
switches and compose the resulting algebraic tensors.

By Lemma~\ref{lem:toric-cm}, choose
a rational decomposition of \(H=H^1(A,\Q)\) into rank-one CM-field
modules, retaining separate copies of repeated summands.  Choose a
Galois CM field \(E\) containing all the fields and their conjugates.
Over \(E\), \(H\) is the direct sum of its embedding lines.  Since cup
product gives \(H^*(A,\Q)=\Lambda^*H\), the exterior monomials in
these lines give a direct-sum decomposition of \(H^*(A,\Qbar)\) into
one-dimensional lines \cite[Proposition~6.4]{MilneShimura2017}.
Every such line is contained in one Hodge bidegree.

Scalar conjugation permutes the monomial
lines, with only the possible sign from reordering exterior factors.
Because \(h\) is rational, a nonzero component on one monomial line
gives a nonzero component on every line in its scalar-Galois orbit.
All these lines must have type \((p,p)\).  The decomposition is
defined over \(E\), so the components of \(h\) have algebraic Betti
coefficients.

Fix a supported monomial and list the complete type functions of its
\(2p\) factors as \(z_1,\ldots,z_{2p}\).  Each \(z_i\) is odd.  In the
conjugate indexed by \(g\), the number of \((1,0)\) factors minus the
number of \((0,1)\) factors is \(\sum_i z_i(g)\).  Its balance is
therefore exactly
\[
 \sum_{i=1}^{2p}z_i=0.
\]
For \(1\leq i\leq p\), set \(x_i=z_i\) and \(y_i=-z_{p+i}\).  Then
\(\sum_i x_i=\sum_i y_i\).  The first \(p\) factors have the complete
types of \(U(x_i)_1\), and the last \(p\) have those of \(U(y_i)_c\).
For any ordered list \(a=(a_1,\ldots,a_p)\) of odd sign functions,
put \(A_a=\prod_{i=1}^pB(a_i)\); for two such lists put
\[
 \mathscr L_{a,b}=
 \bigotimes_{i=1}^pU(a_i)_1\ \otimes\
 \bigotimes_{i=1}^pU(b_i)_c
 \subset H^{2p}(A_a\times A_b,\Qbar).
\]
Repeated factors in these products are kept as separate slots.
Lemma~\ref{lem:cm-line-transfer} reduces the chosen monomial to the
algebraicity of \(\mathscr L_{x,y}\).  We return to \(A\) after
composing the transition tensors.

Regard each list as a matrix with \(p\) rows and one column for each
representative of \(G_E/\langle c\rangle\).  At any column, equality
of the two sums is equality of the numbers of plus signs.  Among the
rows where the current \(x\)-entry differs from the desired
\(y\)-entry, there are equally many mismatches of the forms
\((+1,-1)\) and \((-1,+1)\).  Pair these two sets of rows and swap the
two current entries in every pair, making the opposite swap at the
conjugate coordinate.  This corrects the column, leaves the other
columns unchanged, and preserves oddness of every row.
Successively correcting the columns gives a finite sequence
\[
 x=x^{(0)},\ x^{(1)},\ \ldots,\ x^{(N)}=y
\]
in which consecutive lists differ by one such two-row swap.

At a single swap the old rows \(a,b\) and new rows \(a',b'\) satisfy
\(a+b=a'+b'\) at every coordinate.  Theorem~\ref{thm:switch} supplies
the pure algebraic tensor
\[
 U(a)_1\otimes U(b)_1\otimes U(a')_c\otimes U(b')_c.
\]
Every unchanged row is joined to its new copy by
\(U(v)_1\otimes U(v)_c\), an all-conjugate \((1,1)\) line and hence an
algebraic two-factor tensor.  Pull these classes to their distinct
old and new slots and multiply.  This gives a nonzero algebraic
generator of \(\mathscr L_{x^{(j)},x^{(j+1)}}\).  If the original
lists already agree, the two-factor tensors alone generate
\(\mathscr L_{x,x}\).  It remains to compose consecutive transition
tensors without losing their algebraicity or nonvanishing.

For three ordered lists \(a,b,e\), suppose that
\(\mathscr L_{a,b}\) and \(\mathscr L_{b,e}\) have pure algebraic
generators with algebraic Betti coefficients.  Write them as
\[
 \begin{split}
 t_{ab}&=\alpha_1\otimes\cdots\otimes\alpha_p\otimes
          \beta_1\otimes\cdots\otimes\beta_p,\\
 t_{be}&=\gamma_1\otimes\cdots\otimes\gamma_p\otimes
          \delta_1\otimes\cdots\otimes\delta_p,
 \end{split}
\]
where \(\beta_i\in U(b_i)_c\) and \(\gamma_i\in U(b_i)_1\).  Identify the
middle copies and choose an ample divisor class \(\ell_i\) on each
\(B(b_i)\).  With \(r_{ab},r_{be},r_{ae}\) denoting the projections
from \(A_a\times A_b\times A_e\), and with the \(\ell_i\) also denoting
their pullbacks, cup product and pushforward give
\begin{equation}\label{eq:transition-composition}
 \begin{split}
 (r_{ae})_*\left(
 r_{ab}^*t_{ab}\smile r_{be}^*t_{be}
       \smile\prod_{i=1}^p\ell_i^{\,s-1}\right)
 &=
 (-1)^{p(p-1)/2}\left(\prod_{i=1}^p q_i\right)t_{ae},\\
 q_i&=\int_{B(b_i)}\beta_i\wedge\gamma_i\wedge\ell_i^{\,s-1},
 \end{split}
\end{equation}
where \(t_{ae}=\alpha_1\otimes\cdots\otimes\alpha_p\otimes
\delta_1\otimes\cdots\otimes\delta_p\).  The sign is obtained by regrouping
\(\beta_1\cdots\beta_p\gamma_1\cdots\gamma_p\) into
\(\beta_1\gamma_1\cdots\beta_p\gamma_p\) in the indicated slot order.

Each \(q_i\) is nonzero.  Complex conjugation exchanges the two
middle embedding lines, so \(\gamma_i\) is a nonzero scalar multiple of
\(\bar\beta_i\).  Hodge--Riemann positivity for the ample class makes
the pairing with \(\bar\beta_i\) nonzero, whether \(\beta_i\) is holomorphic
or antiholomorphic.  This requires no compatibility between \(\ell_i\)
and \(E\).  Moreover \(q_i\in\Qbar^\times\), since its vectors have
algebraic Betti coefficients and \(\ell_i\) is rational.  The
codimension on the left of Equation~\eqref{eq:transition-composition}
is
\[
 p+p+p(s-1)-ps=p.
\]
Thus it is an algebraic class of the required degree, and division by
the nonzero algebraic coefficient shows that the endpoint line is
algebraic.  Iterating along the sequence of lists proves algebraicity
of \(\mathscr L_{x,y}\).

Finally choose the line transfers in the direction from the auxiliary
\(U\)'s to \(H^1(A,\Q)\).  Each is an algebraic-coefficient linear
combination of rational pullbacks along maps from \(A\) to the
corresponding \(B(v)\).  Apply their exterior product on \(A^{2p}\)
and pull back along the diagonal \(A\to A^{2p}\).  The result is the
cup product of the selected original factors.  They occupy distinct
embedding or multiplicity lines in the original exterior monomial,
so this cup product is nonzero.  We have proved that every supported
monomial line is algebraic.  Summing the components of \(h\) and
using Equation~\eqref{eq:cycle-descent} proves the rational assertion.

Finite products are covered because they are themselves CM,
as observed after Lemma~\ref{lem:toric-cm}.
\end{proof}

The proof uses tensors on auxiliary products and contracts their
intermediate slots before returning to \(A\).  It does not assert
generation of the Hodge ring on a fixed power by cup products of
classes in codimensions one and two.

\begin{lemma}[Choice of the distinguished embedding]
 \label{lem:switch-normalization}
It suffices to prove Theorem~\ref{thm:switch} when \(s\geq2\) and
\(v_i(1)=1\) for all four functions.
\end{lemma}

\begin{proof}
If \(v_1\) and \(v_2\) are nowhere simultaneously positive, oddness
also rules out simultaneous negativity.  Hence \(v_2=-v_1\).
The relation gives \(v_4=-v_3\), and the first two and the last two
slots of Equation~\eqref{eq:switch-line} are separately
all-conjugate \((1,1)\) pairs.

Otherwise choose \(g\in G_E\) at which \(v_1(g)=v_2(g)=1\).  The
relation makes all four signs positive there.  Choose a scalar
automorphism extending \(g\).  It sends the labels in
Equation~\eqref{eq:switch-line} to \(g,g,gc,gc\).  The complete type
function of the line labeled \(g\) is the right translate
\[
 v_i^g(\tau)=v_i(\tau g).
\]
It is odd, satisfies \(v_i^g(1)=1\), and the four translated functions
satisfy the same relation.  Centrality of \(c\) in \(G_E\) makes the
complete type of the \(gc\)-line equal to \(-v_i^g\).  The
transfers of Lemma~\ref{lem:cm-line-transfer} therefore carry the
normalized switch for the \(v_i^g\)'s to the lines labeled
\(g,g,gc,gc\) on the original factors.  Applying the inverse scalar
automorphism gives Equation~\eqref{eq:switch-line}.

When \(s=1\), there is only one odd sign function positive at the
identity, so the normalized tensor is a product of ordinary pairs.
\end{proof}

For the remaining construction we work in the normalized case, so
all four lines \(U(v_i)_1\) are holomorphic.  The next section
isolates a surface-period condition sufficient to prove the switch.
==== END The-rational-Hodge-conjecture-for-CM-abelian-varieties//sections/02-tensors.tex ====

==== BEGIN The-rational-Hodge-conjecture-for-CM-abelian-varieties//sections/03-surface.tex ====
\section{A surface criterion for the switch}\label{sec:surface}

To prove Theorem~\ref{thm:switch}, we will construct four degree-one
classes on one surface with a nonzero cup product.  The criterion
below explains why this is sufficient.  The four maps give an
algebraic surface class that tests the switch line nontrivially; four
divisor classes then convert that test into a vector on the line
itself.

\begin{proposition}[Surface criterion]\label{prop:surface-switch}
Let \(E\) be a Galois CM field with conjugation \(c\), and let
\(v_1,v_2,v_3,v_4:G_E\to\{1,-1\}\) be odd sign functions satisfying
\(v_1+v_2=v_3+v_4\).  Put \(\lambda=(1,1,c,c)\).
Suppose that \(S\) is a smooth connected projective complex surface
and that there are rational Hodge maps
\[
 h_i:U(v_i)\longrightarrow H^1(S,\Q),\qquad 1\leq i\leq4,
\]
and nonzero vectors
\(\alpha_i\in U(v_i)_{\lambda_i}\subset U(v_i)_{\Qbar}\) such that
\begin{equation}\label{eq:surface-period}
 \int_S h_1(\alpha_1)\smile h_2(\alpha_2)
             \smile h_3(\alpha_3)\smile h_4(\alpha_4)\ne0.
\end{equation}
Here each \(h_i\) is extended to \(\Qbar\)-coefficients.  Then the
line in Equation~\eqref{eq:switch-line} is algebraic.
\end{proposition}

\begin{proof}
By Lemma~\ref{lem:weight-one-maps}, there are positive integers \(m_i\)
and morphisms \(f_i:S\to B(v_i)\) with \(f_i^*=m_i h_i\).  Therefore
\begin{equation}\label{eq:surface-morphism-period}
 k:=\int_S f_1^*\alpha_1\smile f_2^*\alpha_2
              \smile f_3^*\alpha_3\smile f_4^*\alpha_4
 \in\Qbar^\times.
\end{equation}
Nonvanishing follows from Equation~\eqref{eq:surface-period}, since
the new value is multiplied by \(\prod_i m_i\).  The value is
algebraic because the pullbacks, cup product, and evaluation on
\([S]\) are rational operations on Betti cohomology and the
\(\alpha_i\) have algebraic Betti coefficients.

Put \(A=\prod_iB(v_i)\), \(n=\dim A=4s\), and \(f=(f_i):S\to A\).
Let
\[
 z=f_*[S]\in\CH^{n-2}(A),\qquad
 Z=\operatorname{cl}(z)\in\mathcal Z^{2n-4}(A).
\]
For \(t_\alpha=\alpha_1\otimes\cdots\otimes\alpha_4\) in the
four-\(H^1\) Künneth summand of \(H^4(A,\Qbar)\), the projection
formula gives
\begin{equation}\label{eq:surface-functional}
 \int_A Z\smile t_\alpha=k.
\end{equation}
Thus \(Z\) gives a nonzero functional on the switch line; it is not
itself a vector on that line.  In particular \(f(S)\) cannot have
dimension less than two, since then the proper pushforward of the
two-cycle \([S]\) and the displayed pairing would be zero.

For each \(i\), take a copy \(B(v_i)'\) with its transported
\(E\)-action, and choose a nonzero vector
\(\beta_i\in U(v_i)'_{c\lambda_i}\) with algebraic Betti coefficients.
Under the Künneth inclusion define the pure class
\[
 D_i=\alpha_i\otimes\beta_i
 \in H^2\bigl(B(v_i)\times B(v_i)',\Qbar\bigr).
\]
If \(\sigma\in\Gal(\Qbar/\Q)\) restricts to \(g\in G_E\), then
Equation~\eqref{eq:cm-label-conjugation} puts its two factors on the
lines labeled
\[
 g\lambda_i,\qquad gc\lambda_i=cg\lambda_i.
\]
These have opposite Hodge types.  This uses centrality of \(c\) in
\(G_E\) only, and remains valid when the scalar coordinates of
\(\alpha_i\) and \(\beta_i\) lie outside \(E\).  Every scalar
conjugate of \(D_i\) therefore has type \((1,1)\).  Scalar descent
and the Lefschetz \((1,1)\) theorem give
\begin{equation}\label{eq:surface-divisor-kernels}
 D_i\in
 \mathcal Z^2\bigl(B(v_i)\times B(v_i)'\bigr)\otimes_{\Q}\Qbar.
\end{equation}
The entire cohomology class \(D_i\) is algebraic here.  It was defined
in the pure \(H^1\otimes H^1\) summand before applying descent, so
there is no need to project an arbitrary divisor class onto that
summand.

Set \(A'=\prod_iB(v_i)'\).  Let \(p:A\times A'\to A\) and
\(p':A\times A'\to A'\) be the projections, and let
\(q_i:A\times A'\to B(v_i)\times B(v_i)'\) select the \(i\)-th old
and new slots.  Write
\[
 K=\prod_{i=1}^4q_i^*D_i,\qquad
 t_\beta=\beta_1\otimes\cdots\otimes\beta_4.
\]
Order the old factors first and the new factors second in the
Künneth identification for \(A\times A'\).  Moving the new
degree-one factor of \(D_i\) past the later old degree-one factors
makes \(3+2+1=6\) sign changes.  Hence
\[
 K=(-1)^6\,t_\alpha\otimes t_\beta=t_\alpha\otimes t_\beta.
\]
In particular \(K\) has degree four in the old factors and degree
four in the new factors.  Integration over \(A\) consequently gives
the exact formula
\begin{equation}\label{eq:surface-to-tensor}
 p'_*\bigl(p^*Z\smile K\bigr)
   =\left(\int_A Z\smile t_\alpha\right)t_\beta
   =k\,t_\beta.
\end{equation}
Any other Künneth component of \(Z\) contributes zero to this
integration.  The codimension of the left side is
\((n-2)+4-n=2\), equivalently its cohomological degree is
\((2n-4)+8-2n=4\).

By Equation~\eqref{eq:surface-divisor-kernels}, the left side of
Equation~\eqref{eq:surface-to-tensor} is algebraic with
\(\Qbar\)-coefficients.  Since \(k\in\Qbar^\times\), division by \(k\)
proves algebraicity of the line generated by \(t_\beta\), whose
labels are \((c,c,1,1)\).  Ordinary scalar complex conjugation fixes
the rational cycle span on the same \(A'\) and changes these labels
to \((1,1,c,c)\).  Identifying the copied factors with the original
ones gives the asserted line on \(A\).
\end{proof}

The theta construction produces complex classes, and the arithmetic
argument will place them in spans of rational Hodge images.  The
following consequence applies to those classes.

\begin{corollary}[Surface criterion for complex spans]
\label{cor:surface-span}
Let \(E\), \(v_1,\ldots,v_4\), and \(\lambda=(1,1,c,c)\) be as in
Proposition~\ref{prop:surface-switch}, and let \(S\) be a smooth
connected projective complex surface.  Suppose that, for each
\(1\leq i\leq4\), a class \(\eta_i\in H^1(S,\C)\) belongs to
\[
 \operatorname{span}_{\C}
 \{h(U(v_i)_{\lambda_i}):
       h\in\Hom_{\mathrm{Hdg}}(U(v_i),H^1(S,\Q))\}
\]
and that
\[
 \int_S\eta_1\smile\eta_2\smile\eta_3\smile\eta_4\ne0.
\]
Then the line in Equation~\eqref{eq:switch-line} is algebraic.
\end{corollary}

\begin{proof}
Fix a nonzero algebraic Betti generator of each selected line.
Each \(\eta_i\) is a finite complex linear combination of its images
under rational Hodge maps.  Expanding the nonzero cup product selects
a nonzero term of the form in Equation~\eqref{eq:surface-period}.
Thus Proposition~\ref{prop:surface-switch} applies.  This argument
uses only finite sums; all four maps have the same target
\(H^1(S,\Q)\).
\end{proof}
==== END The-rational-Hodge-conjecture-for-CM-abelian-varieties//sections/03-surface.tex ====

==== BEGIN The-rational-Hodge-conjecture-for-CM-abelian-varieties//sections/04-theta.tex ====
\section{Theta differentials and a nonzero mixed period}\label{sec:theta}

The surface criterion of Proposition~\ref{prop:surface-switch} asks for
a nonzero period made from four one-forms on one surface.  We construct
the analytic part of that input using theta series attached to auxiliary
Hermitian spaces of dimensions one and two, called multiplicity spaces.
For a multiplicity line, a component of the first Taylor tensor is a
holomorphic one-form.  Varying the finite theta input makes its values
span a cotangent fiber, so two such constructions give a nonzero wedge.
For their product input, the second Taylor tensor for the orthogonal
sum of the two lines has a double-alternating projection that is a
nonzero multiple of that wedge.  This total two-form allows us to change
the two multiplicity lines while keeping their sum and its finite theta
input fixed.

For the two sign pairs in the normalized switch, we will obtain total
two-forms \(\xi,\xi'\) on one fixed compact ball quotient \(X\).
On a common principal cover \(\pi:X'\to X\), they have finite
expansions into holomorphic one-forms:
\[
 \pi^*\xi=\sum_r a_r\Theta_1(\phi_{1,r})\wedge\Theta_2(\phi_{2,r}),
 \qquad
 \pi^*\xi'=\sum_s b_s\Theta_3(\phi_{3,s})\wedge\Theta_4(\phi_{4,s})
\]
such that
\begin{equation}\label{eq:theta-mixed-period-goal}
 \int_{X'} \pi^*\xi\wedge\overline{\pi^*\xi'}\ne0.
\end{equation}
The comparison of \(\xi'\) with \(\xi\) keeps the rational
symplectic space, the rational special-unitary group, the total finite
theta input, and the compact quotient fixed.  Only afterward will we
factor the rational endpoints and pass to \(X'\), on which the
individual one-forms in both sums descend.  At this stage they give
complex cohomology classes.  Section~\ref{sec:arithmetic} will provide
projective principal levels, and Section~\ref{sec:cm-sources} will
identify the rational CM source maps.

\subsection{The ball and the sign data}\label{subsec:theta-ball}

Let \(E\subset\Qbar\) be a Galois CM field, put \(F=E^+\), and assume
\(s=[F:\Q]\ge2\).  Write \(G_E=\Gal(E/\Q)\) and let \(c\in G_E\)
be complex conjugation.  Fix a set \(\Sigma\subset G_E\) of
representatives modulo \(c\), including the distinguished embedding
\(\mu=1\).  The construction for a single multiplicity line takes an
odd sign function \(v:G_E\to\{1,-1\}\) with \(v(1)=1\).  We index
its signs by
\begin{equation}\label{eq:theta-sign-inversion}
 d(\tau)=v(\tau^{-1}),\qquad
 d(c\tau)=-d(\tau),\qquad d(\mu)=1.
\end{equation}
Thus \(d(g^{-1})=v(g)\), the indexing needed by the arithmetic
calculation.  For the normalized switch, fix four such functions with
\(d_1+d_2=d_3+d_4\).  We first construct the one-form for an arbitrary
one of them, and then compare the pairs \((d_1,d_2)\) and
\((d_3,d_4)\).  All Hermitian forms in this section are linear in
their first argument.

Choose an \(E/F\)-Hermitian space \(V\) of \(E\)-dimension three,
of signature \((2,1)\) at \(\mu\) and positive definite at every
other real place of \(F\).  For example, take a diagonal form with two
totally positive coefficients and a third coefficient negative at
\(\mu\) and positive elsewhere; weak approximation in \(F\) supplies
the last coefficient.  Put
\[
 G=\SU(V),\qquad
 D=\{\,L\subset V_\mu:\dim_\C L=1,\ h_V|_L<0\,\}.
\]
The group \(G\) is defined over \(F\), and \(D\) is the complex
two-ball.  We also write \(L\) for the tautological negative line
bundle over \(D\), and set \(Q=\underline{V_\mu}/L\).  The
Grassmannian tangent calculation gives
\begin{equation}\label{eq:theta-cotangent}
 T_D=\Hom(L,Q),\qquad \Omega_D^1=L\otimes Q^*.
\end{equation}

Fix an \(\mathcal O_E\)-lattice \(\Lambda\subset V\); all principal congruence
levels below are matrix levels for this lattice.  Their quotients of
\(D\) are compact.  To see the relevant anisotropy, an isotropic
vector defined over \(E\) would still be isotropic at a definite real
completion, which is impossible.  Equivalently, \(G\) has no
nontrivial \(F\)-split torus: a nonzero weight space of such a torus
in \(V\) would be isotropic, since invariance of \(h_V\) forces
\(h_V(x,x)=0\) for a vector of nontrivial split weight.  Thus \(G\)
is \(F\)-anisotropic.  The arithmetic lattice and compactness
theorems therefore make its arithmetic subgroups cocompact
lattices in \(G(F\otimes_\Q\R)\)
\cite{BorelHarishChandra1962}; see also the formulation in
\cite[Theorem~3.3 and Proposition~3.5]{MilneShimura2017}.  Here
\[
 G(F\otimes_\Q\R)\simeq \SU(2,1)\times\prod_{\tau\ne\mu}\SU(3).
\]
Projection past the compact factors preserves discreteness and
cocompactness.  Indeed, a sequence in an arithmetic subgroup whose
active projections approach the identity has, after taking a
subsequence, convergent compact components; differences would then
approach the identity in the full discrete subgroup.  A sufficiently
deep principal matrix level is torsion-free.  Its action on \(D\) is
free, because a discrete subgroup has finite intersection with a
point stabilizer.  Hence its quotient is a smooth connected compact
K\"ahler surface.  Later we will impose a principal level for which
the reference moduli construction also proves projectivity.

