\documentclass[11pt]{amsart} \usepackage{amsmath,amssymb,amsthm} \usepackage{url} \usepackage{longtable} \usepackage{tikz} \usetikzlibrary{calc} \setlength{\emergencystretch}{3em} \newtheorem{theorem}{Theorem}[section] \newtheorem{proposition}[theorem]{Proposition} \newtheorem{lemma}[theorem]{Lemma} \newtheorem{corollary}[theorem]{Corollary} \newtheorem*{theoremA}{Theorem A} \newtheorem*{theoremB}{Theorem B} \newtheorem*{corollaryC}{Corollary C} \theoremstyle{definition} \newtheorem{definition}[theorem]{Definition} \newtheorem{example}[theorem]{Example} \theoremstyle{remark} \newtheorem{remark}[theorem]{Remark} \newcommand{\Q}{\mathbb Q} \newcommand{\Z}{\mathbb Z} \newcommand{\C}{\mathbb C} \newcommand{\R}{\mathbb R} \newcommand{\PP}{\mathbb P} \newcommand{\LL}{\mathbb L} \newcommand{\End}{\operatorname{End}} \newcommand{\Hom}{\operatorname{Hom}} \newcommand{\Nm}{\operatorname{Nm}} \newcommand{\disc}{\operatorname{disc}} \newcommand{\rk}{\operatorname{rk}} \newcommand{\Tr}{\operatorname{Tr}} \newcommand{\age}{\operatorname{age}} \newcommand{\Hur}{\mathrm{Hur}} \newcommand{\cHn}{\mathcal H_n} \newcommand{\cM}{\mathcal M} \newcommand{\cD}{\mathcal D} \newcommand{\cI}{\mathcal I} \newcommand{\cS}{\mathcal S} \newcommand{\cT}{\mathcal T} \newcommand{\fd}{\mathfrak d} \newcommand{\Vp}{V_{\rm par}} \newcommand{\hpar}{h_{\rm par}} \title[Weil classes on non-split sixfolds]{Algebraicity of Weil classes on abelian sixfolds of Weil type for $\Q(\sqrt{-3})$ with discriminant $-2$} \author{Benjamin Severn} \email{benzsevern@gmail.com} \subjclass[2020]{Primary 14C30; Secondary 14K20, 14H30, 14D07} \keywords{Hodge conjecture, Weil classes, abelian varieties of Weil type, complex reflection groups, period maps, isomonodromic deformations} \date{21 October 2026 (draft version 0.16)} \begin{document} \begin{abstract} Let $K=\Q(\sqrt{-3})$. We prove that the Weil classes of every polarized abelian sixfold of Weil type for $K$ with Hermitian discriminant $-2$ are algebraic. This discriminant is not split, so the theorems of Schoen and of Markman on split Weil classes do not apply. The statement also follows from a companion paper, which treats every discriminant through split eightfolds and a degeneration; the proof here is independent of it, uses no eightfolds and no degeneration, takes every cycle from Schoen's theorem on cyclic covers, and rests on exact certificates only. The sixfolds come from Galois covers of the projective line with group the complex reflection group $G_{32}$, branched over thirteen points: the multiplicity space of the reflection representation in the first cohomology of such a cover is the first cohomology of an abelian sixfold of Weil type. On every such sixfold the Weil classes are algebraic: they are obtained from Schoen's algebraic Weil classes on fifty-one cyclic quotients of the cover, through a correspondence that Milne's theorem on Lefschetz classes makes algebraic, and the choice of quotients is governed by an integral identity among characters, checked by an exact computation. The family of these sixfolds fills an open set of the nine-dimensional moduli space: along an explicit isomonodromic deformation of one explicit cover, a $9\times9$ minor of the codifferential of the period map has a nonzero Taylor coefficient of order seven, an exact rational number. A Hilbert-scheme and Baire-category argument then reaches the whole moduli space. The discriminant of the family is computed exactly. For the members whose Hodge group is the full special unitary group, every Hodge class on every power is algebraic. \end{abstract} \maketitle %% ------------------------------------------------------------------ \section{Introduction} %% ------------------------------------------------------------------ \subsection{Weil classes} Let $A$ be a complex abelian variety of dimension $2n$ with an action $\iota\colon K\to\End(A)\otimes\Q$ of an imaginary quadratic field $K$, such that the two embeddings of $K$ in $\C$ each occur with multiplicity $n$ on the tangent space. Then the $K$-line \[ W_K(A)=\textstyle\bigwedge^{2n}_KH^1(A,\Q)\subset H^{2n}(A,\Q) \] is a two-dimensional space of Hodge classes of type $(n,n)$, the \emph{Weil classes} of $A$ \cite{Weil}. For a general such $A$ they are not in the ring generated by divisor classes \cite[\S5]{vanGeemen}. A polarization compatible with $\iota$ determines a $K$-Hermitian form $H$ of signature $(n,n)$ on $H_1(A,\Q)$, and its discriminant $\disc H\in\Q^\times/\Nm K^\times$ is an invariant of the polarized rational Hodge structure. The polarized abelian $2n$-folds of Weil type for $K$ with a fixed discriminant form one family, parametrized by a quotient of the Hermitian symmetric domain