\subsection{Positive Lagrangians and differential bundles}\label{subsec:theta-lagrangians}

Fix \(\delta\in E^\times\) with \(c(\delta)=-\delta\).
For a nondegenerate \(E/F\)-Hermitian space \(P\) of \(E\)-dimension
\(n\in\{1,2\}\), let
\[
 \mathbb V_P=\Res_{E/F}(V\otimes_E P).
\]
It is an \(F\)-space of dimension \(6n\).  The product of the two
Hermitian forms is a well-defined Hermitian form on \(V\otimes_E P\),
and we use the \(F\)-valued form
\begin{equation}\label{eq:theta-symplectic-form}
 \psi_P(v\otimes p,v'\otimes p')
  =\Tr_{E/F}\!\bigl(\delta\,h_V(v,v')h_P(p,p')\bigr).
\end{equation}
Interchanging the arguments conjugates the Hermitian product and
changes the sign of \(\delta\), so \(\psi_P\) is alternating.  It
is nondegenerate: the tensor-product Hermitian form is nondegenerate,
and the trace pairing on \(E/F\) is nondegenerate.  More explicitly,
if the Hermitian pairing of \(x\ne0\) with some \(y\) is \(b\ne0\),
then its pairing with \(e y\), for \(e\in E\), is \(c(e)b\);
trace duality supplies \(e\) for which
\(\Tr_{E/F}(\delta c(e)b)\ne0\).  Thus \((\mathbb V_P,\psi_P)\) is
a symplectic space over \(F\).  Its underlying rational vector space
carries the alternating form \(\Tr_{F/\Q}\psi_P\); this is the
restriction of scalars used in complexifications over \(\Q\) below.
Symplectic coordinates and transformations will still be taken over
\(F\).

First take \(P=W_i\) to be a Hermitian line associated with a sign
function \(d_i\) as in \eqref{eq:theta-sign-inversion}.  Complexifying
the trace pairing identifies the \(c\tau\) summand complex-linearly
with the dual of the \(\tau\) summand.  Define a subbundle
\(\mathcal F_i\subset\mathbb V_P\otimes_\Q\C\) over \(D\) at the
active pair of embeddings by
\begin{equation}\label{eq:theta-active-lagrangian}
 (\mathcal F_i)_\mu=L\otimes(W_i)_\mu,\qquad
 (\mathcal F_i)_{c\mu}=Q^*\otimes(W_i)_\mu^*.
\end{equation}
Here \(Q^*=\operatorname{ann}(L)\subset V_\mu^*\), embedded in
the conjugate summand by the Hermitian pairing.  It is the
holomorphic annihilator bundle, not an antiholomorphically varying
orthogonal complement in \(V_\mu\).  At
\(\tau\in\Sigma\setminus\{\mu\}\), take the full \(c\tau\) summand
when \(d_i(\tau)=1\), and the full \(\tau\) summand when
\(d_i(\tau)=-1\); take zero on the other summand.  For a rational
orthogonal sum \(P=W_1\perp W_2\), take the direct sum of these
prescriptions and write \(\mathcal F_P=\mathcal F_1\oplus\mathcal F_2\).

\begin{lemma}\label{lem:theta-positive-lagrangian}
The Hermitian forms on the lines \(W_i\) can be chosen over \(E/F\)
so that \(\mathcal F_P\) is a holomorphic positive Lagrangian for
\(\psi_P\), equivariantly for \(G(F\otimes_\Q\R)\).  Its determinant
and an equivariant square root have the form
\begin{equation}\label{eq:theta-halfdet}
 \det\mathcal F_P\simeq L^{2n}\otimes C_P,\qquad
 (\det\mathcal F_P)^{1/2}\simeq L^n\otimes C_P^{1/2},
\end{equation}
where \(C_P\) is a constant complex line on which
\(G(F\otimes_\Q\R)\) acts trivially.
\end{lemma}

\begin{proof}
Write \(\tau(\delta)=it_\tau\), where \(t_\tau\in\R^\times\).
At the real place represented by \(\tau\in\Sigma\), write
\(\psi_{P,\tau}\) for the complex bilinear extension of the
real-place form.  Our positivity convention is
\[
 \psi_{P,\tau}(u,\overline u)/i>0
 \qquad\bigl(0\ne u\in
       (\mathcal F_P)_\tau\oplus(\mathcal F_P)_{c\tau}\bigr).
\]
On the \(\tau\) summand its sign is the sign of
\(t_\tau h_Vh_P\); on the \(c\tau\) summand the sign is the
opposite.  At \(\mu\), choose the coefficient of each line form to
have sign \(-\operatorname{sgn}(t_\mu)\).  The negative line \(L\)
on the \(\mu\) side and the dual of the positive plane on the
\(c\mu\) side then both have positive sign.  At a definite place
\(\tau\ne\mu\), choose the line coefficient to have sign
\[
 -d_i(\tau)\operatorname{sgn}(t_\tau).
\]
The chosen full summand has positive sign in either case.
These finitely many strict sign conditions on an element of \(F\)
are simultaneously realizable by weak approximation.

At the active pair the two subspaces are annihilators under the
complexified pairing; at every other pair one side is full and the
other zero.  Thus \(\mathcal F_P\) is isotropic of half the total
dimension.  The active annihilator description proves holomorphy,
and the compact-place summands are constant over \(D\).
Equivariance follows from the action on \(L\) and its annihilator
at \(\mu\), and from preservation of the full embedding summands
at the compact places.

Finally
\[
 \det Q^*=L\otimes\det(V_\mu)^*.
\]
For general \(P\) the active determinant is therefore
\[
 \det(L\otimes P_\mu)\otimes\det(Q^*\otimes P_\mu^*)
 \simeq
 L^{2n}\otimes\bigl(\det(V_\mu)^*\bigr)^{\otimes n}
       \otimes\det(P_\mu)^*.
\]
All factors other than \(L^{2n}\), including the determinants at
the compact places, are constant lines.  Their special-unitary
actions are trivial, because they involve powers of determinants
of \(V_\tau\) or \(V_\tau^*\), while \(G\) acts trivially on
\(P\).  This proves the first isomorphism in
\eqref{eq:theta-halfdet}; choosing a square root of the constant
line gives the second.
\end{proof}

We choose nonzero complex vectors in these constant lines when
identifying tensors with differential forms.  Such choices only
rescale those forms by nonzero constants.  No rational trivialization
is asserted.  Also, the constant lines are trivial for \(G=\SU(V)\);
they need not be trivial under the larger unitary group.

The determinant calculation gives the differential targets for the
theta tensors.  For a multiplicity line \(W_i\), the relevant target is
\begin{equation}\label{eq:theta-one-target}
\begin{aligned}
 (\det\mathcal F_i)^{1/2}\otimes(\mathcal F_i)_{c\mu}
 &\simeq L\otimes Q^*\otimes(\text{constant line})\\
 &\simeq \Omega_D^1\otimes(\text{constant line}).
\end{aligned}
\end{equation}
For \(W=W_1\perp W_2\), define the double-alternating map
\[
 \operatorname{alt}_2:
 \Sym^2(Q^*\otimes W_\mu^*)\longrightarrow
       \Lambda^2Q^*\otimes\Lambda^2W_\mu^*
\]
by
\[
 \operatorname{alt}_2\bigl((a\otimes u)(b\otimes v)\bigr)
       =(a\wedge b)\otimes(u\wedge v).
\]
Here the product on the left is multiplication in the symmetric
algebra.  Exchanging its two factors changes both exterior signs,
so the map is well-defined.  Including the half-determinant line gives
\begin{equation}\label{eq:theta-top-target}
 (\det\mathcal F_W)^{1/2}\otimes
       \Lambda^2Q^*\otimes\Lambda^2W_\mu^*
       \simeq K_D\otimes(\text{constant line}),
\end{equation}
because \(K_D=\det\Omega_D^1=L^2\otimes\Lambda^2Q^*\).
The first theta tensor will give the one-form in
\eqref{eq:theta-one-target}; the double-alternating part of the second
will give the two-form in \eqref{eq:theta-top-target}.

The reason to combine the two lines is geometric.  At the active pair,
their total positive Lagrangian has the pieces
\[
 L\otimes W_\mu,\qquad Q^*\otimes W_\mu^*.
\]
These depend only on \(W_\mu\), independently of its decomposition
into lines.  At a compact place, let \(W_\tau^+\) and \(W_\tau^-\)
be the sums of the multiplicity lines with respective signs \(+1\)
and \(-1\) under \(d_i\).  The total positive pieces there are
\[
 V_\tau\otimes W_\tau^-,\qquad
 V_\tau^*\otimes(W_\tau^+)^*;
\]
the second dual is embedded as
\(\operatorname{ann}(W_\tau^-)\subset W_\tau^*\).
The relation \(d_1+d_2=d_3+d_4\) says that the two sign pairs have
the same multiset at each place.  Thus the original local lines can
be relabeled with signs \(d_3,d_4\) without changing these total
pieces.  Weak approximation will produce a rational decomposition
with the new signs and nearby compact-place data, while leaving the
active total Lagrangian unchanged.  We will apply this approximation
in Subsection~\ref{subsec:theta-approximation} to one fixed theta
input.  To compare the resulting forms on a compact quotient, we
first establish their covariance and congruence descent.

\subsection{Rational theta covariance}\label{subsec:theta-covariance}

Choose symplectic coordinates over \(F\) on \(\mathbb V_P\),
written \((p,q)\in F^a\oplus F^a\), where \(a=3n\), with
\[
 \psi_P((p,q),(p',q'))=p^tq'-q^tp'.
\]
The Schr\"odinger position coordinates are \(q\).  A positive
Lagrangian at a real place has a unique graph frame
\[
 e_Z=\binom ZI,\qquad Z=Z^t,\qquad \operatorname{Im}Z>0.
\]
Indeed, it cannot meet the \(p\)-axis nontrivially, because
\(\psi_P(u,\bar u)\) vanishes there.  Projection to the
\(q\)-axis is therefore an isomorphism.  Isotropy says \(Z^t=Z\),
and, for a graph vector,
\[
 \psi_{P,\tau}(e_Zu,\overline{e_Zu})/i
       =2u^t(\operatorname{Im}Z)\bar u,
\]
which proves the last condition.  The positive Lagrangians above
thus give a period map \(D\to\prod_{\tau\in\Sigma}\mathfrak H_a\).

Let \(\phi\in\mathcal S(\A_{F,f}^a)\) be a finite-adelic
Schwartz--Bruhat function.  For
\(Z=(Z_\tau)_\tau\in\prod_\tau\mathfrak H_a\) and
\(z=(z_\tau)_\tau\in\bigoplus_\tau\C^a\), put
\begin{equation}\label{eq:theta-series}
 \vartheta_\phi(Z,z)
  =\sum_{x\in F^a}\phi(x)
    \exp\!\left(
       \pi i\sum_{\tau\in\Sigma}x_\tau^tZ_\tau x_\tau+
       2\pi i\sum_{\tau\in\Sigma}x_\tau^tz_\tau\right).
\end{equation}
The rational transformation law for this series includes an exponential
quadratic in \(z\), which mixes its ordinary Taylor coefficients.
The following completion cancels that factor and gives coefficients
with a tensor transformation law.  Define
\begin{equation}\label{eq:theta-completion}
 \begin{split}
 \widehat\vartheta_\phi(Z,z)
  &=\exp\!\left(\frac{\pi}{2}\sum_{\tau\in\Sigma}
       z_\tau^t(\operatorname{Im}Z_\tau)^{-1}z_\tau\right)
       \vartheta_\phi(Z,z),\\
 \widehat\vartheta_\phi(Z,z)
  &=\sum_{k\ge0}\langle T_{k,\phi}(Z),z^{\otimes k}\rangle .
 \end{split}
\end{equation}
Thus \(T_{k,\phi}\) includes the factor \(1/k!\) in the \(k\)-th
derivative.  We use multiplication in the symmetric algebra for
these homogeneous coefficients.

The series and every fixed order of its derivatives converge
uniformly on compact subsets of the Siegel and \(z\) domains.
Indeed, the rational points in the support of \(\phi\) lie in
finitely many translates of a full lattice in
\((F\otimes_\Q\R)^a\).  On a compact set the imaginary parts have
a common positive lower bound, and the summands, after a fixed
number of differentiations, are bounded by a polynomial in
\(\|x\|\) times
\[
 C\exp(-c\|x\|^2+C'\|x\|).
\]
The lattice sum of this bound converges.  The completion and its
derivatives obey the same bounds on compact sets.  The same
conclusion holds after any fixed number of derivatives in smooth
parameters on a compact subset of a smooth parameter domain:
there the period map and all derivatives of it needed for that
order are bounded, and its imaginary parts have a common positive
lower bound.  Compactness of the image of a period map alone,
without bounds on its parameter derivatives, would not imply this
last assertion.

We next specify the rational Weil-representation input, including
the cover being used.  The circle extension is obtained from the
metaplectic double cover by extending its central \(\{1,-1\}\)
to the group of unit complex scalars.  Take the global additive character of
\(\A_F/F\) whose infinite component is
\(t\mapsto\exp(2\pi i\Tr_{F/\Q}t)\).  Weil's rational section in
the adelic circle extension leaves the summation functional
\[
 \lambda(\Phi)=\sum_{x\in F^a}\Phi(x)
\]
invariant.  In characteristic zero his pseudosymplectic group is
the symplectic group, and his adelic twofold subgroup contains this
rational section
\cite[Ch.~III, nos.~31 and 41, Eq.~(39) and Theorem~6;
      Ch.~IV, no.~45, pp.~200--201]{Weil1964}.
Consequently, for a rational symplectic transformation we may
choose an infinite double-cover lift and its paired finite lift
whose adelic product is the rational lift.  A change of the
infinite sign requires the same change of the finite sign.
These paired lifts give rational covariance; they are not an
assertion that the symplectic double cover splits on an arbitrary
adelic unitary group.

\begin{lemma}\label{lem:theta-tensor-law}
Let \(g\in\Sp(\mathbb V_P)(F)\).  At each real place write
\[
 g=\begin{pmatrix}A&B\\C&D_0\end{pmatrix},\qquad
 J=CZ+D_0,\qquad gZ=(AZ+B)J^{-1}.
\]
Choose an infinite metaplectic lift with factor \(j(g,Z)\) satisfying
\(j(g,Z)^2=\prod_{\tau\in\Sigma}\det J_\tau\), and let
\(\widetilde g_f\) be its paired finite lift.  Then
\begin{equation}\label{eq:theta-jacobi-law}
 \vartheta_{\omega_f(\widetilde g_f)\phi}(gZ,J^{-t}z)
  =j(g,Z)\exp\!\left(\pi i\sum_{\tau\in\Sigma}
             z_\tau^tJ_\tau^{-1}C_\tau z_\tau\right)
             \vartheta_\phi(Z,z).
\end{equation}
The completed coefficients satisfy
\begin{equation}\label{eq:theta-taylor-law}
 T_{k,\omega_f(\widetilde g_f)\phi}(gZ)
   =j(g,Z)\Sym^k(J)\,T_{k,\phi}(Z).
\end{equation}
They are thus tensors valued in
\[
 (\det\mathcal F_P)^{1/2}\otimes\Sym^k\mathcal F_P.
\]
In particular the positive Lagrangian, not its dual, occurs in
the symmetric factor.  The tensor \(T_{1,\phi}\) is holomorphic.
\end{lemma}

\begin{proof}
We first check the infinite formula, keeping track of its signs.
At one real place write
\[
 f_{Z,z}(x)=\exp(\pi ix^tZx+2\pi ix^tz),\qquad K(g,Z)=J^{-1}C.
\]
The matrix \(K(g,Z)\) is symmetric; this follows from
\(JC^t=CJ^t\), a symplectic identity.  The formula to prove is
\begin{equation}\label{eq:theta-gaussian-law}
 \omega_\infty(\widetilde g)f_{Z,z}
  =j(g,Z)^{-1}\exp(-\pi iz^tK(g,Z)z)
       f_{gZ,J^{-t}z}.
\end{equation}
For symmetric \(B\), the upper unipotent
\(n(B)=\left(\begin{smallmatrix}I&B\\0&I\end{smallmatrix}\right)\)
acts by multiplication by \(\exp(\pi ix^tBx)\), which gives
\eqref{eq:theta-gaussian-law} with \(j=1\).  For
\(m(A)=\diag(A,A^{-t})\), the chosen lift acts by
\[
 \omega_\infty(\widetilde m(A))b(x)
       =j(m(A))^{-1}b(A^tx),\qquad
 j(m(A))^2=(\det A)^{-1}.
\]
For \(\det A>0\) the positive choice has
\(j(m(A))^{-1}=|\det A|^{1/2}\).  For negative determinant
the metaplectic phase is retained in \(j\).  Since
\(m(A)Z=AZA^t\) and \(J^{-t}z=Az\), this is the required formula.

For
\(w=\left(\begin{smallmatrix}0&-I\\I&0\end{smallmatrix}\right)\),
choose the lift
\[
 b\longmapsto e^{-\pi ia/4}
        \int_{\R^a}b(y)e^{-2\pi ix^ty}\,dy.
\]
The Gaussian integral is
\[
 \int_{\R^a}f_{Z,z}(y)e^{-2\pi ix^ty}\,dy
  =\det(-iZ)^{-1/2}
       \exp\!\bigl(-\pi i(x-z)^tZ^{-1}(x-z)\bigr),
\]
where the square root of \(\det(-iZ)\) equals \(1\) at \(Z=iI\).
Taking
\[
 j(w,Z)=e^{\pi ia/4}\det(-iZ)^{1/2}
\]
gives \(j(w,Z)^2=\det Z\) and proves
\eqref{eq:theta-gaussian-law} for \(w\).

These generators give the formula for the whole real symplectic
group.  To check that the displayed exponent also composes
correctly, let \(J_2=J(g_2,Z)\) and
\(J_1=J(g_1,g_2Z)\).  The symplectic identities give
\[
 J(g_1g_2,Z)=J_1J_2,\qquad
 K(g_1g_2,Z)=K(g_2,Z)+J_2^{-1}K(g_1,g_2Z)J_2^{-t}.
\]
For the second identity, use
\(A_2-(g_2Z)C_2=J_2^{-t}\) to write
\(C_{12}=C_1J_2^{-t}+J_1C_2\), and multiply by
\((J_1J_2)^{-1}\).  Together with the metaplectic group law for
\(j\), this is exactly the composition law in
\eqref{eq:theta-gaussian-law}.  Multiplying the formula over the
real places gives its infinite adelic version.

Apply rational invariance of \(\lambda\) to
\(\phi\otimes\prod_\tau f_{Z_\tau,z_\tau}\).  By
\eqref{eq:theta-gaussian-law}, the transformed sum is
\[
 j(g,Z)^{-1}
 \exp\!\left(-\pi i\sum_\tau z_\tau^tJ_\tau^{-1}C_\tau z_\tau\right)
 \vartheta_{\omega_f(\widetilde g_f)\phi}(gZ,J^{-t}z).
\]
Equating this with the original sum gives
\eqref{eq:theta-jacobi-law}.  In particular the finite action on
the left is \(\omega_f(\widetilde g_f)\), with no inverse.
If one instead keeps the finite input on the left fixed, the input
on the right is acted on by \(\omega_f(\widetilde g_f)^{-1}\).

For the completion, put \(Y=\operatorname{Im}Z\) and
\(Y'=\operatorname{Im}(gZ)\) at one place.  From
\(Y'=\bar J^{-t}YJ^{-1}\), and then
\(\bar J^t=J^t-2iYC^t\), we obtain
\begin{equation}\label{eq:theta-completion-identity}
 J^{-1}(Y')^{-1}J^{-t}=Y^{-1}-2iJ^{-1}C.
\end{equation}
Here \(C^tJ^{-t}=J^{-1}C\) by symmetry.  Thus the completion
factor at \((gZ,J^{-t}z)\) differs from that at \((Z,z)\) by
\(\exp(-\pi iz^tJ^{-1}Cz)\).  This cancels the exponential in
\eqref{eq:theta-jacobi-law}, so
\[
 \widehat\vartheta_{\omega_f(\widetilde g_f)\phi}(gZ,J^{-t}z)
       =j(g,Z)\widehat\vartheta_\phi(Z,z).
\]
Comparison of homogeneous terms yields
\eqref{eq:theta-taylor-law}.  Finally,
\[
 g e_Z=e_{gZ}J.
\]
Hence \(J\) is the transformation matrix for vectors in the
positive Lagrangian; this proves the asserted tensor target and
its orientation.  The completion has no linear term in \(z\), so
\(T_{1,\phi}\) is the ordinary holomorphic theta gradient.
For comparison, the first-order law agrees with the classical
theta-gradient formula in \cite[\S9.1]{MollerZagier2016}.
\end{proof}

For a multiplicity line \(W_i\), the \(c\mu\)-projection of
\(T_{1,\phi}\) therefore lies in the cotangent target
\eqref{eq:theta-one-target}.  Rational covariance alone does not
supply a congruence subgroup fixing a given finite input.
Subsection~\ref{subsec:theta-splittings} obtains such stabilizers
from an honest smooth finite-adelic action.

\begin{remark}\label{rem:theta-adapted-coordinates}
We will use the lemma to change rational Schr\"odinger coordinates.
Suppose a rational symplectic matrix \(S\) takes adapted coordinate
vectors to the reference coordinate vectors.  Then
\(Z_{\mathrm{ref}}=S Z_{\mathrm{ad}}\), and the input in the
adapted coordinates is
\[
 \phi_{\mathrm{ad}}
    =\omega_f(\widetilde S_f)^{-1}\phi_{\mathrm{ref}}.
\]
Equation~\eqref{eq:theta-taylor-law} compares the resulting tensors.
Changing the sign of the paired lift changes both the infinite
factor and the finite input sign, so their comparison is unchanged.
For coordinates adapted to a rational symplectic direct sum, the
period matrix is block diagonal and the completed theta series of
a product input is the product of the two completed series.
\end{remark}

\subsection{Unitary splittings and congruence descent}\label{subsec:theta-splittings}

The preceding lemma uses paired lifts of rational symplectic
elements.  An honest action on finite inputs, and therefore a
congruence stabilizer for each input, requires a different statement.
We use the unitary splitting theorem in the following precise form.
For a quadratic extension \(E/F\), a Hermitian space \(H\) of
dimension \(m\), a skew-Hermitian space \(K\) of dimension \(n\),
and a global additive character, choose unitary Hecke characters
\(\chi_H,\chi_K\) of \(E\) with
\[
 \chi_H|_{\A_F^\times/F^\times}=\epsilon_{E/F}^{\,m},\qquad
 \chi_K|_{\A_F^\times/F^\times}=\epsilon_{E/F}^{\,n}.
\]
The corresponding unitary actions lift to the \emph{circle-scalar
extension} of the adelic symplectic Weil representation.  The
global lift is assembled from local lifts and agrees with the
canonical rational symplectic section on rational unitary points.
Its finite part acts smoothly on finite Schwartz functions.
Changing the splitting characters twists the unitary actions by
characters composed with their determinants.  Here
\(\epsilon_{E/F}\) is the quadratic idele class character.

The local splitting is due to Kudla \cite{Kudla1994}.  We use the
circle-extension formulation and its local and adelic compatibility
in \cite[\S\S3.1--3.1.1 and 4.1--4.1.1]{BoradeEtAl2025ToriPeriods}.
In the present CM setting, adelic automorphy and the determinant-twist
rule are also recorded in \cite[\S7.2 and Remark~7.3]{BMM2016}.
Throughout this invocation the extension is the circle pushout:
we do not assert that the double cover splits on every unitary
group.  We will compare this circle lift with an actual double
lift only after restriction to \(G=\SU(V)\).
The split-place normalization below will also enter the local Hecke
calculation of Subsection~\ref{subsec:active-parameter}.

\begin{lemma}\label{lem:unitary-splitting}
For the symplectic space \((\mathbb V_P,\psi_P)\), restriction of
the unitary splitting to \(G(\A_F)\) yields an honest finite-adelic
Weil action \(\omega_{P,f}\).  It is compatible with the rational
covariance in Lemma~\ref{lem:theta-tensor-law} and is independent
of the splitting characters.  At infinity its half-determinant
action is the natural action on \(L^n\) in
\eqref{eq:theta-halfdet}.

At a split finite place, choose as Schr\"odinger positions for a
multiplicity line one standard three-dimensional embedding
summand.  The resulting \(\SL_3\)-action is inverse linear
substitution.  More generally, the normalized linear action
associated with a transformation \(h\) of the position space is
\begin{equation}\label{eq:theta-linear-action}
 (\rho(h)b)(x)=|\det h|^{-1/2}b(h^{-1}x).
\end{equation}
A bare metaplectic lift outside \(\SL_3\) is the normalized
operator \(\rho(h)\) times its lift scalar; that scalar must be
retained in any use of such a lift.

Every finite list of inputs is fixed under \(\omega_{P,f}\) by
all sufficiently deep principal lattice congruence subgroups.
\end{lemma}

\begin{proof}
First we check that the required splitting characters exist.
For the idele class groups \(C_F\) and \(C_E\), the natural map
\(C_F\to C_E\) is injective: if an \(F\)-idele becomes a principal
\(E\)-idele, its generator is fixed by \(c\) and hence belongs to
\(F\).  This map is also proper.  The absolute idele norm on
\(C_E\) restricts to the square of the norm on \(C_F\), whose
norm-one kernel is compact.  A compact subset of \(C_E\) has
norms in a compact subinterval of \(\R_{>0}\); its preimage is
therefore a closed subset of a compact norm interval times the
compact norm-one kernel of \(C_F\).  Thus \(C_F\) is a closed
locally compact subgroup of \(C_E\).  The character-extension
theorem for a closed subgroup of a locally compact abelian group
extends each unitary character \(\epsilon_{E/F}^{\,r}\) from
\(C_F\) to \(C_E\)
\cite[\S4.3, Corollary~4.42]{Folland2015}.  Apply this with
\(r=m,n\).

The splitting theorem applies to our form by replacing \(h_P\)
with the skew-Hermitian form \(\delta h_P\).  With both forms linear
in the first variable, the trace product in
\cite[\S\S2.1 and 3.1]{BoradeEtAl2025ToriPeriods} is exactly
\eqref{eq:theta-symplectic-form}.  We take the same global additive
character as in Lemma~\ref{lem:theta-tensor-law}.  The determinant twists from
changing the characters vanish on \(G=\SU(V)\).  The product of
the finite local splittings is therefore an honest action
\(\omega_{P,f}\) on \(G(\A_{F,f})\), independent of those
characters.

We compare its infinite circle lift with the double lift needed
for the differential bundles.  At the active maximal compact
subgroup \(S(\U(2)\times\U(1))\), let \(\ell\) be the character
on the negative line.  The determinant of the positive plane is
\(\ell^{-1}\), so the determinant character on
\(\mathcal F_P\) is \(\ell^{2n}\).  At every other real place
the determinant character is trivial on \(\SU(3)\).  On the
maximal compact \(\U(a)\) of each real symplectic factor, the
metaplectic double cover is the determinant-square-root cover.
For the product of the real factors its square is the product of
those determinant characters.  Its pullback to the maximal
compact subgroup of \(G(F\otimes_\Q\R)\) therefore splits with
square-root character \(\ell^n\).

A connected real reductive group retracts onto its maximal compact
subgroup.  The preceding determinant computation says that every
fundamental-group loop of \(G(F\otimes_\Q\R)\) has even
determinant winding in the symplectic group.  The lift from the
universal covering group consequently descends to a group lift
of \(G(F\otimes_\Q\R)\) into the double cover.  Its
half-determinant action is the one on \(L^n\) in
\eqref{eq:theta-halfdet}.  The original circle lift and this
double lift, pushed out to the circle extension, differ by a
continuous character of \(G(F\otimes_\Q\R)\).  This group is
connected and semisimple.  The differential of such a character
is zero, and connectedness makes the character trivial.
The two lifts therefore agree.  The rational compatibility of
the global splitting now pairs \(\omega_{P,f}(\gamma)\) with
this infinite lift for every \(\gamma\in G(F)\), proving the
claimed rational covariance.

At a split finite place the two embedding summands are dual
stable Lagrangians.  If \(h\) is the action on the position
summand, the symplectic matrix is \(\diag(h^{-t},h)\).
The Schr\"odinger linear formula gives
\eqref{eq:theta-linear-action}, up to the scalar of the chosen
bare lift.  On \(\SL_3(F_v)\), the normalized substitution lift
and the unitary lift have a character as their ratio.  This
character is trivial because \(\SL_3(F_v)\) is perfect:
elementary matrices generate it, and for distinct \(i,j,k\),
\[
 e_{ij}(u)=[e_{ik}(u),e_{kj}(1)].
\]
The determinant factor is also \(1\) there.  This proves the
split-place assertion without removing scalars from lifts outside
\(\SL_3\).

Finally the finite Weil action is smooth, so finitely many
Schwartz functions have a common open stabilizer.  Intersect it
with the fixed lattice stabilizer in \(G(\A_{F,f})\).
Principal lattice congruence neighborhoods form a basis at the
identity in that stabilizer, so a sufficiently deep principal
one lies in the intersection.  This proves the last assertion.
\end{proof}

Restrict the tensors of Lemma~\ref{lem:theta-tensor-law} to the
period map from \(D\).  If a finite list of inputs is fixed by an
open compact \(K_f\subset G(\A_{F,f})\), the lemma just proved
and \eqref{eq:theta-halfdet} give their bundle covariance for
\(\Gamma=G(F)\cap K_f\).  Passing to a torsion-free principal
subgroup makes all of them descend to one compact quotient.
This uses stabilizers in the smooth finite-adelic representation;
it makes no claim that an arbitrary finite-index subgroup of a
ball lattice is congruence.

\subsection{Theta one-forms span cotangent fibers}\label{subsec:theta-one-forms}

For a line \(W_i\), project \(T_{1,\phi}\) to the \(c\mu\)
summand of the positive Lagrangian.  Using our fixed complex
trivialization of the constant line in \eqref{eq:theta-one-target},
denote the resulting one-form on
\(D\) by \(\Theta_i(\phi)\).

\begin{proposition}\label{prop:theta-one}
For every finite input \(\phi\), the form \(\Theta_i(\phi)\) is
holomorphic on \(D\) and descends to every sufficiently deep
principal congruence quotient.  At any fixed point of \(D\), the
values \(\Theta_i(\phi)\), as \(\phi\) varies over finite
Schwartz inputs, span the cotangent fiber.  Finitely many actual
inputs already span that fiber and descend at a common sufficiently
deep principal level.
\end{proposition}

\begin{proof}
Holomorphy follows from that of \(T_{1,\phi}\), the holomorphic
embedding projection, and \eqref{eq:theta-cotangent}.  Descent
follows from Lemma~\ref{lem:unitary-splitting}.

Fix a point \(x_*\in D\) and its period point \(Z\).  Here \(a=3\).  For
\(0\ne x_0\in\mathcal O_F^a\) and positive integers \(k\), take
\[
 \phi_k=\mathbf1_{\,x_0+k\widehat{\mathcal O}_F^{\,a}}.
\]
Its rational support is exactly \(x_0+k\mathcal O_F^a\).
The first Taylor tensor is the ordinary gradient, so in the
chosen graph frame
\[
 T_{1,\phi_k}(Z)
  =2\pi i\sum_{x\in x_0+k\mathcal O_F^a}
      (x_\tau)_{\tau\in\Sigma}
       \exp\!\left(\pi i\sum_\tau x_\tau^tZ_\tau x_\tau\right).
\]
Under the Minkowski embedding \(\mathcal O_F^a\) is a full
lattice.  If \(\rho>0\) is the minimum norm of a nonzero lattice
vector, then, for \(m\ne0\) in this lattice and
\(k\ge2\|x_0\|/\rho\),
\[
 \|x_0+km\|\ge k\|m\|/2.
\]
The Gaussian bound used above, with any polynomial factor
absorbed into a weaker Gaussian, bounds the tail by a constant
times a convergent lattice sum of the form
\(\sum_{m\ne0}\exp(-c k^2\|m\|^2)\).  That sum tends to zero.
The same argument is uniform for \(Z\) in a compact
neighborhood.  We have proved
\begin{equation}\label{eq:theta-sparse-limit}
 T_{1,\phi_k}(Z)\longrightarrow
  2\pi i\,(x_{0,\tau})_{\tau\in\Sigma}
       \exp\!\left(\pi i\sum_\tau
                       x_{0,\tau}^tZ_\tau x_{0,\tau}\right).
\end{equation}

The scalar exponential is nonzero.  As \(x_0\) varies, the
vectors \((x_{0,\tau})_\tau\) span the real position space,
and their complex span is the entire first-tensor fiber in this
frame.  The complex span of the actual values \(T_{1,\phi}(Z)\)
is a subspace of a finite-dimensional space and hence is closed.
It contains all the limits in \eqref{eq:theta-sparse-limit},
so it is the full fiber.  Projection onto
\((\det\mathcal F_i)^{1/2}\otimes(\mathcal F_i)_{c\mu}\)
is surjective; therefore the projected values span
\(\Omega_D^1|_{x_*}\).  Choose finitely many actual inputs whose
values contain a basis, and then apply the stabilizer assertion
of Lemma~\ref{lem:unitary-splitting}.

The inputs \(\phi_k\) in this limit need not have a common
congruence stabilizer as \(k\) varies.  The limit takes place
in one fiber over \(D\); a common level is chosen only after
finitely many actual inputs have been selected.
\end{proof}

On a compact K\"ahler quotient a holomorphic one-form is closed
and harmonic.  A nonzero such form therefore has nonzero
cohomology class.  These are complex classes.  Nothing in this
proposition identifies a rational CM source for them.

\subsection{Rank-two tensors, wedges, and fixed-data variation}\label{subsec:theta-rank-two}

Take \(W=W_1\perp W_2\), so \(n=2\).  Project \(T_{2,\phi}\)
to \(\Sym^2((\mathcal F_W)_{c\mu})
=\Sym^2(Q^*\otimes W_\mu^*)\), and then apply
\(\operatorname{alt}_2\).  Its target is the canonical
bundle in \eqref{eq:theta-top-target}, up to a constant line.
After trivializing that line, denote the resulting differential
two-form by \(\Xi_W(\phi)\).  At a rational decomposition the product
formula for theta series will express it as a finite sum of wedges
of the one-forms just constructed.  We then vary the compact-place
data of the total space to compare two such expressions.

We now specify exactly what may vary in this construction.
Keep fixed the rational Hermitian space \(W\), the symplectic
space \((\mathbb V_W,\psi_W)\), the rational embedding
\(G\to\Sp(\mathbb V_W)\), reference rational Schr\"odinger
coordinates, the global Weil and unitary splitting choices, and
the resulting finite action \(\omega_{W,f}\).  Also fix a single
total finite input \(\phi\) in these reference coordinates.
At a compact place \(\tau\ne\mu\), let
\[
 W_\tau=W_\tau^+\perp W_\tau^-
\]
range over orthogonal decompositions with
\[
 \dim W_\tau^\pm=\#\{i\in\{1,2\}:d_i(\tau)=\pm1\}.
\]
Require the form on \(W_\tau^+\) to have sign
\(-\operatorname{sgn}(t_\tau)\), and the form on
\(W_\tau^-\) the opposite sign.  Let \(\mathcal T\) be the
product of these parameter spaces.  A definite local factor is
a point; when the two signs differ, the local factor is the
domain of lines of one prescribed sign in a Hermitian plane of
signature \((1,1)\).

For \(t\in\mathcal T\), the two compact-place pieces of the
positive Lagrangian are
\[
 V_\tau\otimes W_\tau^-,
 \qquad
 V_\tau^*\otimes(W_\tau^+)^*.
\]
In the second expression \((W_\tau^+)^*\) is embedded as
\(\operatorname{ann}(W_\tau^-)\subset W_\tau^*\); the
orthogonal decomposition identifies it with the dual of
\(W_\tau^+\).  The active pieces remain, exactly,
\[
 L\otimes W_\mu,\qquad Q^*\otimes W_\mu^*.
\]
They depend only on \(W_\mu\), not on any decomposition of it.
These prescriptions give positive Lagrangians for all
\(t\in\mathcal T\).  The same projection as in
\eqref{eq:theta-top-target}, with a local smooth choice of
constant-line trivialization, defines a smooth form of type
\((2,0)\), denoted \(\Xi_W(\phi;t)\), even when \(t\) is not
induced by rational lines.  We do not claim it is holomorphic
for such an intermediate parameter.

\begin{proposition}\label{prop:theta-top}
The rank-two construction has the following properties.
\begin{enumerate}[label=(\roman*)]
\item At period data induced by a rational orthogonal
  decomposition \(W=W_1\perp W_2\), the form \(\Xi_W(\phi)\)
  is holomorphic.  On a sufficiently deep common principal
  quotient it is a finite sum, with constant coefficients, of
  forms \(\Theta_1(\phi_{1,r})\wedge\Theta_2(\phi_{2,r})\).
\item There is an input which is a product in adapted rational
  coordinates and for which \(\Xi_W(\phi)\) is nonzero.
\item Fix \(W\), the total finite input \(\phi\), and a
  torsion-free principal quotient \(X\) whose finite level fixes
  \(\phi\) under \(\omega_{W,f}\).  With locally smooth
  constant-line trivializations, the map
  \[
    t\longmapsto\Xi_W(\phi;t)
  \]
  is continuous in the \(C^\infty\) topology of differential
  forms on this fixed compact quotient.  The same finite level
  works throughout the parameter neighborhood.
\end{enumerate}
\end{proposition}

\begin{proof}
For (i), the rational orthogonal decomposition gives a rational
symplectic direct sum
\(\mathbb V_W=\mathbb V_{W_1}\perp\mathbb V_{W_2}\).
Choose rational coordinates adapted to it.  By
Remark~\ref{rem:theta-adapted-coordinates}, the total reference
input becomes another finite Schwartz input in the adapted
coordinates.  Such an input is a finite sum
\[
 \sum_r\phi_{1,r}\otimes\phi_{2,r}.
\]
Indeed, a locally constant compactly supported function on the
product of two finite-adelic vector spaces is invariant under a
sufficiently small product compact open subgroup.  Only finitely
many of its cosets meet the support, and each coset is a product
coset.

For a product input the completed theta series multiply, since
both the sum and its completion split in the adapted coordinates.
If \(T_j^{(i)}\) denotes the degree-\(j\) coefficient for the
\(i\)-th line, the coefficient of total degree two is
\begin{equation}\label{eq:theta-quadratic-product}
 T_2^{(1)}T_0^{(2)}
       +T_1^{(1)}T_1^{(2)}
       +T_0^{(1)}T_2^{(2)}.
\end{equation}
The coefficient of the mixed product is \(1\), by the
\(1/k!\) normalization in \eqref{eq:theta-completion}.
The two unmixed terms have both multiplicity factors in the
same one-dimensional \((W_i)_\mu^*\); they vanish under
\(\operatorname{alt}_2\).  The mixed term becomes the wedge
of the two projected first tensors.  With compatible
constant-line trivializations it is exactly
\(\Theta_1(\phi_{1,r})\wedge\Theta_2(\phi_{2,r})\);
other trivializations multiply it by a fixed nonzero scalar.
Rational changes of coordinates are already accounted for by
\eqref{eq:theta-taylor-law}.  Moreover, two identifications
with the differential bundle used here have a ratio invariant
under the transitive infinite special-unitary action on \(D\),
so that ratio is constant.  This proves the wedge expansion
and its holomorphy.  We have proved holomorphy only after the
double-alternating projection: the whole completed second
Taylor tensor need not be holomorphic.  The finitely many
individual inputs descend on a further common principal level
by Lemma~\ref{lem:unitary-splitting}.

For (ii), fix a point of \(D\).  Proposition~\ref{prop:theta-one}
provides an input for \(W_1\) with a nonzero cotangent value and
an input for \(W_2\) whose value is outside the line it spans.
Their wedge is nonzero at that point.  The product of these
inputs in adapted coordinates has a nonzero double-alternating
second tensor by the calculation above.  Transporting this
product back to the reference coordinates gives a total finite
input with \(\Xi_W(\phi)\ne0\).

For (iii), no rational coordinate change is made as \(t\)
varies.  The rational symplectic space, its rational \(G\)-action,
the finite representation, and the total input all remain
fixed.  The changed compact-place Lagrangians remain invariant
under \(\SU(3)\), since all changes are in the multiplicity
factor.  Their determinant characters on \(\SU(3)\) are
trivial.  The active pieces have not changed, so their
determinant character remains \(\ell^4\).  The infinite
comparison in Lemma~\ref{lem:unitary-splitting} therefore gives
the same half-determinant action \(L^2\) at every \(t\).
Consequently the same subgroup fixing \(\phi\) gives
equivariance and descent at every parameter.  There is no
finite translation, Hecke choice, or change of level in this
variation.

It remains to check uniformity on the quotient.  The constant
determinant factors in \eqref{eq:theta-halfdet} vary smoothly
with the compact-place subspaces.  On a small parameter
neighborhood choose smooth nonzero frames and their square
roots, agreeing with the chosen frames at the center.
Cover the fixed compact \(X\) by finitely many charts with
precompact lifts to \(D\).  On the closure of each lifted chart
times a sufficiently small compact parameter neighborhood, the
period map is smooth, its image is a compact subset of the
Siegel domain, and every fixed order of its derivatives is
bounded.  The imaginary parts thus have a common positive
lower bound.  The rational support of the fixed \(\phi\)
continues to lie in one fixed finite union of lattice cosets.
The Gaussian estimates following
\eqref{eq:theta-completion} now give uniform convergence of
every fixed order of derivatives in the chart coordinates and
the parameters.  Projection to \eqref{eq:theta-top-target}
and the chosen smooth trivializations preserve these bounds.
The unchanged equivariance glues these convergent local
expressions on \(X\).  This is continuity in the \(C^\infty\)
topology.  Compactness of \(X\) is used here; no estimate at a
cusp or boundary is needed.
\end{proof}

\subsection{Approximation while the total theta input stays fixed}\label{subsec:theta-approximation}

We now apply the last proposition to the sign switch.  The
approximation occurs in \(\mathcal T\), before the rational
coordinate changes used to factor either endpoint.

\begin{corollary}\label{cor:theta-switch-continuity}
Suppose \(d_1+d_2=d_3+d_4\), with \(d_i(\mu)=1\) for
all \(i\).  Start with \(W=W_1\perp W_2\) as above, let
\(t_0\) be its compact-place data, and choose a total input
\(\phi\) for which
\(\xi=\Xi_W(\phi;t_0)\) is a nonzero holomorphic top form
on a sufficiently deep compact principal quotient \(X\).
There are rational orthogonal decompositions
\(W=W_3\perp W_4\) with line signs \(d_3,d_4\) whose
total compact-place positive Lagrangians are arbitrarily close
to those at \(t_0\).  The total active positive Lagrangian is
exactly unchanged.

For a sufficiently close rational choice \(t'\), the form
\(\xi'=\Xi_W(\phi;t')\) is holomorphic on the same quotient
\(X\), with the same total finite input, and
\begin{equation}\label{eq:theta-nonzero-pairing}
 \int_X\xi\wedge\overline{\xi'}\ne0.
\end{equation}
Both endpoints admit the wedge expansions of
Proposition~\ref{prop:theta-top} on a further common principal
cover, and their pairing remains nonzero there.
\end{corollary}

\begin{proof}
At a compact place \(\tau\ne\mu\), equality of the two sign
sums says exactly that the multisets
\(\{d_1(\tau),d_2(\tau)\}\) and
\(\{d_3(\tau),d_4(\tau)\}\) coincide.  Reorder the two
original local lines so that the first has the sign required
by \(d_3(\tau)\), and choose a nonzero vector on it.
After fixing an \(E\)-basis of \(W\), weak approximation in
\(E\), coordinate by coordinate at these finitely many
complex places, gives a single \(w\in W(E)\) arbitrarily
close to all the chosen local vectors.  Nondegeneracy and
the signs of their Hermitian norms are open conditions.
For a sufficiently close approximation,
\[
 W_3=Ew,\qquad W_4=W_3^\perp
\]
is therefore a rational orthogonal decomposition.  Its first
line has the prescribed compact-place signs for \(d_3\);
orthogonal inertia gives the complementary signs for \(d_4\).
At \(\mu\) the form on \(W_\mu\) is definite, so both new
lines automatically have the same required sign there.