of $SU(n,n)$, of dimension $n^2$. The class $(-1)^n$ is called \emph{split}. Let $K=\Q(\sqrt{-3})$. The rational primes $p\equiv2\pmod3$ are inert in $K$, so a norm from $K$ has even valuation at each of them. For sixfolds the split class is $-1$, and the class $-2$ is not split: $-2$ and $-2/(-1)=2$ have odd valuation at the inert prime $2$, so neither is a norm. \subsection{What was known} For abelian fourfolds of Weil type the Weil classes are algebraic for every $K$ and every discriminant (Markman \cite{Markman}). For sixfolds the split case is known: Schoen for $\Q(\sqrt{-3})$ \cite{Schoen98}, and Markman for every $K$ \cite{Markman}. The companion papers treat further non-split cases by other routes: \cite{SevernQ3} proves algebraicity for sixfolds of every discriminant for $K=\Q(\sqrt{-3})$, through split eightfolds and Schoen's product argument, and \cite{SevernQi} treats the class $-3$ for $\Q(i)$. In particular Theorem~A below is also a consequence of \cite{SevernQ3}. The present paper gives an independent proof. It uses no eightfolds, no product argument and no boundary degeneration; every cycle it uses comes from Schoen's theorem on cyclic covers, transported by a correspondence; and its three certificates (the character identity, the dominance and the discriminant) are exact. The method needs only a finite group with a suitable representation and an integral character identity, so it applies to class data where the degeneration of \cite{SevernQ3} is not available; Remark~\ref{rem:witt} explains why no degeneration of the $G_{32}$ family reaches the class $-2$. Sections~\ref{sec:weil}, \ref{sec:covers} and \ref{sec:spread}, the set-up of Section~\ref{sec:supply}, Lemma~\ref{lem:schoenR}, Lemma~\ref{lem:cocycle} and Lemma~\ref{lem:sl} adapt the corresponding parts of \cite{SevernQ3} and \cite{SevernQi}. They are stated here with proofs, so that this paper can be read alone. \subsection{Results} \begin{theoremA} Let $K=\Q(\sqrt{-3})$. The Weil classes of every polarized abelian sixfold of Weil type for $K$ with Hermitian discriminant $-2$ in $\Q^\times/\Nm K^\times$ are algebraic. \end{theoremA} \begin{theoremB} Let $A$ be an abelian sixfold as in Theorem~A whose Hodge group is $SU_H$, the special unitary group of its Hermitian form. Then every Hodge class on every power $A^m$, $m\ge1$, is algebraic. \end{theoremB} The hypothesis of Theorem~B holds outside a countable union of proper closed subvarieties of the moduli space \cite[Theorem~6.11]{vanGeemen}. For $m=1$ it is the Hodge conjecture for $A$ itself: by \cite[Theorem~6.12]{vanGeemen} the Hodge classes of $A$ are then generated by divisor classes and the Weil classes. \begin{corollaryC}[Corollary~\ref{cor:typeIII}] Let $A$ be a simple abelian sixfold of type III whose endomorphism algebra $D$ contains $K$, and whose induced Hermitian discriminant for $K$ is $-2$. Then $W_K(A)$ is algebraic. If moreover the PEL family of $A$ is effective in Abdulali's sense (its arithmetic group is torsion free and its generic member has endomorphism algebra $D$; Section~\ref{sec:typeIII}), then the Hodge conjecture holds for every power of every general member of that family, that is, every member whose Hodge group is that of the generic member. This includes families with $D=(-1,-1)_\Q$ for which no imaginary quadratic subfield of $D$ gives a split discriminant (Remark~\ref{rem:typeIII}). \end{corollaryC} \subsection{The construction in one page}\label{subsec:map} \emph{The family.} Let $G\subset GL_4(K)$ be the Shephard--Todd group $G_{32}$, of order $155\,520$, generated by the four matrices of Appendix~\ref{app:tuple}, and $\rho$ its natural representation on $K^4$, with character field $K$. Its reflections have order three, with nontrivial eigenvalue $\omega$ (class $\mathcal R_1$) or $\omega^2$ (class $\mathcal R_2$), $\omega=e^{2\pi i/3}$. Let $\cI$ be the class of elements with eigenvalues $(1,1,-1,-1)$. We consider Galois covers $C\to\PP^1$ with group $G$, branched over thirteen points, with local monodromy in $\mathcal R_1$ at six of them, in $\mathcal R_2$ at six, and in $\cI$ at one. Such a cover has genus $505\,441$, by Riemann--Hurwitz. By the Chevalley--Weil formula, $\rho$ occurs in $H^1(C)$ with multiplicity six and Hodge numbers $(3,3)$ (Lemma~\ref{lem:sixfold}); the multiplicity space is the first cohomology of an abelian sixfold $A_\rho(C)\subset J(C)$ of Weil type for $K$, the image of an endomorphism of $J(C)$ given by an element of $\Q[G]$. Without the point in $\cI$ the multiplicity would be four, giving fourfolds. Thirteen points give a family of dimension $13-3=10$, one more than the dimension $9$ of the moduli space. \emph{Cycles (Section~\ref{sec:supply}).