The total compact-place Lagrangians converge to the original
ones.  Reordering the original lines only relabels their total
positive summands.  For a nondegenerate local vector the
orthogonal projection onto its line is
\[
 z\longmapsto h_W(z,w)h_W(w,w)^{-1}w.
\]
It varies continuously with \(w\), so both the line and its
orthogonal complement converge to the chosen original ones.
The selected signed subspaces, and hence the total positive
Lagrangians, converge.  At the active pair the total pieces
are always \(L\otimes W_\mu\) and \(Q^*\otimes W_\mu^*\),
so they are unchanged, not merely close.  The approximating
rational lines may have small active norms or large arithmetic
denominators.  No bound on these is needed: the active total
period data are fixed, and the continuity assertion concerns
the total tensor in the original reference coordinates.

All the hypotheses of Proposition~\ref{prop:theta-top}(iii)
now hold with the same rational \(W\), the same \(G\), the
same total finite \(\phi\), and the same compact \(X\).
Thus
\[
 \Xi_W(\phi;t')\longrightarrow \Xi_W(\phi;t_0)
\]
smoothly on \(X\).  The limit \(\xi\) is a nonzero
holomorphic top form.  In local complex coordinates
\(\xi=f\,dz_1\wedge dz_2\), the form
\(\xi\wedge\bar\xi\) is a positive multiple of
\(|f|^2\,dx_1\wedge dy_1\wedge dx_2\wedge dy_2\).
Consequently
\(\int_X\xi\wedge\bar\xi>0\).  Continuity of the integral
gives \eqref{eq:theta-nonzero-pairing} for all sufficiently
close rational choices.  At each such endpoint holomorphy
follows from Proposition~\ref{prop:theta-top}(i), not from a
holomorphy assertion along the intermediate parameter path.

Factor the two endpoints only now.  There are finitely many
line inputs in their two wedge expansions.  Choose a principal
subgroup contained both in the level of \(X\) and in their
finite stabilizers.  It gives a finite unramified cover
\(\pi:X'\to X\) on which all individual factors descend.
The covering formula is
\begin{equation}\label{eq:theta-cover-pairing}
 \int_{X'}\pi^*\xi\wedge\overline{\pi^*\xi'}
       =\deg(\pi)\int_X\xi\wedge\overline{\xi'}\ne0.
\end{equation}
Both quotients are compact and torsion-free, so there is no
cusp term or ramification correction.  This proves the last
assertion.
\end{proof}

Expansion of the two pulled-back endpoint forms on \(X'\) in
\eqref{eq:theta-cover-pairing} gives
\eqref{eq:theta-mixed-period-goal}; in particular at least one
pair of individual wedge terms has nonzero mixed period.
This is the analytic nonvanishing needed for
Proposition~\ref{prop:surface-switch}.
To apply that criterion
we must still express the relevant one-form classes as complex
linear combinations of images of the selected CM lines under
rational Hodge maps, on a common projective principal cover.
That arithmetic step is separate from the approximation just
proved.
==== END The-rational-Hodge-conjecture-for-CM-abelian-varieties//sections/04-theta.tex ====

==== BEGIN The-rational-Hodge-conjecture-for-CM-abelian-varieties//sections/05a-moduli.tex ====
\section{Projective ball quotients and a Frobenius identity}\label{sec:arithmetic}

The theta construction has supplied complex one-form classes on compact
ball quotients.  To use the surface criterion, we must identify their
rational sources: for the sign function \(v\), the relevant class must
lie in the complex span of the images of \(U(v)_1\) under rational Hodge
maps from \(U(v)\).  This section and Section~\ref{sec:cm-sources}
prove the following extraction theorem.  We first recall its input
data from the theta construction.

Retain the notation of Section~\ref{sec:theta}.  Thus \(E\subset\Qbar\)
is Galois CM, \(F=E^+\), \(G_E=\Gal(E/\Q)\), and \(c\in G_E\) is
complex conjugation.  We assume \(s=[F:\Q]\geq2\).  The Hermitian
\(E/F\)-space \(V\) has dimension three, signature \((2,1)\) at
\(\mu=1\), and is positive definite at the other real places.
Write \(G=\SU(V)\), let \(D\) be its ball of negative lines, and retain
the fixed \(\mathcal O_E\)-lattice \(\Lambda\subset V\).  Fix an odd
function \(v:G_E\to\{1,-1\}\) with \(v(1)=1\), and put
\[
d(\tau)=v(\tau^{-1}).
\]
For one multiplicity line \(P\) with these compact-place signs, write
\(\Theta(\phi)\) for its theta one-form and abbreviate its finite
special-unitary action to \(\omega_f=\omega_{P,f}\).  We work with a
finite collection of inputs at a common principal level.  The auxiliary
prime will be chosen only after that level is fixed.

\Needspace{11\baselineskip}
\begin{theorem}[Degree-one CM extraction]\label{thm:cm-extraction}
Let \(v:G_E\to\{1,-1\}\) be odd with \(v(1)=1\), and use the
multiplicity-line theta construction prescribed by
\(d(\tau)=v(\tau^{-1})\). Given a finite collection of its finite
adelic Schwartz inputs, there is a common sufficiently deep principal
ball quotient \(S_0=\Gamma\backslash D\) on which every associated
theta one-form class belongs to
\begin{equation}\label{eq:cm-extraction-span}
\operatorname{span}_{\C}\left\{
f\bigl(U(v)_1\bigr):
f\in\Hom_{\mathrm{Hdg}}\bigl(U(v),H^1(S_0,\Q)\bigr)
\right\}.
\end{equation}
The maps in this formula are rational Hodge maps, and their images
are viewed after extension to \(\C\). The conclusion persists after
passage to a deeper principal level.
\end{theorem}

The quotient \(S_0\) will be a smooth connected projective surface.
Corollary~\ref{cor:finite-family-extraction} gives the version needed
for the surface criterion: finitely many families with different sign
functions satisfy these span conclusions on one projective principal
cover of any prescribed torsion-free principal base quotient.  The reference
moduli construction depends only on \(V\), so the same geometry
serves every multiplicity line.

We begin by realizing the principal quotients as components of a full
projective moduli space of polarized abelian varieties with \(E\)-action.
Keeping all components makes the isogeny correspondences into algebraic
Hecke operators on the product of their Albanese varieties.  Dense
ordinary loci then let us prove an integral Frobenius polynomial for
these operators.  Section~\ref{sec:cm-sources} combines that polynomial
with the active theta parameter to determine the Hodge types of the
generalized Hecke spaces containing the theta classes.  Scalar descent
then proves the extraction theorem.

\subsection{A projective moduli space for the reference three-space}\label{subsec:reference-moduli}

The reference abelian varieties use \(V\), not \(V\otimes_E P\).
For a negative line \(L\in D\), prescribe on the complexified homology
\[
\mathcal F_\mu=L,\qquad
\mathcal F_c=\operatorname{ann}(L)\subset V_\mu^*,
\]
where the Hermitian form identifies the conjugate summand with the
complex-linear dual.  At each other conjugate pair choose, once and
for all, the whole summand on one side and zero on the other.  These
annihilator prescriptions together form a Lagrangian.  An imaginary element
\(\delta_{\mathrm{ref}}\in E^\times\), whose signs are chosen by weak
approximation in \(F\), makes them positive for
\[
\psi_{\mathrm{ref}}(x,y)
 =\Tr_{E/\Q}\bigl(\delta_{\mathrm{ref}}h_V(x,y)\bigr).
\]
Multiplying \(\delta_{\mathrm{ref}}\) by a positive integer makes the
form integral on \(\Lambda\).

Here the homology filtration is the kernel of the Lie quotient.  If
\(J\) is the complex structure on real homology, then
\(z=u+iJu\) lies in that filtration and
\[
\psi_{\mathrm{ref}}(z,\overline z)/i
   =-2\psi_{\mathrm{ref}}(u,Ju).
\]
Thus \(-\psi_{\mathrm{ref}}\) is the Riemann form in the positivity
convention of Section~\ref{sec:theta}.  The corresponding abelian
varieties have dimension \(3s\), and their Lie ranks are
\begin{equation}\label{eq:reference-lie-ranks}
r_\mu=2,\qquad r_c=1,\qquad
\{r_\tau,r_{c\tau}\}=\{0,3\}
\quad\bigl(\tau\notin\{1,c\}\bigr).
\end{equation}
Fix their polarization type.

Choose \(N\) sufficiently divisible to give a fine abelian moduli
problem, a torsion-free principal lattice level, and a level fixing
the chosen finite theta inputs.  We will also require the level to
kill all possible unitary determinant roots of unity.  A marking
means a full ordered basis of \(N\)-torsion; we allow every pairing
matrix for that basis.  In particular, transporting it through an
isogeny of degree prime to \(N\) is allowed even when the isogeny
rescales the polarization pairing.

We retain all components so that the full isogeny correspondences can
be defined without assuming that the reference component is preserved.
Over a number field \(K_0\subset\Qbar\) containing \(E\), consider the
reduced scheme \(S\) of tuples
\[
(B,\lambda,\iota_E,\eta_N).
\]
Here \(B\) is an abelian variety of dimension \(3s\), \(\lambda\) is
a polarization of the fixed type,
\(\iota_E:\mathcal O_E\to\End(B)\) is a unital action whose Rosati
adjoint is \(c\), \(\eta_N\) is the full marking, and each Lie
summand has the particular rank fixed above (with values displayed
in \eqref{eq:reference-lie-ranks}).  We retain all connected
components of this tuple space.

\begin{lemma}\label{lem:reference-moduli}
The scheme \(S\) is of finite type, smooth, and projective over \(K_0\).
Its complex connected components are compact ball quotients.  For
sufficiently divisible \(N\), the component arising from \(\Lambda\)
is the quotient of \(D\) by its principal special-unitary level.
\end{lemma}

\begin{proof}
The Lie-rank condition need not identify a single rational Hermitian
isomorphism class.  We therefore construct the whole finite-type
action locus, identify its complex components as ball quotients, and
then prove projectivity and identify the component of \(\Lambda\).

\emph{The finite-type action locus.}
The fine-moduli theorem for polarized abelian varieties gives a
quasi-projective scheme with universal polarization and full level at
sufficiently large invertible level
\cite[Chapter~7, Definitions~7.1--7.2, Proposition~7.6, and
Theorem~7.9]{MumfordFogarty1982}.
In that theorem the polarization degree is fixed and the level is an
ordered torsion basis, with no prescribed pairing matrix.  After
inverting the polarization degree, the polarization kernel is finite
étale, so its geometric elementary-divisor type is locally constant.
We may therefore take the open-and-closed part of the required type.
If necessary, first apply the theorem to a sufficiently large divisor
of \(N\) prime to the polarization degree and then add the remaining
full \(N\)-basis as a finite étale level cover.  These constructions
work over a ring of integers with finitely many primes inverted, as
well as in characteristic zero.

We next impose the action by a finite-type construction.  Choose
finitely many ring generators \(b\) of \(\mathcal O_E\) and
parameterize their graphs in relative Hilbert schemes of the square
of the universal abelian scheme.  The degrees of the graphs in
question are bounded.  Indeed, put \(\alpha=1+\bar b b\).  It is a
nonzero totally positive element of \(F\).  Choose a positive integer
\(m\) with \(m\alpha^{-1}\in\mathcal O_E\).  In every such action
\(\alpha\) is an isogeny, and
\[
\deg(\alpha)\deg(m\alpha^{-1})=m^{6s}.
\]
The second factor is a positive integer, so \(\deg(\alpha)\) is
bounded independently of the point.

To translate this into a Hilbert bound, the normalized Poincaré bundle
on the universal abelian scheme times its dual, pulled back by
\((\id,\lambda)\), is a relatively ample bundle inducing \(2\lambda\).
The tensor product of its two pullbacks to the square restricts on the graph of
\(b\) to a bundle inducing \(2\lambda(1+\bar b b)\).  Its polarization
degree is bounded by the preceding identity.  Riemann--Roch on an
abelian variety of dimension \(3s\) gives the Hilbert polynomial
\(\chi(\mathcal L)n^{3s}\), with
\(\chi(\mathcal L)^2=\deg(\phi_{\mathcal L})\).  Only finitely many
polynomials can therefore occur.  On their graph loci impose the
homomorphism, ring, Rosati, and Lie-rank identities.  This gives a
finite-type action locus.

Over the characteristic-zero polarized base this locus is unramified.
One form of rigidity suffices: an infinitesimal difference of two
homomorphisms between fixed abelian schemes would give a homomorphism
from a proper connected abelian fiber to an additive tangent group,
and must vanish.  The locus is also proper over that base.  Given a
discrete valuation ring mapping to the polarized base and a lift of
its generic point to the action locus, each
generator homomorphism extends uniquely between the fixed abelian
schemes by the Néron mapping property
\cite[\S1.2, Definition~1 and Proposition~8]{BoschLutkebohmertRaynaud1990}.
Its graph remains a graph of the same Hilbert polynomial.  The ring
and Rosati identities extend by uniqueness.  Since the base contains
\(E\), the action algebra splits into idempotents in characteristic
zero, and their Lie summands have locally constant ranks.  Thus the
extended point lies in the same action locus.  The valuative criterion
gives properness.  A proper finite-type unramified morphism is
quasi-finite and hence finite, so the action locus is finite over the
polarized base.

\emph{The complex components.}
We now identify the reduced complex structure of this algebraic locus.
In a marked period chart,
the integral matrices of the action are locally constant.  Such a
matrix acts on the complex torus exactly when it preserves the
homology filtration; the resulting holomorphic endomorphism of a
polarized complex torus is algebraic.  Hence the reduced local locus
is precisely the locus of positive filtrations stable under the fixed
matrices.  Trace duality and the Rosati identity express the
alternating form as an imaginary trace multiple of an
\(E/F\)-Hermitian form on the rational homology, of \(E\)-dimension
three.  Isotropy makes the two filtrations in a conjugate pair
annihilators.  Positivity and \eqref{eq:reference-lie-ranks} leave one
negative line at the active place and constant data at every other
place.  The local locus is therefore a smooth complex two-ball.

This description gives whole ball quotients, not only local ball
charts.  Fix a marked polarized \(E\)-lattice occurring in \(S\).
Every positive filtration in its compatible ball gives a complex
torus with the fixed integral Riemann form, and hence an abelian
variety; the preserving integral matrices give its algebraic
\(\mathcal O_E\)-action.  Conversely every tuple with these lattice
data gives such a filtration.  Its connected component is thus the
quotient of the whole ball by the marked lattice stabilizer.

\emph{Projectivity and the principal level.}
For each such rational Hermitian space, a vector isotropic over \(E\)
would remain isotropic at a definite real place.  Since \(s\geq2\),
there is no such vector, and its special-unitary group is
\(F\)-anisotropic.  The arithmetic lattice and compactness theorems
give compact ball quotients
\cite{BorelHarishChandra1962}; see
\cite[Theorem~3.3 and Proposition~3.5]{MilneShimura2017}.
The level removes stabilizers.  There are finitely many components
because \(S\) is of finite type.  The finite action locus is
quasi-projective, and its reduced complex space is smooth with compact
components.  It is therefore proper, and a proper quasi-projective
scheme is projective.  Smoothness and properness descend from the
complex base change to \(K_0\).

Finally the initial lattice \(\Lambda\) is among these data.  Even
if it is not free, its determinant module is a fractional
\(\mathcal O_E\)-ideal.  The determinant of a unitary lattice
automorphism is consequently a unit of \(\mathcal O_E\) of norm one.
All its complex conjugates have modulus one, so it is a root of unity.
A sufficiently divisible principal level kills every nontrivial such
determinant.  Its stabilizer is then precisely the principal
special-unitary lattice level, as claimed.
\end{proof}

\subsection{Good models and ordinary points on every component}\label{subsec:good-ordinary-models}

Enlarge \(K_0\) to a number field \(K\subset\Qbar\) over which all
connected components \(S_j\) are geometrically connected and have
\(K\)-points.  Index the original ball component by \(S_0\).
Use those points as origins for the Albanese maps
\[
a_j:S_j\longrightarrow A_j=\Alb(S_j),\qquad A_S=\prod_j A_j.
\]
We regard each \(A_j\) as a factor of \(A_S\), with zero in the other
factors.  The Albanese pullbacks give an isomorphism of rational Hodge
structures
\begin{equation}\label{eq:albanese-h1}
H^1(A_S,\Q)\simeq H^1(S,\Q).
\end{equation}

Because the Frobenius identity will be checked on ordinary points and
then on Albanese differences, we need density on every component and
generation of each reduced Albanese factor by those differences.

\begin{lemma}\label{lem:good-ordinary-primes}
After one finite extension of \(K\) and removal of finitely many
primes, the \(S_j\) have smooth proper models with geometrically
connected fibers, the \(A_j\) extend to abelian schemes, and the
\(a_j\) extend to the models.  The model of \(S\) lies in the full
tuple parameter scheme, retaining the \(\mathcal O_E\)-action and its
graph coordinates.  Differences of Albanese images generate every
geometric fiber of each \(A_j\).

At every remaining rational prime \(q\) completely split in \(K\),
the ordinary locus of the universal abelian scheme is dense in each
geometric component of the special fiber of \(S\).
\end{lemma}

\begin{proof}
\emph{Models and Albanese generation.}
Spread the finitely many schemes, morphisms, and universal data over
an open part of the ring of integers of \(K\).  One can obtain the
proper model of \(S\) by taking its closure in a projective
compactification of the full parameter scheme used in the proof of
Lemma~\ref{lem:reference-moduli}.  Its intersection with the boundary
has proper closed image on the integer base.  That image misses the
generic point because \(S\) is proper and lies in the parameter
scheme.  Removing the finitely many primes in the image leaves the
closure inside the full parameter scheme.  Thus its geometric points
are still distinguished by their polarized, acted-on, marked tuples,
including the graph coordinates.  Further shrinking gives smooth
proper models with geometrically connected component fibers, good
abelian models, and the extended Albanese maps.

For a component \(S_j\), let
\(\Delta_j=a_j(S_j)-a_j(S_j)\subset A_j\).  It is closed,
irreducible, symmetric, and contains zero.  The successive sum
images \(n\Delta_j\) are nested irreducible closed sets.  Their
dimensions eventually stabilize; then equality of two successive
sets implies stability under addition by \(\Delta_j\).  Symmetry
shows that the stable image is the abelian subvariety generated by
\(\Delta_j\), which is \(A_j\) by the Albanese property.  Thus for
some fixed \(n\) the proper sum map
\[
(S_j\times S_j)^n\longrightarrow A_j
\]
is surjective.  Spread this map together with the other data.  Its
closed image contains the generic fiber of the flat abelian scheme;
that fiber is dense, so the image is the whole abelian scheme.
Consequently the sum map is surjective on every geometric fiber.

\emph{Ordinary points.}
For each characteristic-zero component, we now choose a point whose
reduction at the split primes retained below is ordinary.  In its
rational Hermitian three-space choose, by weak approximation, an \(E\)-line
negative at \(\mu\), and diagonalize its orthogonal complement.
At the period whose negative line is this coordinate line, the three
rational orthogonal projectors split the rational homology
into three \(E\)-lines and preserve the filtration.  Integral
multiples of them are endomorphisms of the corresponding abelian
variety.

These points and endomorphisms are defined over \(\Qbar\).  Indeed,
add their graphs as extra parameters with the Hilbert polynomials
at the selected point.  This is a finite-type parameter scheme over
a number field.  Locally the projector matrices are fixed by
rigidity.  A negative line preserved by all three projectors must
be the unique negative coordinate line, so the selected enhanced
point is isolated.  An isolated complex point of a finite-type
scheme over a number field has algebraic coordinates.  Adjoin the
finitely many fields of definition to \(K\), spread these points
and endomorphisms, and exclude their projector and isogeny
denominators.

Now let \(q\) be a remaining completely split prime.  At each
selected point the integral projectors and the idempotents of
\(\mathcal O_E\otimes\Z_q\) split the \(q\)-divisible group into
pieces of height one.  Their heights are the corresponding
characteristic-zero Tate ranks and stay constant under specialization.
A height-one \(q\)-divisible group over an algebraically closed
field has dimension zero or one.  In the first case it is étale;
in the second its dual has dimension zero, so it is multiplicative.
Thus each selected reduction is ordinary.  Ordinariness is open,
and a smooth geometrically connected fiber is geometrically
irreducible.  This nonempty open is dense on each component.
Chebotarev's theorem supplies infinitely many completely split
rational primes after the finite exclusions.
\end{proof}

Choose such a rational prime \(q\), also prime to the polarization
degree, \(N\), the discriminant of \(E\), and all exceptional local
data of the finite theta inputs.  Fix an embedding
\[
\iota:\Qbar\hookrightarrow\overline{\Q}_q
\]
and normalize \(\nu_\iota\) by \(\nu_\iota(q)=1\).  The resulting
completion of \(K\) is \(\Q_q\), with residue field \(\Fq\).
For \(\tau\in G_E\), let \(\mathfrak p_\tau\) be the prime of \(E\)
defined by \(\iota\circ\tau\).  Thus for \(x\in E^\times\),
\[
\nu_\iota(\tau(x))=\operatorname{val}_{\mathfrak p_\tau}(x).
\]
Select one prime from each conjugate pair, including
\begin{equation}\label{eq:active-prime}
w=\mathfrak p_c,\qquad \bar w=\mathfrak p_1.
\end{equation}
Write \(\mathcal J\) for this set of selected blocks.  The
\(w\)-block has height three and Lie rank one; every other selected
block has height three and Lie rank zero or three.  The opposite
blocks are their polarization duals.  These rank assertions persist
on the model because the corresponding idempotent Lie and torsion
summands are locally free.  Choosing \(w=\mathfrak p_c\), rather
than \(\mathfrak p_1\), will fix the conjugate-valuation signs.

\subsection{Algebraic Hecke operators on the Albanese}\label{subsec:algebraic-hecke}

Choose bases for the selected \(q\)-adic lattices and the opposite
dual lattices.  Since the polarization is prime to \(q\), the split
local similitude group and its integral subgroup are
\[
\mathscr G_q=
\left(\prod_{j\in\mathcal J}\GL_3(\Q_q)\right)\times\Q_q^\times,
\qquad
\mathscr K_q=
\left(\prod_{j\in\mathcal J}\GL_3(\Z_q)\right)\times\Z_q^\times.
\]
The last coordinate is the common polarization multiplier.
Use the raw spherical Hecke algebra with
\(\operatorname{vol}(\mathscr K_q)=1\).  Its integral basis consists
of double-coset characteristic functions; its rational extension
will act on the Albanese up to isogeny.  A double coset is specified
by a multiplier exponent \(h\in\Z\) and, on each selected block,
an unordered triple of elementary exponents
\((a_{j,1},a_{j,2},a_{j,3})\).
For a representative \(g_j\) with those exponents, our target
position is the overlattice \(g_j^{-1}\Lambda_j\); branches are
represented modulo the integral subgroup on the left.

First suppose that
\begin{equation}\label{eq:positive-hecke-range}
h\geq0,\qquad 0\leq a_{j,k}\leq h.
\end{equation}
On the \(j\)-block of \(B[q^h]\), choose a subgroup of the indicated
elementary divisors.  On the opposite block choose its annihilator
under the perfect \(q^h\)-torsion pairing.  The total subgroup \(H\)
is maximal isotropic: each conjugate pair contributes order
\(q^{3h}\), so its total order is \(q^{3sh}\).
Its quotient \(f:B\to B'=B/H\) is required
to have polarization
\begin{equation}\label{eq:hecke-polarization}
f^\vee\lambda'f=q^h\lambda.
\end{equation}
We verify that this is an algebraic construction in families,
including the polarization homomorphism.

Over the characteristic-zero parameter scheme the torsion and the
subgroups are finite étale.  The translation quotient exists as an
fppf scheme quotient for the free finite-locally-free action
\cite[Tags~07S6 and~07S7]{StacksProject}.  Its group law descends,
and the finite étale quotient of the abelian scheme is again an
abelian scheme.

On a complex fiber the polarization in
\eqref{eq:hecke-polarization} is seen directly on lattices.  If
\(\Lambda_j\subset\Lambda_j'\subset q^{-h}\Lambda_j\) is the
selected target overlattice and the multiplier is \(q^h\), its
opposite lattice is
\[
q^{-h}(\Lambda_j')^\vee.
\]
The \(q^h\)-scaled alternating form is integral and perfect on this
pair at \(q\), by the annihilator condition.  At every other prime
the \(q\)-isogeny changes no elementary-divisor valuation.  The
scaled form is positive, so it gives a Riemann polarization on the
quotient of the original type.  More generally, the field-level
isotropic descent criterion gives the unique symmetric polarization
homomorphism on every geometric fiber
\cite[Chapter~XI, Proposition~11.25]{EdixhovenVanDerGeerMoonen2011Polarizations}.

Here is the relative descent from this fiber calculation.  Since
\(H\subset B[q^h]\), multiplication by \(q^h\) descends to
\(g:B'\to B\), with
\[
gf=[q^h]_B,\qquad fg=[q^h]_{B'}.
\]
Put \(\beta=g^\vee\lambda g:B'\to B'^\vee\).
On each geometric fiber the preceding polarization satisfies
\(\beta=q^h\lambda'\), so \(\beta\) kills \(B'[q^h]\).
The correspondence
parameter scheme is smooth and reduced, and \(B'[q^h]\) is finite
étale over it, hence reduced.  Vanishing on its geometric points is
therefore vanishing of the relative morphism.  The fppf quotient by
\(B'[q^h]\) now factors \(\beta\) uniquely through \([q^h]_{B'}\).
It gives a relative homomorphism \(\lambda'\) with
\(\beta=q^h\lambda'\).  Its fibers are the symmetric polarizations
just constructed, and cancellation of isogenies gives
\eqref{eq:hecke-polarization}.  This also proves uniqueness.  We
have descended the polarization homomorphism, not a particular
line bundle.

The \(\mathcal O_E\)-action descends because \(H\) is stable.
Cancelling \(f\) in \eqref{eq:hecke-polarization} shows that its
Rosati adjoint remains \(c\).  The characteristic-zero Lie ranks
are unchanged, and \(f\) transports the full \(N\)-basis.  Thus
the quotient is a point of the same full tuple space.  The subgroup
choices form a finite étale scheme over \(S\); the quotient
construction is its algebraic target map.  Distinct kernels remain
distinct branches even when their target tuples are isomorphic.

\begin{lemma}\label{lem:hecke-albanese}
The rational spherical Hecke algebra of
\((\mathscr G_q,\mathscr K_q)\) acts in \(\End_K(A_S)\otimes\Q\) and,
by pullback, on \(H^1(S,\Q)\).  A basis operator in the positive
range \eqref{eq:positive-hecke-range} is an integral homomorphism
defined over \(K\).  Its pullback on one-forms is the sum of
pullbacks over its target branches.
\end{lemma}

\begin{proof}
Let \(C\to S\) be one of the finite étale source correspondences.
On a source component, compose each target branch with the
appropriate Albanese map and sum in \(A_S\).  This sum descends
from a cover trivializing the branches, because it is invariant
under their permutations.  It is a morphism \(n_j:S_j\to A_S\).
Subtracting \(n_j\) at the basepoint and applying the Albanese
universal property gives an integral homomorphism
\(A_j\to A_S\).  On differences it has the exact formula
\begin{equation}\label{eq:hecke-albanese-differences}
T\bigl(a_j(x)-a_j(y)\bigr)
 =\sum_{z\in C_x}a_{j(z)}(t(z))
  -\sum_{z\in C_y}a_{j(z)}(t(z)),
\end{equation}
where \(t\) is the target map, \(j(z)\) indexes the component
containing \(t(z)\), fibers are counted with their branches, and
\(a_{j(z)}\) is included in \(A_S\).  These
homomorphisms on the factors define the operator on \(A_S\).
Their pullback on one-forms is the branch sum, since a constant
translation contributes no differential.

Composition has the raw spherical convolution multiplicities.
For a selected lattice \(\Lambda_j\), a branch is an overlattice
\(g_j^{-1}\Lambda_j\) with a representative modulo the integral
subgroup on the left.  If its cumulative multiplier is \(q^h\),
the opposite target lattice is forced to be
\(q^{-h}(\Lambda_j')^\vee\).  Consequently a chain of quotient
choices is precisely a chain of selected overlattices, as in
spherical convolution.  The opposite lattices, polarizations, and
transported markings agree for the same composite.  The numbers
of branches landing in each target component are locally constant
on a connected source.  Thus the component basepoint translations
cancel when \eqref{eq:hecke-albanese-differences} is composed.

Consider the central position with all selected exponents equal to
one and multiplier exponent two.  It is quotient by \(B[q]\);
under the multiplication-by-\(q\) identification its target has
the same polarization and action and marking \(q\eta_N\).  It is
therefore a finite-order marking automorphism, since \(q\) is
invertible modulo \(N\).  Adding a sufficiently large power of
this position to any integer exponent position puts it in
\eqref{eq:positive-hecke-range}.  Define its operator by clearing
that power with the inverse marking automorphism.  The positive
convolution identity shows that this definition is independent
of the clearing power.  This gives the full rational spherical
action.  The spherical algebra is commutative, as follows also
from the Satake isomorphism \cite[\S3]{Gross1998}.
\end{proof}
==== END The-rational-Hodge-conjecture-for-CM-abelian-varieties//sections/05a-moduli.tex ====

==== BEGIN The-rational-Hodge-conjecture-for-CM-abelian-varieties//sections/05b-frobenius.tex ====
\subsection{A Frobenius polynomial from ordinary neighbors}\label{subsec:frobenius-neighbors}

The Hecke action is now algebraic.  We next relate its reduction to
Frobenius by counting quotient isogenies at ordinary points of the
universal family.  This calculation depends on the reference
three-space and its Lie ranks; it is independent of the theta
multiplicity line.

For \(i=1,2,3\), let \(T_i\) be the full moduli Hecke operator with
\(i\) active exponents equal to one, all other selected exponents
zero, and multiplier exponent zero.  These are the active minuscule
cosets in the full similitude algebra.  They need not be positive in
\eqref{eq:positive-hecke-range}; Lemma~\ref{lem:hecke-albanese}
defines them by central clearing.

Let \(C_0\) be the central position with multiplier exponent one
and all active exponents zero.  On a nonactive selected block,
take all three exponents to be one if its Lie rank is three, and
zero if its Lie rank is zero.  Put
\begin{equation}\label{eq:integral-D-operators}
D_i=C_0^iT_i,\qquad 1\leq i\leq3.
\end{equation}
Here \(D_i\) has multiplier exponent \(i\), active exponents zero
or one, and nonactive selected exponents zero or \(i\).
It is a positive raw operator, hence an integral homomorphism of
\(A_S\).  The quotient branches defining \(D_i\) can therefore be
studied by specializing the isogenies constructed in
Subsection~\ref{subsec:algebraic-hecke}.

Let \(\widetilde A_S/\Fq\) be the good reduction at the fixed
completion of \(K\), and let \(\Phi\) be its \(q\)-power Frobenius
endomorphism.  The \(K\)-defined integral raw Hecke homomorphisms
extend over \(\Z_q\) by the Néron property; rational combinations
extend after tensoring the endomorphism ring with \(\Q\).
Their reductions are defined
over \(\Fq\), so they commute with \(\Phi\).  Use the same
letters \(D_i\) for these reductions.

\begin{proposition}\label{prop:albanese-congruence}
In the integral endomorphism ring of \(\widetilde A_S\),
\begin{equation}\label{eq:albanese-frobenius-polynomial}
\Phi^3-D_1\Phi^2+qD_2\Phi-q^3D_3=0.
\end{equation}
\end{proposition}

\begin{proof}
Let \(k=\overline{\Fq}\), and let \(x\) be an ordinary geometric
point of the special fiber of \(S\).  Its active \(q\)-divisible
group has the canonical splitting
\[
G_w=M_0\oplus E_0,
\]
where \(M_0\) is multiplicative of height one and \(E_0\) is
étale of height two.  Over the perfect field \(k\), the reduced
subgroup at each finite torsion level is the canonical étale
part; it is preserved by the endomorphism idempotents.  Each
nonactive selected block is wholly multiplicative or wholly
étale according to its Lie rank.

Three quotient constructions will account for the neighbors.
Write \(X_0\) for the quotient with active kernel \(M_0[q]\)
and multiplier \(q\), \(Y_0\) for the sum of quotients with
active kernel a line \(\ell\subset E_0[q]\) and multiplier
\(q\), and \(Z_0\) for the quotient with active kernel
\(E_0[q]\) and multiplier \(q^2\).  For multiplier \(q^h\),
prescribe the connected \(q^h\)-torsion on every nonactive
selected block and the forced annihilator on every opposite block.
We will prove that these quotients define full tuples of \(S\),
that \(X_0\) is Frobenius, and that the specialized neighbor sums are
\[
D_1:\ X_0+qY_0,\qquad
D_2:\ X_0Y_0+q^2Z_0,\qquad
D_3:\ X_0Z_0.
\]
The displayed formulas are the identities to establish; at this
point the quotient constructions have not yet been identified with
operations on \(S\).  Once they are, their cancellation will give
the asserted polynomial on Albanese differences.

\smallskip
\noindent\emph{Specialization multiplicities.}
Lift \(x\), using smoothness, to \(R=W(k)\), and write
\(\mathcal G_w\) for the lifted active \(q\)-divisible group.
The ring \(R\) is a complete henselian discrete valuation ring.
In the lifted active
\(q\)-torsion, the \(q^2\) open-and-closed special label
components lift uniquely by lifting idempotents of its finite
coordinate algebra.  Each lifted component is finite flat of
rank \(q\).  The identity-specializing component is a subgroup,
and multiplication respects the lifted labels by uniqueness.
Thus there is an exact sequence
\[
0\longrightarrow\mathcal H^0
\longrightarrow\mathcal G_w[q]
\longrightarrow(\Fq^2)_R\longrightarrow0
\]
with \(\mathcal H^0\) finite flat of rank \(q\).  No splitting
of this sequence over \(R\) is asserted.

Pass to a finite extension of \(R\) over which the generic torsion
is split.  Its active fiber is a three-dimensional \(\Fq\)-space
with distinguished line \(\mathcal H^0_{\mathrm{gen}}\).
For a fixed \(b\)-dimensional subspace of the label quotient,
the generic subspaces disjoint from that line and mapping onto
the label subspace are its graph lifts.  They form a torsor
under the \(b\)-dimensional space of maps to the line, so there
are \(q^b\) of them.  A generic subspace containing the line is
the unique full inverse image of its label subspace.

The schematic closure of a generic subgroup in the finite flat
torsion scheme is finite torsion-free over the new DVR, hence
flat.  It is still a subgroup, since the subgroup identities
extend from the generic fiber on flat schemes.  Intersection
with a label component is an idempotent direct summand of its
coordinate algebra, so its rank is its generic rank.  A graph
has rank one on each of its labels.  A finite flat rank-one
algebra over a local ring is that ring, and its special fiber
is the unique reduced point of that label component.  All
graph lifts over a label subspace therefore specialize to the
same étale subgroup of \(E_0[q]\).  A subspace containing the
distinguished line fills the corresponding rank-\(q\) components
and specializes to \(M_0[q]\) plus that étale subgroup.

The active kernels of \(D_i\) are \(i\)-dimensional subspaces of
the generic \(q\)-torsion.  The preceding count gives:
\[
\begin{array}{c|c|c}
\text{operator}&\text{special active kernel}&\text{multiplicity}\\ \hline
D_1&M_0[q]&1\\
   &\ell\subset E_0[q],\ \dim_{\Fq}\ell=1&q\text{ for each }\ell\\
D_2&M_0[q]\oplus\ell,\ \dim_{\Fq}\ell=1&1\text{ for each }\ell\\
   &E_0[q]&q^2\\
D_3&M_0[q]\oplus E_0[q]&1 .
\end{array}
\]
In the first two cases the total number is \(q^2+q+1\):
there are \(q+1\) étale lines.  The nonactive selected kernels
are uniquely their connected \(q^i\)-torsion for \(D_i\).

\smallskip
\noindent\emph{Full tuples and Frobenius.}
We now show that the counted kernels give targets in \(S\), and
identify the resulting quotient operations.  Properness of the
model of \(S\) extends each generic neighbor
over a finite valuation-ring extension.  The universal source
and target are abelian schemes over that new DVR.  Apply the
Néron mapping property there, anew, to extend the generic
isogeny \cite[\S1.2, Proposition~8]{BoschLutkebohmertRaynaud1990}.
The polarization identity extends by separatedness.  On each
fiber the kernel of the extension is contained in the finite
kernel of \(f^\vee\lambda'f=[q^i]\lambda\), so it has finite
kernel and is an isogeny between equal-dimensional abelian
varieties.  Since source and target polarizations have the
same degree, its degree is \(q^{3si}\) on every fiber.
Properness and fiberwise quasi-finiteness make the extension
finite.  Fiberwise flatness of isogenies between the smooth fibers
and the relative flatness criterion make it flat;
see \cite[\S7.3, Lemma~1 and Proposition~6; \S2.4,
Proposition~2]{BoschLutkebohmertRaynaud1990}.
Its finite flat kernel is therefore the
schematic closure of its generic kernel.

Finite-locally-free Cartier duality commutes with base change.
For a subgroup \(H\subset G\) in a finite group with a perfect
pairing, the annihilator is
\[
\operatorname{ann}(H)
 =\ker(G^D\longrightarrow H^D)=(G/H)^D.
\]
It is finite locally free and commutes with the above
specialization.  Thus the opposite generic kernels specialize
to the annihilators of the selected kernels just counted.
The corresponding nonactive kernels are already prescribed.
Equal total kernels give canonically isomorphic abelian
quotients.  Their descended \(\mathcal O_E\)-actions and
transported markings are unique, and their polarizations are
unique by cancelling \(f\) in \(f^\vee\lambda'f=q^i\lambda\).
The model retains these full tuples, so equal quotients are
equal target points of \(S\).  In particular every counted
kernel has such a target.  It is ordinary, since ordinariness
is invariant under isogeny.

The constructions \(X_0,Y_0,Z_0\) are therefore operations on
the ordinary points of the same full moduli space.  We use them
as multiset operations on points; no global special-fiber
algebraic correspondence is required.

The operation \(X_0\) is the \(q\)-power Frobenius on full tuples.
Indeed its total kernel is the connected \(q\)-torsion on every
block.  On the active pair, duality pairs the multiplicative
part with the opposite étale part and the étale part with the
opposite multiplicative part, so the annihilator of \(M_0[q]\)
is exactly the opposite connected part; the pure nonactive
pairs have the same property.  The total kernel is therefore
the kernel of relative Frobenius
\(F_B:B\to B^{(q)}\).  Naturality of the polarization and
Frobenius--Verschiebung duality give
\[
F_B^\vee\lambda^{(q)}F_B
 =V_{B^\vee}F_{B^\vee}\lambda=[q]\lambda.
\]
Here \(V_{B^\vee}\) denotes Verschiebung.
The action on the target is \(\iota_E^{(q)}\).  For the constant
label group \(T=(\Z/N\Z)^{6s}\), naturality gives
\[
F_B\eta_N=\eta_N^{(q)}F_T.
\]
Relative Frobenius on \(T\) fixes its labels, and \(F_B\) is an
isomorphism on \(N\)-torsion.  Thus the transported marking is
exactly the twisted marking, with no multiplication-by-\(q\)
marking change.  Since the model and universal data are defined
over \(\Fq\), this is its coordinate-\(q\)-power operation on
geometric points, not its inverse.

Frobenius commutes with \(Y_0\) and with \(Z_0\) as multiset
operations.  For any of their quotient maps \(f:B\to B'\),
\[
F_{B'}f=f^{(q)}F_B.
\]
Frobenius is an isomorphism on the étale part and permutes its
line choices bijectively; this square also respects the
polarizations, actions, and markings.  On the active block the
composite \(X_0Y_0\) has kernel \(M_0[q]\oplus\ell\) with
multiplier \(q^2\), and \(X_0Z_0\) has kernel
\(M_0[q]\oplus E_0[q]\) with multiplier \(q^3\).

For clarity, the opposite kernels of these composites are
indeed forced by their selected kernels and multipliers.  On
the \(q\)-divisible groups, write \(G,G'\) for source and target
and define the polarization adjoint
\(f^\dagger=\lambda^{-1}f^D\lambda'\); the polarizations are
isomorphisms there because their degrees are prime to \(q\).
For a composite with multiplier \(q^h\),
\[
f^\dagger f=[q^h],\qquad ff^\dagger=[q^h].
\]
The second identity follows by cancellation of the fppf
epimorphism \(f\).  As fppf subgroup sheaves,
\(\ker f=f^\dagger(G'[q^h])\).  For \(u,v\in G'[q^h]\),
adjunction gives
\[
e_{q^h}\bigl(f^\dagger u,f^\dagger v\bigr)
 =e'_{q^h}\bigl(u,ff^\dagger v\bigr)=1,
\]
so the image is isotropic in \(G[q^h]\).  The endomorphism
idempotents split its kernel into conjugate-block parts.
If \(K_j,K_{\bar j}\) are the parts in one conjugate pair of
height-three blocks, then
\[
K_{\bar j}\subseteq K_j^\perp,\qquad
\operatorname{rank}(K_j)\operatorname{rank}(K_{\bar j})
 \leq q^{3h}.
\]
Cartier duality gives equal kernel ranks for \(f\) and \(f^\dagger\).
Their composite has kernel rank \(q^{6sh}\), so
\(\operatorname{rank}\ker f=q^{3sh}\).
Multiplying the pairwise inequalities gives equality, and hence
equality in each pair.  Thus \(K_{\bar j}=K_j^\perp\).
For \(X_0Z_0\), for example, the annihilator of the active
\(q\)-torsion in \(q^3\)-torsion is the opposite block's
\(q^2\)-torsion.  The pure nonactive connected kernels likewise
compose to connected \(q^h\)-torsion.  Uniqueness of the
polarizations and markings now identifies the composite targets
with the counted targets.  We have proved the multiset identities
\begin{equation}\label{eq:ordinary-neighbor-counts}
\begin{aligned}
D_1&:\quad X_0+qY_0,\\
D_2&:\quad X_0Y_0+q^2Z_0,\\
D_3&:\quad X_0Z_0.
\end{aligned}
\end{equation}

\smallskip
\noindent\emph{The integral Albanese identity.}
The neighbor counts are now identities of multisets of full tuples.
We pass from them to endomorphisms by applying the Albanese difference
formula.  Let \(\mathcal O\) be the set of ordinary \(k\)-points of the
union of components and extend \(X_0,Y_0,Z_0\) linearly to the
free abelian group \(\Z[\mathcal O]\).  Define
\[
\alpha:\Z[\mathcal O]\longrightarrow\widetilde A_S(k),
\qquad \alpha([x])=a_{j(x)}(x).
\]
The component Albanese maps reduce over \(\Fq\), so
\(\Phi\alpha=\alpha X_0\).  For ordinary \(x,y\) in one source
component, lift them and specialize the integral identity
\eqref{eq:hecke-albanese-differences}.  With
\[
N_1=X_0+qY_0,\qquad
N_2=X_0Y_0+q^2Z_0,\qquad
N_3=X_0Z_0,
\]
it gives
\[
D_i\bigl(a_j(x)-a_j(y)\bigr)
 =\alpha\bigl(N_i([x]-[y])\bigr).
\]
This is an equality in the actual group of geometric points,
with all branch multiplicities retained.  No rationalization
of a finite-field point group is used.

Frobenius preserves each component and its ordinary locus.
Applying the same formula to \(X_0^m x,X_0^m y\) shows that
the endomorphism in \eqref{eq:albanese-frobenius-polynomial}
on this difference is
\[
\alpha\!\left(
(X_0^3-N_1X_0^2+qN_2X_0-q^3N_3)([x]-[y])
\right).
\]
The operator inside this expression vanishes:
\[
\begin{aligned}
&X_0^3-N_1X_0^2+qN_2X_0-q^3N_3\\
&\qquad=-qY_0X_0^2+qX_0Y_0X_0
        +q^3Z_0X_0-q^3X_0Z_0=0.
\end{aligned}
\]
Only the two proved commutations with \(X_0\) are used.
Ordinary points are dense in every smooth component by
Lemma~\ref{lem:good-ordinary-primes}.  Therefore the morphism
obtained by applying the endomorphism to
\(a_j(x)-a_j(y)\) vanishes on all of the product of that
component with itself.  The finite-sum generation in the same
lemma makes the endomorphism zero on each Albanese factor.
It is zero on \(\widetilde A_S\) after base change to \(k\);
faithful base change gives the integral identity over \(\Fq\).
\end{proof}

For the cohomological application, we record the Frobenius convention.  On
\(H^1_{\mathrm{cris}}(\widetilde A_S/\Z_q)[1/q]\), crystalline
Frobenius is the pullback \(\varphi=\Phi^*\) of this \(q\)-power
endomorphism: it is defined by absolute-Frobenius pullback, and
the Witt Frobenius on \(W(\Fq)=\Z_q\) is the identity because
\(q\) is prime \cite[Tag~07N0]{StacksProject}.  In degree-one
cohomology pullback is additive in abelian homomorphisms.
It reverses composition, but the displayed endomorphisms
commute.  Thus \eqref{eq:albanese-frobenius-polynomial} gives
the same operator polynomial with \(\varphi\) and the Hecke
pullbacks, not a reciprocal polynomial.  This is the identity
used in Proposition~\ref{prop:active-hodge-types}.
==== END The-rational-Hodge-conjecture-for-CM-abelian-varieties//sections/05b-frobenius.tex ====

==== BEGIN The-rational-Hodge-conjecture-for-CM-abelian-varieties//sections/05b-satake-frobenius.tex ====
\section{CM sources of the theta one-forms}\label{sec:cm-sources}

We now prove Theorem~\ref{thm:cm-extraction}, which identifies the
rational CM sources of the theta classes.  Section~\ref{sec:arithmetic}
constructed an algebraic Hecke action and a cubic relation with
Frobenius.  We calculate the active Hecke characters of the theta
classes, insert them into that relation, and use degree-one Weil
weight to isolate one Frobenius slope on every full character space.
The Hecke projectors preserve the Hodge filtration, so these slopes
determine the Hodge types of all scalar conjugates.  Scalar descent
will then give the required rational Hodge maps.

Retain the fields, moduli space, Albanese product, and Hecke operators
of Section~\ref{sec:arithmetic}.  In particular, the split good prime
\(q\) was chosen after fixing the finite theta inputs and their
common principal level.

\subsection{The active spherical parameter}\label{subsec:active-parameter}

We calculate the active Hecke characters in two steps.  First we
identify the geometric Hecke action on theta inputs as the inverse
spherical action.  We then compute that action on local inputs fixed
by \(\SL_3(\Z_q)\), which form a cyclic module for a dilation
operator.  Rational scalar covariance will
make its eigenvalues algebraic and encode the compact-place signs in
their \(q\)-adic valuations.  A lattice count then identifies the
spherical parameter, with the raw Hecke normalization fixed.

Let \(\mathcal H_w^0\) be the rational subalgebra supported on
positions with multiplier exponent zero, every nonactive selected
block trivial, and active exponent sum zero.  It is the spherical
algebra of \(\SL_3(\Q_q)\).

\begin{lemma}\label{lem:active-hecke-theta}
Every double coset in \(\mathcal H_w^0\) has all its branches
within each component of \(S\).  On the original ball component,
its pullback action on theta classes is the inverse spherical
action on the active local Weil input: a local double coset acts
through its inverse.
\end{lemma}

\begin{proof}
As a quasi-isogeny, such a position leaves the rational polarized
period fixed.  It changes only the active lattice and its opposite
dual, keeping the marking and all other local data.  This is also
the description obtained by clearing the central shift in
Lemma~\ref{lem:hecke-albanese} and undoing its marking change.
A determinant-exponent-zero lattice position has a representative
in \(\SL_3(\Q_q)\).  Adjusting determinant units in Smith bases
shows that its double cosets and branches are exactly those for
\(\SL_3(\Z_q)\backslash\SL_3(\Q_q)/\SL_3(\Z_q)\).

For each component apply strong approximation to the
special-unitary group of its rational Hermitian space.  This
group is simply connected and absolutely almost simple over \(F\),
and its factor at \(\mu\) is noncompact.  Strong approximation
away from all infinite places therefore makes its \(F\)-points
dense in its finite adelic points
\cite[Theorem~4.16]{MilneShimura2017}.
Put \(K_q=\SL_3(\Z_q)\).  Write \(\Lambda^{(j)}\) for the marked
lattice of the component under consideration.  For a branch with
new active lattice \(g^{-1}\Lambda_w^{(j)}\), write
\((\Lambda^{(j)})'\) for the full changed lattice and choose a rational representative
\(\gamma\) whose active component lies in \(K_qg\) and whose other
finite components are integral and marking-preserving.  Then
\(\gamma(\Lambda^{(j)})'=\Lambda^{(j)}\), and \(\gamma\) identifies
the target tuple with the transformed negative period line in the
same component.

On the original component, Lemmas~\ref{lem:theta-tensor-law}
and~\ref{lem:unitary-splitting} give
\[
\gamma^*\Theta\bigl(\omega_f(\gamma)\phi\bigr)=\Theta(\phi).
\]
Thus fixed-input pullback by the branch acts through
\(\omega_f(\gamma)^{-1}\).  Away from \(w\) the chosen level fixes
the input.  At \(w\) our good-prime choice makes it invariant
under \(K_q\); if \(\gamma_w=kg\), its inverse action on that
input is therefore \(\omega_f(g^{-1})\).  Summing the branches
proves the inverse spherical formula.  The sum is raw: no
division by its degree and no additional power of \(q\) occurs.
\end{proof}

We now compute this inverse spherical action on the local inputs.

At \(w\), choose the Schrödinger position space to be the
three-dimensional \(w\)-embedding summand, with lattice \(\Z_q^3\).
By Lemma~\ref{lem:unitary-splitting}, the action of
\(\SL_3(\Q_q)\) is inverse linear substitution.  Every finite input
is a finite sum of tensors separating this place from the other
finite places.  Our choice of \(q\) makes the input invariant under
\(\SL_3(\Z_q)\); averaging a separated decomposition over that
compact group gives one whose local factors are invariant.

The invariant Schwartz functions are radial: \(\SL_3(\Z_q)\) is
transitive on vectors with a fixed minimum coordinate valuation.
Local constancy at zero and compact support therefore make this
space the span of
\[
b_j=\mathbf 1_{q^j\Z_q^3},\qquad j\in\Z.
\]
The functions \(b_j\) are linearly independent.  Define the
normalized dilation
\begin{equation}\label{eq:radial-dilation}
(\mathcal D_q b)(x)=q^{3/2}b(q^{-1}x).
\end{equation}
Then \(\mathcal D_qb_j=q^{3/2}b_{j+1}\), so the radial space is a
cyclic module for \(\C[\mathcal D_q,\mathcal D_q^{-1}]\).

\begin{lemma}\label{lem:scalar-radial-relation}
Fix the input at all finite places other than \(w\).  There are
\(M>0\), an element \(r\in E^\times\), and a nonzero algebraic
number \(\omega\) such that
\[
r\bar r=1,\qquad
\operatorname{val}_{w}(r)=M,\qquad
\operatorname{val}_{\bar w}(r)=-M,
\]
with zero valuation at every other finite prime, and the map from
radial inputs to theta one-form classes factors through
\(\mathcal D_q^M=\omega\).  In the covariance convention of
Lemma~\ref{lem:theta-tensor-law},
\begin{equation}\label{eq:radial-scalar-weight}
\omega^2=\zeta\prod_{\tau\in\Sigma}\tau(r)^{-3d(\tau)}
\end{equation}
for a root of unity \(\zeta\).  Every root of \(X^M-\omega\) has
absolute value one under every complex embedding.
\end{lemma}

\begin{proof}
Principalize a positive power \(w^m=(a)\) and put \(r=a/\bar a\).
It has norm one and the required valuations with \(M=m\).
At every other finite place it is a unit.  Powers of an element of
a compact unit group enter every prescribed open subgroup: its
image in the finite quotient by that subgroup has finite order.
Smoothness of the bare local Weil representation gives an open
stabilizer of the fixed input in the metaplectic group; its image
meets the scalar unit group in an open subgroup.
Replacing \(r\) by a common positive power therefore places it
in the stabilizers needed at the finitely many exceptional places
of the fixed input; replace \(M\) accordingly.

Multiplication by \(r\) on \(V\otimes_E P\) preserves its
\(F\)-symplectic form because \(r\bar r=1\).  We use its
\emph{bare rational symplectic} covariance, with paired
metaplectic lifts, not the value of a chosen unitary splitting
character.  For a transformation \(h\) of a local position space,
the normalized linear action is
\[
(\rho(h)b)(x)=|\det h|_q^{-1/2}b(h^{-1}x),
\]
up to the finite Weil phase of the bare lift.  The linear-subgroup
formula, rational summation invariance, and the double-cover
control are in
\cite[I, no.~13; III, no.~41; IV, nos.~43--45]{Weil1964}.
At \(w\), the position action is the scalar \(r_w\), of valuation
\(M\).  Its unit part fixes a radial function, so the finite
action there is \(\mathcal D_q^M\), up to a root of unity.
At almost every other place the input is the unramified lattice
vector and \(r\) is integral.  At the remaining places the
chosen power fixes the input under a suitable bare lift;
the paired lift differs only by its finite phase.  Their product
is again a root of unity.  An arbitrary lift in the circle
pushout would not justify this conclusion, which is why the bare
rational action is used here.

At infinity, multiplication by \(r\) acts on the \(\tau\)-embedding
summand as the scalar \(\tau(r)\): this is the defining scalar
action after tensoring the \(E\)-space through \(\tau\).
On the \(c\tau\)-summand it acts as
\(\tau(\bar r)=\tau(r)^{-1}\).  It consequently fixes every
chosen positive Lagrangian and the point of \(D\).  In the
identity \(g e_Z=e_{gZ}J\) of
Lemma~\ref{lem:theta-tensor-law}, \(J\) is this action on the
homology filtration.  The squared scalar on the projected first
tensor is the scalar on \(\det\mathcal F_P\) times the square of
the scalar on its projected \(c\)-summand.
At \(\mu\), the filtration has scalar exponents \(1,-1,-1\).
Its determinant contributes \(-1\), and the squared projected
conjugate factor contributes \(-2\).  The total is \(-3\).
At a compact pair, the selected full rank-three summand
contributes \(-3d(\tau)\), with no projected tangent factor.
Thus the squared infinite scalar is
\(\prod_{\tau\in\Sigma}\tau(r)^{-3d(\tau)}\).
This calculation includes the scalar action on the determinant's
constant line, which is trivial only on \(\SU(V)\).

The covariance formula has the finite transformed input on its
left and the tensor action on its right.  Since \(r\) fixes the
period, it gives
\(\Theta(\mathcal D_q^M b)=\omega\Theta(b)\), with the finite
phase absorbed in \(\omega\), and its square is exactly
\eqref{eq:radial-scalar-weight}.  This equality of one-forms
implies the asserted equality of cohomology classes.

The right side of \eqref{eq:radial-scalar-weight} is algebraic,
so \(\omega\) is algebraic.  Every complex embedding of \(E\)
sends \(r\) to a number of modulus one, since \(E\) is CM and
\(r\bar r=1\).  Applying any scalar conjugation to the displayed
identity shows that every complex conjugate of \(\omega\) has
modulus one.  The same is then true of each root of \(X^M-\omega\).
\end{proof}

For a finite separated sum of inputs, take the common stabilizers
of all its other-place factors in this proof.  The same \(r,M,\omega\)
then works for that sum.

We use the half-modulus Satake convention, namely the parameter
attached to normalized parabolic induction.  We do not apply the
additional central Hecke/Tate shift of Gross's Section~8.  In particular the
trivial representation of \(\GL_2\) has parameter
\(\{q^{1/2},q^{-1/2}\}\).  On \(\SL_3\), a parameter is an unordered
triple of nonzero scalars modulo a common scalar.
We say that an active character occurs in a theta class when its
simultaneous generalized eigenspace in the finite-dimensional
Hecke span has a nonzero projection of that class.

\begin{proposition}\label{prop:theta-satake}
Every active spherical character occurring in a theta class of
the multiplicity line \(P\) has half-modulus projective parameter
\begin{equation}\label{eq:active-satake-triple}
\{q^{1/2},q^{-1/2},e\},
\end{equation}
where \(e=e_0^{-1}\) for a root \(e_0^M=\omega\) from
Lemma~\ref{lem:scalar-radial-relation}.  It is algebraic and all
its complex conjugates have modulus one.  For every
\(\sigma\in\Gal(\Qbar/\Q)\),
\begin{equation}\label{eq:active-conjugate-valuations}
\nu_\iota(\sigma e)
 =-\frac32d\bigl((\sigma|_E)^{-1}\bigr)
 =-\frac32v(\sigma|_E).
\end{equation}
A scalar-conjugate active character has a representative of the
form \eqref{eq:active-satake-triple} with third entry
\(e'=\pm\sigma e\).
\end{proposition}

\begin{proof}
The active spherical action commutes with \(\mathcal D_q\).
The radial Laurent module modulo \(\mathcal D_q^M-\omega\)
is a direct sum of one-dimensional spaces
\(\mathcal D_q=e_0\), because \(X^M-\omega\) has distinct
nonzero roots.  It suffices to compute the spherical character
on one of them.  This decomposition concerns the radial
quotient only; it asserts no semisimplicity for the ambient
cohomological Hecke action.

For this computation, extend the substitutions to
\[
\rho(g)b(x)=|\det g|_q^{-1/2}b(g^{-1}x),
\qquad g\in\GL_3(\Q_q),
\]
and put \(K_3=\GL_3(\Z_q)\).  This is a computational extension
of the active \(\SL_3\)-action, not a claim of theta equivariance
for the full moduli similitude group.  Let \(\mathsf T_i\) be
the raw double coset of
\(\diag(q,\ldots,q,1,\ldots,1)\), with \(i\) entries \(q\),
and \(\operatorname{vol}(K_3)=1\).  Its right cosets correspond
to sublattices between \(q\Z_q^3\) and \(\Z_q^3\) of index
\(q^i\).  For \(i=1,2\) there are \(q^2+q+1\) such lattices.
The numbers containing a fixed primitive vector are respectively
\(q+1\) and \(1\).  Since \(\rho(g)b_0\) is
\(q^{i/2}\) times the characteristic function of its sublattice,
we obtain
\begin{equation}\label{eq:radial-minuscule-actions}
\begin{aligned}
\mathsf T_1b_0&=q^{1/2}\bigl((q+1)b_0+q^2b_1\bigr),\\
\mathsf T_2b_0&=q\bigl(b_0+(q^2+q)b_1\bigr),\\
\mathsf T_3b_0&=\mathcal D_qb_0.
\end{aligned}
\end{equation}
For example, vectors in \(q\Z_q^3\) belong to every sublattice,
while a primitive vector is tested by its nonzero class in
\(\Fq^3\); the displayed counts are the numbers of the
corresponding planes or lines containing that class.

On the quotient \(\mathcal D_q=e_0\) we have
\(b_1=q^{-3/2}e_0b_0\).  Substitution into
\eqref{eq:radial-minuscule-actions} gives the values
\[
q\,e_1(e_0,q^{1/2},q^{-1/2}),\qquad
q\,e_2(e_0,q^{1/2},q^{-1/2}),\qquad
e_3(e_0,q^{1/2},q^{-1/2}),
\]
where \(e_i\) is the \(i\)-th elementary symmetric polynomial.
In the half-modulus Satake isomorphism the raw minuscule cosets
have transforms \(q e_1,q e_2,e_3\); these generate the spherical
algebra, with \(e_3\) invertible.  See
\cite[Equation~(3.14) and \S6]{Gross1998}, where the
\(\GL_n\) formula is attributed to Tamagawa.  The character just
computed therefore has parameter
\(\{e_0,q^{1/2},q^{-1/2}\}\).

The geometric action on theta inputs is inverse by
Lemma~\ref{lem:active-hecke-theta}.  Inversion sends the
half-modulus parameter to its reciprocal, without a degree
factor.  One can check this on the same generators:
\(\mathsf T_1^\vee=\mathsf T_3^{-1}\mathsf T_2\),
\(\mathsf T_2^\vee=\mathsf T_3^{-1}\mathsf T_1\), and
\(\mathsf T_3^\vee=\mathsf T_3^{-1}\), where \(f^\vee(g)=f(g^{-1})\).
Their transforms are the corresponding elementary symmetric
functions of the reciprocal triple.  After interchanging the
first two entries, the active projective parameter is therefore
\(\{q^{1/2},q^{-1/2},e_0^{-1}\}\).

The determinant-zero \(\GL_3\) and \(\SL_3\) double cosets agree
as in Lemma~\ref{lem:active-hecke-theta}.  Their Satake unipotent
integrals and modulus factors agree as well.  The half-sum of
positive roots is \((1,0,-1)\), integral on the relevant
cocharacter lattices; the Satake comparison is consequently
defined over rational coefficients
\cite[Proposition~3.6]{Gross1998}.  Scalar conjugation of an
active numerical character therefore conjugates its projective
parameter.  It may change the two entries \(q^{1/2},q^{-1/2}\)
by their common sign.  Removing that sign projectively leaves
third entry \(\pm\sigma e\).

It remains to calculate the valuation.  Put \(g=\sigma|_E\).
By the definition of the prime labels,
\[
\nu_\iota(\sigma\tau(r))
 =\operatorname{val}_{\mathfrak p_{g\tau}}(r).
\]
This is nonzero only when \(g\tau=1\) or \(c\).  Among the
representatives \(\Sigma\), the contributing one is \(g^{-1}\)
with valuation \(-M\), or \(cg^{-1}\) with valuation \(M\).
Here \(c\) is central in \(G_E\), and oddness of \(d\) gives
in either case
\[
\sum_{\tau\in\Sigma}d(\tau)\nu_\iota(\sigma\tau(r))
   =-M d(g^{-1}).
\]
Taking valuations in \eqref{eq:radial-scalar-weight} and in
\(e_0^M=\omega\) yields
\(\nu_\iota(\sigma e_0)=\tfrac32d(g^{-1})\).
Since \(e=e_0^{-1}\) and \(d(g^{-1})=v(g)\), this proves
\eqref{eq:active-conjugate-valuations}.  The unit-modulus
assertion follows from Lemma~\ref{lem:scalar-radial-relation}.
\end{proof}

The active character is now known, including the valuation of every
scalar conjugate.  The full Hecke character may still multiply its
three Satake entries by a common scalar.  The following finite-order
relations will control that scalar when we apply the Frobenius
polynomial of Proposition~\ref{prop:albanese-congruence}.

\begin{lemma}\label{lem:central-finite-order}
The central operators \(T_3\) and \(C_0\) have finite order on
the full tuple space \(S\) and on \(A_S\).
\end{lemma}

\begin{proof}
Each central position has a unique lattice choice.  For \(T_3\),
use the norm-one element \(r\) constructed in
Lemma~\ref{lem:scalar-radial-relation}; only its valuations and
norm are needed here.  The scalar \(r\) takes the \(T_3^M\)-target
lattice back to the source lattice on the active and opposite
blocks.  It preserves every polarized period
because it is an \(E\)-scalar of norm one.  Its denominators are
supported at \(q\), so a further power is one on the \(N\)-marking.
That power of \(T_3\) is the identity on every tuple.

For \(C_0\), let \(I\) be the product of the primes where its
full exponent is one.  There is exactly one from each conjugate
pair, including \(\bar w\), and \(I\bar I=(q)\).  Choose
\(I^m=(a)\).  Then
\[
a\bar a=q^m u
\]
for a totally positive unit \(u\in\mathcal O_F^\times\).
The scalar \(b=a^2/u\) has exactly the valuations of \(I^{2m}\)
and satisfies
\[
b\bar b=q^{2m}.
\]
It identifies the corresponding powered target lattice with
the source, with exactly the required polarization multiplier.
A further power fixes the marking.  This argument works on
every rational Hermitian component because an \(E\)-scalar
preserves each of its embedding filtrations.  Thus a power of
\(C_0\) is also the identity on the full tuple space.  The
induced Albanese homomorphisms have finite order as well.
\end{proof}
==== END The-rational-Hodge-conjecture-for-CM-abelian-varieties//sections/05b-satake-frobenius.tex ====

==== BEGIN The-rational-Hodge-conjecture-for-CM-abelian-varieties//sections/05c-cm-types.tex ====
\subsection{From Frobenius slopes to Hodge types}\label{subsec:slopes-to-types}

We now combine the Frobenius polynomial with the theta parameters to
determine the Hodge types of the relevant Hecke spaces. The polynomial
will force a single Newton slope on each full generalized character
space. To turn this into a Hodge type, we also need its projector to
preserve the Hodge filtration. We first construct these projectors and
explain the filtered argument.

Let \(\mathbb T\) be the image of the full spherical Hecke algebra in
\(\End_{\Q}H^1(A_S,\Q)\). It is a finite-dimensional commutative
\(\Q\)-algebra; its active subalgebra is the image of
\(\mathcal H_w^0\). A numerical character is a \(\Q\)-algebra
homomorphism \(\chi:\mathbb T\to\Qbar\). Inside a character value,
the symbol of an endomorphism denotes its degree-one pullback.
After a finite extension of algebraic
coefficients, \(\mathbb T\) is a product of local Artinian algebras.
Its product idempotents \(p_\chi\) give the simultaneous generalized
character spaces
\[
H_\chi=p_\chi H^1(A_S,\Qbar).
\]
If \(\eta\) is a character of the active subalgebra, its generalized
space is
\begin{equation}\label{eq:active-generalized-space}
H^{\mathrm{act}}_\eta
=\bigoplus_{\chi|_{\mathrm{act}}=\eta}H_\chi.
\end{equation}
Indeed the image of the active idempotent in a local factor of the full
algebra is either zero or one, as decided by its residue character.
Thus the sum includes every full extension of \(\eta\) and all of its
nilpotent parts. No semisimplicity of the Hecke operators is assumed.

Each Hecke operator is induced by a rational endomorphism of \(A_S\)
defined over \(K\). In degree-one Betti, de Rham, and crystalline
cohomology, pullback is additive in abelian homomorphisms: in the
Künneth decomposition of \(H^1(A\times A)\), pullback by addition has
both components equal to the identity, as restriction to the two axes
shows. Pullback reverses composition, but the Hecke endomorphisms
commute. Pullback on Betti \(H^1\) is faithful by
Lemma~\ref{lem:weight-one-maps}, so polynomial relations in
\(\mathbb T\) are also relations among the corresponding
endomorphisms. The same degree-one functoriality preserves those
relations in de Rham and crystalline cohomology. Thus the \(p_\chi\)
remain idempotents in these degree-one
realizations after extension of coefficients. They need not have
rational coefficients; what matters is that they are
algebraic-coefficient polynomials in extending endomorphisms.

The following lemma applies to precisely these projected summands. Its
use of both complementary projectors is what turns weak admissibility
into equality of the Hodge and Newton numbers on each summand.

\begin{lemma}\label{lem:filtered-hecke-summands}
Let \(\mathcal A/\Z_q\) be an abelian scheme with generic fiber \(A\), and
put
\[
D_A=H^1_{\mathrm{dR}}(A/\Q_q).
\]
Equip \(D_A\) with its Hodge filtration and its good-reduction crystalline
Frobenius \(\varphi_A\). Then \(D_A\) is weakly admissible, its filtration
has only indices \(0\) and \(1\), and endomorphisms extending over \(\Z_q\)
preserve the filtration and commute with \(\varphi_A\).

Let \(L/\Q_q\) be finite. Suppose that an \(L\)-linear idempotent
\(p\) on \(D_A\otimes_{\Q_q}L\) preserves the filtration and commutes
with \(\varphi_A\otimes1\). If every Frobenius slope on its image is
\(t\in\{0,1\}\), then the induced filtration satisfies
\[
F^1\bigl(p(D_A\otimes L)\bigr)=
\begin{cases}
0,&t=0,\\
p(D_A\otimes L),&t=1.
\end{cases}
\]
\end{lemma}

\begin{proof}
Good-reduction crystalline comparison identifies
\[
D_{\mathrm{cris}}\!
 \left(H^1_{\mathrm{\acute et}}(A_{\overline{\Q}_q},\Q_q)\right)
\simeq
H^1_{\mathrm{cris}}(\mathcal A_{\Fq}/\Z_q)[1/q]
\]
and identifies the filtered de Rham realization with \(D_A\).
These identifications are functorial for extending endomorphisms and
compatible with Frobenius and filtration
\cite[Theorem~1.1(i), Theorem~12.1, and \S12.2]{BhattMorrowScholze2018}.\footnote{For the proper formal model and the algebraic/analytic
cohomology identifications, see \cite[\S7.2, p.~161]{Bosch2014FormalRigid}
and \cite[Theorems~9.1 and~9.3]{Scholze2013RigidAnalyticHodge}, read with
\cite{Scholze2016RigidAnalyticHodgeCorrigendum}. Passage from the algebraic
closure to its completion uses \cite[Tag~0DDG]{StacksProject}.
The \'etale comparisons are applied with \(\Z/q^n\)-coefficients,
then passed to inverse limits and rationalized.}
As in Subsection~\ref{subsec:frobenius-neighbors}, crystalline Frobenius is
the pullback of the special fiber's \(q\)-power endomorphism: Witt
Frobenius on \(W(\Fq)=\Z_q\) is the identity
\cite[Tags~07N3 and~07N5]{StacksProject}.
Crystalline representations give weakly admissible filtered modules
\cite[Theorem~9.3.4]{BrinonConrad2009}. The degree-one Hodge filtration
has only indices \(0\) and \(1\), and functoriality gives the assertion
about extending endomorphisms.

The extension by \(L\) changes coefficients, not the geometric base
field. With Frobenius acting trivially on this coefficient factor, the
underlying \(\Q_q\)-filtered Frobenius module
\[
\bigl(D_A\otimes_{\Q_q}L,\ \varphi_A\otimes1,\
       F^iD_A\otimes_{\Q_q}L\bigr)
\]
is a direct sum of \([L:\Q_q]\) copies of \(D_A\), and is weakly
admissible. Since \(p\) and \(1-p\) both preserve the filtration and
commute with Frobenius, their images split it as filtered Frobenius
modules. Weak admissibility gives \(t_H\leq t_N\) on each image.
The Hodge and Newton numbers are additive, and are equal for the whole
module. They are therefore equal on each image separately.

Let \(m=\dim_{\Q_q}p(D_A\otimes L)\). A pure slope \(t\) gives
\(t_N=tm\) after restriction of scalars: all \(\Q_q\)-conjugates of a
\(q\)-adic algebraic number have the same valuation, including when
\(L/\Q_q\) is ramified. On the other hand,
\[
t_H=\dim_{\Q_q}F^1\bigl(p(D_A\otimes L)\bigr),
\]
because the only filtration indices are \(0\) and \(1\). Equality of
these two numbers proves the assertion. The general coefficient
formalism is also described in
\cite[Proposition~2.17(ii) and Corollary~2.22]{Hellmann2011}.
\end{proof}

The requirement on the idempotent is essential. For example,
\(\varphi=\diag(1,q)\) on \(\Q_qe_0\oplus\Q_qe_1\), with only
filtration indices \(0,1\) and \(F^1=\langle e_0+e_1\rangle\), is
weakly admissible, but its slope-one line has induced \(F^1=0\).
Its slope projector does not preserve the filtration. The Hecke
projectors do preserve it, because they are formed from extending
algebraic endomorphisms.

We can therefore obtain a pure Hodge type by proving that a full
generalized Hecke space has slope zero or one. The finite-order central
relation controls the common scalar in a full extension of the active
projective parameter, and the Frobenius polynomial then determines the
slope.

Fix an active character \(\eta\) with projective parameter
\[
\{q^{1/2},q^{-1/2},e'\},
\]
where \(e'=e\) is supplied by Proposition~\ref{prop:theta-satake}, or
\(e'=\pm\sigma e\) is the parameter of a scalar-conjugate character.
Thus \(e'\) is algebraic, all its complex conjugates have modulus one,
and \(\nu_\iota(e')\in\{-3/2,3/2\}\). For a full character \(\chi\)
extending \(\eta\), its active \(\GL_3\) parameter is a scalar
multiple of this triple. Absorb the numerical value of \(C_0\) into
that scalar and write the resulting triple as
\begin{equation}\label{eq:full-satake-triple}
(\alpha_1,\alpha_2,\alpha_3)
=b(q^{1/2},q^{-1/2},e').
\end{equation}
The determinant coordinate is the character value of
\(D_3=C_0^3T_3\). Lemma~\ref{lem:central-finite-order} therefore gives
\begin{equation}\label{eq:satake-central-relation}
b^3e' \text{ is a root of unity}.
\end{equation}
In particular \(b\) is algebraic and every complex conjugate of \(b\)
has modulus one.

Write \(e_i\) for the elementary symmetric polynomials in three
variables. The raw minuscule double cosets, with the integral subgroup
of volume one, have Satake transforms
\(q e_1,\ q e_2,\ e_3\); this is
\cite[Equation~(3.14) and \S6]{Gross1998} for \(\GL_3\).
Since \(D_i=C_0^iT_i\), absorption of the \(C_0\)-value into \(b\)
gives
\begin{equation}\label{eq:raw-D-character}
\chi(D_1)=q e_1(\alpha),\qquad
\chi(D_2)=q e_2(\alpha),\qquad
\chi(D_3)=e_3(\alpha).
\end{equation}
These are numerical character values. On \(H_\chi\) the operators
may still have nilpotent parts.

\begin{proposition}\label{prop:active-hodge-types}
Let \(\eta\) be an active numerical character with projective parameter
\(\{q^{1/2},q^{-1/2},e'\}\) supplied by
Proposition~\ref{prop:theta-satake} or by scalar conjugation, and put
\[
t_\eta=\frac12-\frac13\nu_\iota(e')\in\{0,1\}.
\]
For every full numerical character \(\chi\) extending \(\eta\), the
space \(H_\chi\otimes_{\Qbar}\C\) is contained in \(H^{1,0}(A_S)\)
if \(t_\eta=1\), and in \(H^{0,1}(A_S)\) if \(t_\eta=0\).
Consequently \(H^{\mathrm{act}}_\eta\otimes_{\Qbar}\C\) has the same
pure Hodge type.
\end{proposition}

\begin{proof}
Fix a full character \(\chi\) extending \(\eta\); if there is none,
the assertion is vacuous.
We first use Weil weight to select the Frobenius root. We then apply
Lemma~\ref{lem:filtered-hecke-summands} and compare the resulting
filtration ranks with those on the complex Betti realization.

At the fixed completion \(K_v=\Q_q\), write
\[
D_S=H^1_{\mathrm{dR}}((A_S)_{K_v}/\Q_q)
\]
and let \(\varphi\) be its crystalline Frobenius. If \(\Phi\) is the
\(q\)-power Frobenius endomorphism of \(\widetilde A_S/\Fq\) used in
Proposition~\ref{prop:albanese-congruence}, then
\(\varphi=\Phi^*\) under crystalline comparison. The Hecke
endomorphisms extend over \(\Z_q\); their reductions commute with
\(\Phi\), so their pullbacks commute with \(\varphi\).

By additivity of pullback on abelian degree-one cohomology, the integral identity of
Proposition~\ref{prop:albanese-congruence} gives
\begin{equation}\label{eq:cohomological-frobenius-polynomial}
\varphi^3-D_1^*\varphi^2+qD_2^*\varphi-q^3D_3^*=0.
\end{equation}
Pullback reverses composition, but all operators in this formula
commute.

Choose a prime \(\ell\ne q\) and put
\(V_\ell\widetilde A_S=T_\ell\widetilde A_S\otimes_{\Z_\ell}\Q_\ell\).
Weil's theorem for abelian varieties over \(\Fq\) gives
\[
 P_\Phi(X)=\det\bigl(X-\Phi\mid V_\ell\widetilde A_S\bigr)\in\Z[X],
\]
all of whose complex roots have absolute value \(q^{1/2}\).
The prime-to-\(q\) Tate-module action on endomorphisms is faithful.
Cayley--Hamilton therefore gives \(P_\Phi(\Phi)=0\) in the
endomorphism ring, and degree-one additivity gives
\(P_\Phi(\varphi)=0\).  These statements hold for an arbitrary
abelian variety over \(\Fq\); see
\cite[Lemma~12.2, Proposition~12.9, and Theorem~19.1(c) with its proof]{Milne1986AbelianVarieties}.
No CM or ordinary hypothesis on \(\widetilde A_S\) is involved.
This uses the \(q\)-power endomorphism and its pullback, not a
reciprocal Galois-Frobenius convention.

Transport the algebraic coefficients of \(p_\chi\) and \(\chi\) by
\(\iota\). On the corresponding generalized summand of
\(D_S\otimes_{\Q_q}\overline{\Q}_q\), the commuting Hecke maps and
\(\varphi\) can be simultaneously triangularized. Equations
\eqref{eq:cohomological-frobenius-polynomial}
and~\eqref{eq:raw-D-character} imply that every diagonal Frobenius
eigenvalue is a root of
\[
X^3-qe_1(\alpha)X^2+q^2e_2(\alpha)X-q^3e_3(\alpha)
=\prod_{j=1}^3(X-q\alpha_j).
\]
Thus the candidates, before applying \(\iota\), are
\begin{equation}\label{eq:frobenius-candidates}
bq^{3/2},\qquad bq^{1/2},\qquad qbe'.
\end{equation}
Each is algebraic. If its image under \(\iota\) is a root of the
rational polynomial \(P_\Phi\), the algebraic number itself is a
root of \(P_\Phi\). We may therefore apply the complex
absolute-value condition. The three absolute values are respectively
\[
q^{3/2},\qquad q^{1/2},\qquad q,
\]
because every complex conjugate of \(b\) and \(e'\) has modulus one.
Only \(bq^{1/2}\) remains.

The weight condition has now selected one Newton slope on the entire
generalized summand. Equation~\eqref{eq:satake-central-relation} gives
\begin{equation}\label{eq:active-hodge-slope}
\nu_\iota(bq^{1/2})
=\frac12-\frac13\nu_\iota(e')=t_\eta.
\end{equation}
This conclusion concerns every Frobenius eigenvalue; Jordan blocks do
not change it. Choose a finite extension \(L/\Q_q\) containing the
coefficients of the transported idempotent and numerical character.
The idempotent \(p_\chi\)
is a polynomial in extending Hecke maps, so it preserves the Hodge
filtration and commutes with \(\varphi\). Lemma
\ref{lem:filtered-hecke-summands} makes its filtration pure of index
\(t_\eta\).

It remains to relate this filtered rank to the Hodge type of the same
complex generalized space. Put
\[
V_{\mathrm{dR}}=H^1_{\mathrm{dR}}(A_S/K).
\]
Its filtration and Hecke operators are defined over the fixed field
\(K\). For a numerical character \(\chi\), choose a polynomial
expression for \(p_\chi\), and let \(k_\chi\subset\Qbar\) be the
actual compositum of \(K\) and its algebraic coefficients. Evaluating
that polynomial in the fixed operators gives an idempotent on
\(V_{\mathrm{dR}}\otimes_K k_\chi\). Its induced filtration is its
projected filtration, since the idempotent preserves \(F^\bullet\).
The dimensions of its image and of this filtration are ranks of
matrices over \(k_\chi\);
they are unchanged under the restriction of the fixed embedding
\(\iota\) to \(k_\chi\).

Over the given embedding \(K\subset\C\), Betti--de Rham comparison
intertwines the same operators and polynomial. It identifies these
ranks with the filtration ranks on \(H_\chi\otimes_{\Qbar}\C\).
For a scalar-conjugate numerical character \(\sigma\chi\), repeat
this argument with \(p_{\sigma\chi}\) in the same \(K\)-defined
operators. On Betti cohomology their rational matrices give
\(\sigma H_\chi=H_{\sigma\chi}\). We have therefore computed the
type of the scalar-conjugated Betti space on the same complex variety.
Each conjugate projector is evaluated separately in the fixed
\(K\)-defined operators. This works even when \(K\) intersects the
character field, since it uses ranks over their actual compositum.

Finally the complex Hecke operators preserve both \(H^{1,0}\) and
\(H^{0,1}\). Hence a full projected \(F^1\) means that the whole
generalized space is holomorphic, and zero projected \(F^1\) means
that it is antiholomorphic. Equation~\eqref{eq:active-hodge-slope}
depends only on \(\eta\), so summing all full extensions gives the
assertion for \(H^{\mathrm{act}}_\eta\).
\end{proof}

\subsection{Rational CM sources for the theta classes}\label{subsec:rational-cm-sources}

The preceding proposition determines the type of every vector in each
scalar-conjugate active Hecke space. We can therefore apply scalar
descent to maps with arbitrary algebraic Betti vectors in these spaces.
This will prove the rational-map span asserted in
Theorem~\ref{thm:cm-extraction}.

\begin{proof}[Proof of Theorem~\ref{thm:cm-extraction}]
Use the Albanese identification
\(H^1(A_S,\Q)\simeq H^1(S,\Q)\). Extend a theta class from \(S_0\)
by zero on the other components of \(S\). This is compatible with
the active algebra, because Lemma~\ref{lem:active-hecke-theta}
makes that algebra component-preserving. The class is holomorphic
by Proposition~\ref{prop:theta-one}. Decompose it into the active
generalized character spaces of
Equation~\eqref{eq:active-generalized-space}. For each nonzero component,
let \(\eta\) be its active character and \(e\) its parameter from
Proposition~\ref{prop:theta-satake}. For every
\(\sigma\in\Gal(\Qbar/\Q)\), put \(g=\sigma|_E\). That proposition
and Proposition~\ref{prop:active-hodge-types} give
\begin{equation}\label{eq:cm-sign-to-hodge-type}
\nu_\iota(\sigma e)=-\frac32v(g),\qquad
t_{\sigma\eta}
=\frac12-\frac13\nu_\iota(\sigma e)
=\frac{1+v(g)}2.
\end{equation}
The possible sign in the conjugate parameter \(e'=\pm\sigma e\)
has valuation zero. Consequently the entire active space for
\(\sigma\eta\) is of type \((1,0)\) when \(v(g)=1\) and of type
\((0,1)\) when \(v(g)=-1\). This applies to all of its vectors,
not just to theta vectors. At the identity it gives type \((1,0)\),
in agreement with \(v(1)=1\) and the holomorphy of the theta class.

Each active space is defined over \(\Qbar\) in the rational Betti
structure, because its idempotent has algebraic coefficients.
Choose \(0\ne x\in H^{\mathrm{act}}_\eta\) and
\(0\ne u\in U(v)_1(\Qbar)\). Define a \(\Qbar\)-linear map
\[
h:U(v)_{\Qbar}\longrightarrow H^1(S,\Qbar)
\]
by \(h(u)=x\) and by setting it equal to zero on every other
embedding line of \(U(v)_{\Qbar}\). This is an element of
\[
\Hom_{\Q}\bigl(U(v),H^1(S,\Q)\bigr)\otimes_{\Q}\Qbar.
\]
Scalar conjugation by \(\sigma\) moves its nonzero source line to
\(U(v)_g\) and its target vector to
\(\sigma x\in H^{\mathrm{act}}_{\sigma\eta}\). The source line has
the type prescribed by \(v(g)\), and
Equation~\eqref{eq:cm-sign-to-hodge-type} gives the same type for
the target. All other source lines map to zero. Hence every scalar
conjugate of \(h\) is a Hodge map.

Lemma~\ref{lem:trace-descent}, applied to the complex-linear
subspace of Hodge maps, expresses \(h\) as a \(\Qbar\)-linear
combination of rational Hodge maps. Algebraic-Betti vectors span
each complex active space, so taking complex spans covers every
active component of the original theta class, including classes
with transcendental coefficients. Restriction from \(S\) to
\(S_0\) is a rational Hodge map and gives
Equation~\eqref{eq:cm-extraction-span}.

For a finite collection of inputs, all level choices are made
before the auxiliary prime is chosen, and only finitely many primes
are excluded. The preceding argument therefore applies to that
collection at one sufficiently deep level. If
\(\pi:S'_0\to S_0\) is a deeper principal cover, then
\(\pi^*\) is a rational Hodge map and the theta classes pull back
to their classes on \(S'_0\). Applying \(\pi^*\) to
Equation~\eqref{eq:cm-extraction-span} proves persistence.
\end{proof}

By Lemma~\ref{lem:weight-one-maps}, each rational map in
Equation~\eqref{eq:cm-extraction-span} is, up to a rational multiple,
pullback along a morphism \(S_0\to B(v)\).

\begin{corollary}[Finite-family CM extraction]\label{cor:finite-family-extraction}
Fix \(E,V,D,\Lambda\) and the principal lattice levels of
Section~\ref{sec:arithmetic}.
Let \(v_1,\ldots,v_m:G_E\to\{1,-1\}\) be odd with \(v_j(1)=1\).
For each \(j\), choose a finite family of finite adelic Schwartz inputs
in the multiplicity-line construction
\(d_j(\tau)=v_j(\tau^{-1})\).  Let \(X_{\mathrm{base}}\) be a
principal quotient at a torsion-free level in this framework.
There is a common deeper principal cover \(Y\to X_{\mathrm{base}}\),
with \(Y\) a smooth connected projective surface, on which every
associated theta one-form class from the \(j\)-th family belongs to
\[
\operatorname{span}_{\C}\left\{
f\bigl(U(v_j)_1\bigr):
f\in\Hom_{\mathrm{Hdg}}\bigl(U(v_j),H^1(Y,\Q)\bigr)
\right\}.
\]
The maps are rational Hodge maps, with their images viewed after
extension to \(\C\).  The conclusion persists after pullback to every
deeper principal level.  The auxiliary primes used for different
\(v_j\) may differ.
\end{corollary}

\begin{proof}
Apply Theorem~\ref{thm:cm-extraction} to each finite input family.
Choose a principal level contained in the resulting finitely many
principal levels and in the level of \(X_{\mathrm{base}}\), and deepen
it until Lemma~\ref{lem:reference-moduli} applies.  Its quotient \(Y\)
is smooth, connected, and projective.  The reference moduli construction
depends on \(V\), not on the multiplicity line, so it applies to this
common level.  The persistence in Theorem~\ref{thm:cm-extraction}
pulls all the rational-map spans to \(Y\) and then to every deeper
principal level.  Each application chooses its auxiliary prime only
after its own finite family and level have been fixed; the primes need
not agree.
\end{proof}
==== END The-rational-Hodge-conjecture-for-CM-abelian-varieties//sections/05c-cm-types.tex ====

==== BEGIN The-rational-Hodge-conjecture-for-CM-abelian-varieties//sections/06-assembly.tex ====
\section{Completion of the CM proof}\label{sec:assembly}

The theta construction gives a nonzero period on a compact ball
quotient.  The finite-family extraction corollary expresses the one-form
classes in that period through rational maps from the prescribed CM
modules, after passage to sufficiently deep levels.  We put these
statements on one projective surface and apply the criterion of
Section~\ref{sec:surface}.

\begin{proof}[Proof of Theorem~\ref{thm:switch}]
By Lemma~\ref{lem:switch-normalization}, it suffices to treat the case
\(s\geq2\) and \(v_i(1)=1\) for all four functions.  Use the common
Galois CM field \(E\), the active embedding \(\mu=1\), and the line
signs
\[
 d_i(\tau)=v_i(\tau^{-1}).
\]
They satisfy \(d_1+d_2=d_3+d_4\) and \(d_i(\mu)=1\).  Choose the
rank-three Hermitian space \(V\), the ball \(D\), and the rational
orthogonal decomposition \(W=W_1\perp W_2\) prescribed by \(d_1,d_2\),
as in Section~\ref{sec:theta}.

Proposition~\ref{prop:theta-top}(ii) supplies a total finite theta
input whose form for this decomposition is nonzero.  With that input
fixed, Corollary~\ref{cor:theta-switch-continuity} supplies a second
rational orthogonal decomposition and a nonzero mixed period.
Write \(X'\) for the individual-factor cover
constructed there, and \(\xi_1,\xi_2\) for the two pulled-back endpoint
forms.  On this common compact principal quotient they have finite expansions
\[
 \xi_1=\sum_{a\in I}c_a\,
       \Theta_1(\phi_{1a})\wedge\Theta_2(\phi_{2a}),
 \qquad
 \xi_2=\sum_{b\in J}c'_b\,
       \Theta_3(\phi_{3b})\wedge\Theta_4(\phi_{4b}),
\]
where the constants account for the chosen constant-line
trivializations, and
\[
 \int_{X'}\xi_1\wedge\overline{\xi_2}\ne0.
\]
All four families of individual inputs here are finite.  The quotient
and these expansions may be pulled to any deeper principal level;
the integral is then multiplied by the degree of the cover.

Apply Corollary~\ref{cor:finite-family-extraction} to these four families,
with \(X_{\mathrm{base}}=X'\) and the fixed \(E,V,D,\Lambda\).
It gives a common smooth connected projective principal cover
\(\pi:Y\to X'\) on which all four \(U(v_i)_1\)-span conclusions hold;
the auxiliary primes may differ.

Denote the pulled-back one-form classes by
\[
 \vartheta_{ia}=[\pi^*\Theta_i(\phi_{ia})]\in H^1(Y,\C),
\]
with \(a\in I\) for \(i=1,2\) and \(a\in J\) for \(i=3,4\).
These classes are represented by holomorphic one-forms.  Their cup
products compute the integrals of the wedge products, and the
covering formula gives
\begin{equation}\label{eq:assembly-period}
\begin{split}
 0&\ne\int_Y\pi^*\xi_1\wedge\overline{\pi^*\xi_2}\\
  &=\sum_{a\in I,\ b\in J}c_a\overline{c'_b}\,
    \left\langle
      \vartheta_{1a}\cup\vartheta_{2a}\cup
      \overline{\vartheta_{3b}}\cup\overline{\vartheta_{4b}},[Y]
    \right\rangle .
\end{split}
\end{equation}
Choose \(a,b\) for which the displayed cup-product period is nonzero,
and write its four holomorphic factors as \(\vartheta_i\).

Corollary~\ref{cor:finite-family-extraction} gives
\[
 \vartheta_i\in\operatorname{span}_{\C}
 \{h(U(v_i)_1):
       h\in\Hom_{\mathrm{Hdg}}(U(v_i),H^1(Y,\Q))\},
 \qquad 1\leq i\leq4.
\]
The maps are rational, so they commute with scalar complex
conjugation, which sends \(U(v_i)_1\) to \(U(v_i)_c\).  Consequently
\[
 \overline{\vartheta_i}\in\operatorname{span}_{\C}
 \{h(U(v_i)_c):
       h\in\Hom_{\mathrm{Hdg}}(U(v_i),H^1(Y,\Q))\},
 \qquad i=3,4.
\]
The four classes
\(\vartheta_1,\vartheta_2,\overline{\vartheta_3},
\overline{\vartheta_4}\) thus satisfy
Corollary~\ref{cor:surface-span} on the same projective surface \(Y\),
with the source labels \((1,1,c,c)\) and the nonzero period selected
from Equation~\eqref{eq:assembly-period}.  The normalized switch line
is therefore algebraic.  Lemma~\ref{lem:switch-normalization} returns
every switch to this case or
to a product of algebraic two-factor classes, and proves the theorem.
\end{proof}

\begin{proof}[Proof of Theorem~\ref{thm:cm}]
The four-factor switch theorem is now proved for every Galois CM field.
Proposition~\ref{prop:switch-reduction} consequently places every
rational Hodge class on a CM abelian variety in its rational cycle
span.  Its final scalar descent takes place on the original variety.
Finite products and powers are CM, so the same conclusion covers them.
\end{proof}
==== END The-rational-Hodge-conjecture-for-CM-abelian-varieties//sections/06-assembly.tex ====

==== BEGIN The-rational-Hodge-conjecture-for-CM-abelian-varieties//sections/07-consequences.tex ====
\section{Generalized Hodge and arithmetic consequences}
\label{cc:section}

Theorem~\ref{thm:cm} applies to products of CM abelian varieties and
therefore algebraizes rational Hodge correspondences between them.
We use this to obtain the algebraic supports predicted by the
generalized Hodge conjecture, following Abdulali's effective-realization
argument
\cite[Lemma~1 and Theorems~3--4]{Abdulali2005CM}. Hazama had earlier
reduced this implication to codimension-two Hodge classes on a special
family of CM abelian varieties
\cite[Theorem~8.3]{Hazama2003CM}. We then apply Milne's Hodge-to-Tate
and polarization theorems to obtain the Tate and Hodge standard
conjectures for abelian varieties.
A final application uses CM transport to algebraize specialized Hodge
classes on arbitrary good-reduction abelian varieties.

\subsection{The generalized Hodge conjecture}

A pure rational Hodge structure is \emph{effective} if all its nonzero
Hodge components have nonnegative indices. Recall that
\(V(r)^{p,q}=V^{p+r,q+r}\), so the twist \(V(r)\) has weight
\(\operatorname{wt}(V)-2r\). The Hodge circle of a pure Hodge structure
\(V\) acts on \(V^{p,q}\) by \(z^{p-q}\), for \(|z|=1\). Let \(J(V)\)
be the smallest algebraic subgroup of \(\GL(V)\) over \(\Q\) whose real
points contain that circle. A polarizable \(V\) is \emph{of CM type}
if \(J(V)\) is a torus, with the trivial group allowed. Every Hodge
substructure of the cohomology of a CM abelian variety is of CM type:
its Hodge-circle group is an image of the torus acting on degree-one
cohomology.

For a smooth projective complex variety \(X\), the geometric coniveau
filtration is
\[
 N^rH^k(X,\Q)=
 \sum_{\substack{Z\subset X\text{ closed}\\ \codim_X Z\ge r}}
 \ker\bigl(H^k(X,\Q)\longrightarrow H^k(X\setminus Z,\Q)\bigr).
\]
Grothendieck's corrected generalized Hodge conjecture says that a
rational Hodge substructure \(W\subset H^k(X,\Q)\) lies in
\(N^rH^k(X,\Q)\) whenever \(W(r)\) is effective; see
\cite[\S2]{Grothendieck1969General} and \cite[\S2]{Hazama2003CM}.
Equivalently, one closed subset of
codimension at least \(r\) supports all of \(W\). Indeed, finite
dimensionality permits taking a finite union of the supports needed
for a basis of \(W\).

We will obtain such a support by algebraizing a Hodge map from an
auxiliary CM abelian variety.  To get the required codimension, that
variety must realize an effective twist of \(W\) in cohomology of the
same weight; the twisting index will determine the codimension of the
resulting correspondence.  The next lemma constructs it by expressing
the Hodge indices at each embedding as sums of weight-one indices.

\begin{lemma}[Effective realization of a simple CM Hodge structure]
\label{cc:realization}
Let \(W\) be a nonzero simple polarizable rational Hodge structure of
CM type and weight \(k\). Put
\[
 r_0=\min\{p:W^{p,k-p}\ne0\},\qquad n=k-2r_0.
\]
Then \(n\ge0\), and there is a CM abelian variety \(B\), possibly a
point, and a surjective rational Hodge map
\[
 H^n(B,\Q)(-r_0)\longrightarrow W.
\]
\end{lemma}

\begin{proof}
Set \(U=W(r_0)\). Hodge symmetry shows that \(U\) is effective of
weight \(n\ge0\), with smallest Hodge index zero. If \(n=0\), every
type of \(U\) is \((0,0)\). Simplicity then gives \(U\simeq\Q\), and
we take \(B\) to be a point.

Suppose \(n>0\).  We first show that \(U\) is a line over a CM field;
we will then decompose its Hodge type into weight-one CM types.
Write \(T=J(U)\). Let \(E_0\) be the
\(\Q\)-subalgebra of \(\End_\Q(U)\) generated by the image of \(T\).
After splitting the torus, its action is diagonal by characters, so
\(E_0\) is a commutative finite étale algebra. Its field idempotents
and every subspace linear over one field factor are \(T\)-stable.
Simplicity of \(U\) therefore forces \(E_0\) to be a field \(E\) and
\(\dim_E U=1\).

We check that \(E\) is a CM field. Let the Weil operator \(C_U\) act
as \(i^{p-q}\) on \(U^{p,q}\). By polarizability there is a rational
Hodge pairing \(\psi\), with the conventional weight-dependent sign
absorbed in its definition, for which
\[
 q(x,y)=\psi(C_Ux,y)
\]
is positive definite on \(U_\R\). The adjoint \(\dagger\) for
\(\psi\) sends \(t\in T\) to \(t^{-1}\), hence preserves \(E\).
Every element of \(E\) commutes with \(C_U\), so \(\dagger\) is also
the adjoint for \(q\). On the real finite étale algebra \(E_\R\), it
must fix each primitive idempotent: if an idempotent \(e\) were sent
to a different one, then \(e^\dagger e=0\), contrary to
\(q(ex,ex)=q(x,e^\dagger ex)>0\) for some \(x\). On a complex factor
it must act by complex conjugation, since the identity would give
\(i^\dagger i=-1\), again contrary to positivity. Thus if
\(\dagger\) is trivial on \(E\), the field is totally real. If it is
nontrivial, a real embedding would force \(\dagger\) to be trivial
on \(E\), so there are no real embeddings; the fixed algebra in
\(E_\R\) is then a product of copies of \(\R\). Its fixed field is
therefore totally real, and \(E\) is CM.

The totally real case cannot occur here. The Hodge circle acts
through \(E_\R^\times\); its image in each real scalar factor is a
compact connected subgroup of \(\R^\times\), hence is trivial.
All Hodge types of \(U\) would then have equal indices. Since their
smallest index is zero, this would force \(n=0\). Consequently \(E\)
is CM, and \(\dagger\) is its complex conjugation.

Let \(\Sigma=\Hom(E,\C)\), and write \(\bar\sigma\) for the conjugate
of \(\sigma\). Since \(E\) acts by Hodge endomorphisms and \(U\) is an
\(E\)-line,
\[
 U_\C=\bigoplus_{\sigma\in\Sigma}U_\sigma,\qquad
 U_\sigma\text{ has type }(a_\sigma,n-a_\sigma),
\]
where \(0\le a_\sigma\le n\) and
\(a_\sigma+a_{\bar\sigma}=n\). Choose \(n\) CM types
\(\Phi_1,\ldots,\Phi_n\) of \(E\) with
\[
 \#\{j:\sigma\in\Phi_j\}=a_\sigma.
\]
This is possible independently for each conjugate pair: put
\(\sigma\) into any \(a_\sigma\) lists and \(\bar\sigma\) into the
remaining lists.

For each \(j\), give the rational vector space \(H_j=E\) the
weight-one Hodge structure whose \(\sigma\)-line has type \((1,0)\)
exactly when \(\sigma\in\Phi_j\).  The polarization and Riemann
construction of Section~\ref{sec:tensors}, surrounding
Equation~\eqref{eq:cm-types}, applies to any CM field, with no
Galois hypothesis.  It gives a CM abelian variety \(B_j\) with
\(H^1(B_j,\Q)\simeq H_j\).

Identify \(U\) and the \(H_j\) with \(E\) as \(E\)-lines. The rational
multiplication map
\[
 E^{\otimes_\Q n}\longrightarrow E,\qquad
 x_1\otimes\cdots\otimes x_n\longmapsto x_1\cdots x_n
\]
is surjective. Over \(\C\) it kills tensors with unequal embedding
labels, while the tensor with all labels equal to \(\sigma\) has type
\((a_\sigma,n-a_\sigma)\) and maps to \(U_\sigma\). It is therefore a
Hodge map to \(U\). For \(B=\prod_{j=1}^nB_j\), its source is the
rational Künneth summand \(\bigotimes_j H^1(B_j,\Q)\) of \(H^n(B,\Q)\).
Extending by zero on the other summands and twisting by \((-r_0)\)
proves the lemma.
\end{proof}

\begin{corollary}[Generalized Hodge conjecture for CM abelian varieties]
\label{cc:ghc}
Let \(A\) be a complex CM abelian variety. For integers \(k,r\ge0\)
and a rational Hodge substructure \(W\subset H^k(A,\Q)\) such that
\(W(r)\) is effective, there is a closed algebraic subset \(Z\subset A\)
of codimension at least \(r\) such that
\[
 W\subset\ker\bigl(H^k(A,\Q)\longrightarrow
                         H^k(A\setminus Z,\Q)\bigr).
\]
This includes every finite product of CM abelian varieties and every
power of such a product.
\end{corollary}

\begin{proof}
The zero substructure permits \(Z=\varnothing\). Otherwise,
semisimplicity of polarizable rational Hodge structures reduces the
claim to a simple summand \(W\), after which we take the finite union
of the resulting supports. Let \(r_0\) be its smallest Hodge index.
Then \(r_0\ge r\). Also \(r_0\le k/2\le\dim A\), since \(W\ne0\).
Put \(n=k-2r_0\), take \(B\) from
Lemma~\ref{cc:realization}, and write \(b=\dim B\). Composing the map
of that lemma with \(W\hookrightarrow H^k(A,\Q)\) gives
\[
 f:H^n(B,\Q)(-r_0)\longrightarrow H^k(A,\Q),
 \qquad \im(f)=W.
\]
By Poincaré duality, \(f\) corresponds to a rational Hodge class of
type \((b+r_0,b+r_0)\) in
\[
 H^{2b-n}(B,\Q)\otimes H^k(A,\Q)
 \subset H^{2b+2r_0}(B\times A,\Q).
\]
The product \(B\times A\) is CM. Theorem~\ref{thm:cm} therefore
represents this exact class by
\(\Gamma\in\CH^{b+r_0}(B\times A)_\Q\), whose action in degree \(n\)
is \(f\).

Choose a finite rational cycle representing \(\Gamma\), and let
\(Z\) be the projection of its support to \(A\). Every component of
that cycle has dimension
\[
 \dim(B\times A)-(b+r_0)=\dim A-r_0.
\]
The projection is proper, so \(Z\) is closed and
\(\codim_A Z\ge r_0\ge r\). The correspondence restricts to zero on
\(B\times(A\setminus Z)\); functoriality of its cohomological action
therefore makes its image \(W\) restrict to zero on \(A\setminus Z\).
\end{proof}

\subsection{The Tate conjecture over finite fields}

Let \(A\) be an abelian variety over \(\Fq\), put
\(\bar A=A\times_{\Fq}\overline{\Fq}\), and write
\(G_q=\Gal(\overline{\Fq}/\Fq)\). For a prime
\(\ell\ne\operatorname{char}(\Fq)\), the rational Tate conjecture in
codimension \(r\) asks whether cycles over \(\Fq\) span
\(H^{2r}_{\mathrm{\acute et}}(\bar A,\Q_\ell(r))^{G_q}\).

\begin{corollary}[Tate conjecture for abelian varieties over finite fields]
\label{cc:tate}
For every abelian variety \(A/\Fq\), every integer \(r\ge0\), and every
prime \(\ell\ne\operatorname{char}(\Fq)\), the cycle-class map
\[
 \CH^r(A)\otimes_\Z\Q_\ell\longrightarrow
 H^{2r}_{\mathrm{\acute et}}(\bar A,\Q_\ell(r))^{G_q}
\]
is surjective. Numerical equivalence and \(\ell\)-adic homological
equivalence coincide on rational algebraic cycles on \(\bar A\).
\end{corollary}

\begin{proof}
Apply Milne's Theorem~7.1 and his formulation (0.1)
\cite{Milne1999Tate}, with the correction in
\cite{cc:Milne1999Addenda}: the cofinal auxiliary Galois CM fields
are enlarged to contain a real subfield of degree \(>2\) inert at
the residue prime. The theorem's hypothesis is supplied by
Theorem~\ref{thm:cm}. It gives the stated equality of
equivalence relations and algebraic spanning of the geometric Tate
classes: those fixed over some finite field of definition.

Let \(\alpha\) lie in the displayed \(G_q\)-invariant space. The
geometric conclusion writes \(\alpha\) as a finite
\(\Q_\ell\)-linear combination of cycle classes. All the cycles in
this expression descend to \(A_{\mathbb F_{q^m}}\) for one \(m\);
let \(z_m\in\CH^r(A_{\mathbb F_{q^m}})\otimes_\Z\Q_\ell\) denote
the resulting combination. For the finite projection
\(\pi_m:A_{\mathbb F_{q^m}}\to A\), proper pushforward and base
change for cycle classes give
\[
 \operatorname{cl}_\ell\!\left(\frac1m(\pi_m)_*z_m\right)
 =\frac1m\sum_{\sigma\in
        \Gal(\mathbb F_{q^m}/\Fq)}\sigma(\alpha)=\alpha.
\]
Thus \(m^{-1}(\pi_m)_*z_m\) is the required class over \(\Fq\).
Division by \(m\) is valid in \(\Q_\ell\), even when \(m\) is
divisible by \(\ell\) or by the characteristic.
\end{proof}

Milne's implication compares CM and finite-field abelian motives. It
does not assert that each Tate class lifts to a Hodge class on a
single CM lifting, even up to isogeny
\cite[Introduction, Remark~(a)]{Milne1999Tate}.

\subsection{The Hodge standard conjecture}

Let \(A\) be an abelian variety of dimension \(g\) over an
algebraically closed field \(K\), and let
\(A^r_{\mathrm{num}}(A)_\Q\) be the space of rational
codimension-\(r\) cycles modulo numerical equivalence. A
\emph{rational ample divisor class} is an
\(h\in A^1_{\mathrm{num}}(A)_\Q\) for which some positive integral
multiple is the numerical class of an ample Cartier divisor. For
integers \(0\le r\le g/2\), put
\[
 P_h^r(A)=\ker\!\left(
 h^{g-2r+1}:A^r_{\mathrm{num}}(A)_\Q
 \longrightarrow A^{g-r+1}_{\mathrm{num}}(A)_\Q\right).
\]

\begin{corollary}[Hodge standard conjecture for abelian varieties]
\label{cc:standard}
Let \(K\) be an algebraically closed field and \(A/K\) an abelian
variety of dimension \(g\). For every prime
\(\ell\ne\operatorname{char}(K)\), numerical
and \(\ell\)-adic homological equivalence coincide on rational
algebraic cycles on \(A\). For every rational ample divisor class
\(h\) and every integer \(0\le r\le g/2\), the rational symmetric form
\[
 P_h^r(A)\times P_h^r(A)\longrightarrow\Q,\qquad
 (x,y)\longmapsto(-1)^r\deg(xy\,h^{g-2r})
\]
is positive definite after extension of scalars to \(\R\). Thus the
Hodge standard conjecture for \(\ell\)-adic étale cohomology holds
for abelian varieties in arbitrary characteristic.
\end{corollary}

\begin{proof}
Apply Milne's Theorem~3.3 \cite{Milne2002Polarizations} using
Theorem~\ref{thm:cm}. In Milne's proof, the same Hodge hypothesis
supplies the compatibility and positivity of the relevant
polarizations through Theorem~2.1, read with the author's addendum
\cite{cc:Milne2002Addendum} and the corrected auxiliary-field choice
from \cite{cc:Milne1999Addenda}. Theorem~3.3 gives both the equality
of equivalence relations and positivity on the primitive space of
cycles modulo homological equivalence. The former identifies that
primitive space with the numerical one above.

The case \(g=0\) is immediate. For \(g>0\), choose a positive integer
\(m\) so that \(mh\) is the class of a smooth hyperplane section;
clearing denominators, taking a sufficiently large very ample
multiple, and Bertini's theorem permit this. Multiplication by the
nonzero rational scalar \(m^{g-2r+1}\) does not change the primitive
kernel, and the form for \(mh\) is \(m^{g-2r}\) times the form for
\(h\). This positive factor preserves positive definiteness.
\end{proof}

\subsection{Specialization of Hodge classes}

\begin{remark}[Algebraicity of specialized Hodge classes]
\label{cc:algebraic-specialization}
Theorem~\ref{thm:cm} also combines with the CM transport of
\cite[Corollary~3.4]{OpenAIMilneRationality2026} to give a
specialization statement for abelian varieties that need not be CM.
Fix a prime \(p\) and a \(p\)-adic place of \(\Qbar\).
If \(A/\Qbar\) has good reduction \(A_0/\overline{\mathbb F}_p\)
at that place, every rational Hodge class on \(A\) specializes to
the class of one rational algebraic cycle on \(A_0\), simultaneously
in all prime-to-\(p\) realizations and in crystalline cohomology
after extension to the completed field at the chosen place.
Indeed, the transport writes that specialization as the pullback of
a specialized CM Hodge class by a rational homomorphism of special
fibers. Theorem~\ref{thm:cm} algebraizes the CM class; specialization
and Chow pullback after clearing the homomorphism's denominator give
the required cycle. See
\cite[Corollary~6.4]{OpenAIMilneRationality2026} for the
field-of-definition argument and precise realization conventions,
including \(p=2\). This application uses the CM transport, not the
independent rationality proof of that paper, and is not an input to
Theorem~\ref{thm:cm}. It asserts algebraicity on the special fiber,
not the Hodge conjecture for arbitrary complex abelian varieties.
\end{remark}
==== END The-rational-Hodge-conjecture-for-CM-abelian-varieties//sections/07-consequences.tex ====

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@article{Milne1999Tate,
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@incollection{Milne1986AbelianVarieties,
  author = {Milne, J. S.},
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          unchanged theorem numbering; locators refer to this version}
}

@article{Milne2002Polarizations,
  archiveprefix = {arXiv},
  author = {Milne, J. S.},
  doi = {10.2307/3062126},
  eprint = {math/0103175v3},
  url = {https://arxiv.org/abs/math/0103175v3},
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@misc{Milne2022,
  archiveprefix = {arXiv},
  author = {Milne, J. S.},
  eprint = {2010.08857},
  note = {\href{https://www.jmilne.org/math/articles/HAV.pdf}{Version 1.1, July 12, 2022}},
  primaryclass = {math.AG},
  title = {Hodge classes on abelian varieties},
  url = {https://www.jmilne.org/math/articles/HAV.pdf},
  year = {2022}
}

@incollection{MilneShimura2017,
  address = {Providence, RI},
  author = {Milne, J. S.},
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  note = {\href{https://www.jmilne.org/math/xnotes/svi.pdf}{Revised September 16, 2017}; locators refer to this version},
  pages = {265--378},
  publisher = {American Mathematical Society},
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}

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@article{Scholze2013RigidAnalyticHodge,
  author = {Scholze, Peter},
  doi = {10.1017/fmp.2013.1},
  journal = {Forum of Mathematics, Pi},
  note = {77 pp.; see Corollary 3.17(i), p. 27, Proposition 8.2, p. 66, and Theorems 9.1 and 9.3, pp. 74--75; see also the 2016 corrigendum},
  pages = {e1},
  title = {{$p$}-adic {Hodge} theory for rigid-analytic varieties},
  url = {https://www.cambridge.org/core/journals/forum-of-mathematics-pi/article/padic-hodge-theory-for-rigidanalytic-varieties/9C5E8381BE3CD320ECADDDEAD2616B55},
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@article{Scholze2016RigidAnalyticHodgeCorrigendum,
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  doi = {10.1017/fmp.2016.4},
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  note = {4 pp.; the corrigendum does not change the Section 9 comparisons used here},
  pages = {e6},
  title = {{$p$}-adic {Hodge} theory for rigid-analytic varieties---corrigendum},
  url = {https://www.cambridge.org/core/journals/forum-of-mathematics-pi/article/padic-hodge-theory-for-rigidanalytic-varieties-corrigendum/423022854D448D337C999BCD505D8F75},
  volume = {4},
  year = {2016}
}

@misc{OpenAIMilneRationality2026,
  author = {{OpenAI}},
  title = {{Milne's rationality conjecture for abelian varieties}},
  howpublished = {OpenAI Math Release preprint
                  \href{https://github.com/openai/math/blob/main/preprints/Milnes-rationality-conjecture-for-abelian-varieties-September-23-2026/paper.pdf}{OAI:Milnes-rationality-conjecture-for-abelian-varieties-September-23-2026}},
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@misc{StacksProject,
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@article{Weil1964,
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@misc{cc:Milne1999Addenda,
  author = {Milne, J. S.},
  howpublished = {Author's corrections, \url{https://www.jmilne.org/math/articles/1999b.html}},
  note = {Consulted September 30, 2026},
  title = {Addenda to {Lefschetz} motives and the {Tate} conjecture},
  url = {https://www.jmilne.org/math/articles/1999b.html}
}

@misc{cc:Milne2002Addendum,
  author = {Milne, J. S.},
  howpublished = {Author's addendum, \url{https://www.jmilne.org/math/articles/2002a.html}},
  note = {Consulted September 30, 2026},
  title = {Addendum to {Polarizations} and {Grothendieck}'s standard conjectures},
  url = {https://www.jmilne.org/math/articles/2002a.html}
}
==== END The-rational-Hodge-conjecture-for-CM-abelian-varieties//references.bib ====