} For a subgroup $H'\le G$ and a character $\chi$ of $H'$ of order $m\ge3$, the curve $C/\ker\chi$ is a cyclic cover of $C/H'$. When its branch exponents pair off as $\alpha,-\alpha$, Schoen proved that the Weil classes of the primitive part of its Jacobian are algebraic \cite{Schoen88}. That primitive part is a factor of $J(C)$, and the centralizer $U$ of $G$ in the symplectic group of $H^1(C)$ acts on its Weil classes through characters of the abelianization of $U$; it acts on the Weil line $\det H^1(A_\rho)_\rho$ through a character $e_\rho$. If $e_\rho$ is an integer combination of the characters of such pairs, a product of the corresponding Schoen classes, contracted against a $U$-invariant class, maps onto the Weil classes of $A_\rho(C)$, and Milne's theorem on Lefschetz classes \cite{Milne} makes the contracting class algebraic. For $G_{32}$ an identity with fifty-one pairs holds (Proposition~\ref{prop:lattice}); it is printed in Appendix~\ref{app:witness}. \emph{Dominance (Section~\ref{sec:dominance}).} An explicit cover $C_0$ is the pullback of a rigid rank-four Fuchsian system on $\PP^1$ minus four points along a rational function of degree six. At $C_0$ the codifferential of the period map, a $9\times10$ matrix over $\Q$, has rank only $5$. Moving the twelve reflection points of $C_0$ along a Galois-equivariant rational direction and solving the Schlesinger equations as power series in the deformation parameter $\theta$, all Taylor coefficients lie in $\Q$; a $9\times9$ minor of the codifferential has its first nonzero coefficient at $\theta^7$, an exact rational number (Proposition~\ref{prop:rank}). So the period map has rank $9$ at nearby covers. \emph{Discriminant and spreading (Sections~\ref{sec:disc} and \ref{sec:spread}).} The Hermitian form of $A_\rho(C)$ is computed from the monodromy as a parabolic cup product, exactly over $K$, with determinant $-2/81$ (Proposition~\ref{prop:disc}). The set of points of the moduli space where the Weil classes are algebraic is a countable union of closed algebraic subsets and contains an open set, so it is everything. \subsection{What is new, and the inputs} The mechanisms are known: Schoen's cycles on cyclic covers \cite{Schoen88}, their transfer to a non-abelian factor (van Geemen and Verra \cite[Corollaries~4.9 and~4.10]{vGV}, who use one unramified cyclic subcover of order four and its quaternionic translates), Lefschetz classes \cite{Milne}, and dominance of a family of Prym varieties followed by spreading \cite{Koike}. New here are the $G_{32}$ family, the integral character identity with fifty-one cyclic quotients that combines Schoen's classes into the Weil classes of its factor, an exact certificate of its dominance, and the exact computation of its discriminant. \emph{Published results used:} Schoen's theorem \cite[Theorem~2.0 and Corollary~3.1]{Schoen88}, with a replacement for one step (Lemma~\ref{lem:schoenR}); Milne's theorem on Lefschetz classes \cite[Theorem~3.2]{Milne}; Katz's rigidity theorem \cite[Theorem~1.1.2]{Katz}; Landherr's theorem and van Geemen's description of the moduli \cite{vanGeemen}; the Chevalley--Weil formula and the codifferential of the period map of a family of Galois covers \cite{FGP}; the indices of the maximal subgroups of $PSp_4(3)$ \cite{ATLAS}; Abdulali's theorem on type III \cite[Theorem~4.1]{AbdulaliIII}; the first fundamental theorem for $SL_n$ \cite[\S8.4]{KraftProcesi}; and standard sheaf theory (proper base change, relative Hilbert schemes \cite{VoisinII}). \emph{Finite computations,} each exact and described in the appendices: the order and generation of $G$, and the group-theoretic inputs of the monodromy argument (Appendix~\ref{app:tuple}); the character identity of Proposition~\ref{prop:lattice} (Appendix~\ref{app:witness}); the rank certificate of Proposition~\ref{prop:rank} (Appendix~\ref{app:rank}); and the Hermitian matrix of Proposition~\ref{prop:disc}. All of them run in characteristic zero, without floating point. \subsection{Organization} Section~\ref{sec:weil} recalls abelian varieties of Weil type and their moduli, and Section~\ref{sec:covers} constructs the factor $A_\rho(C)$ and the $G_{32}$ family. Section~\ref{sec:supply} proves that the Weil classes of every member are algebraic. Section~\ref{sec:dominance} proves dominance, Section~\ref{sec:disc} computes the discriminant, and Section~\ref{sec:spread} spreads algebraicity to the moduli space and proves Theorem~A. Section~\ref{sec:typeIII} treats type III and Section~\ref{sec:powers} proves Theorem~B. The appendices give the data. \begin{center} \small \begin{tabular}{lp{0.68\textwidth}} \multicolumn{2}{l}{\emph{Main notation}}\\\hline $K$, $\omega$, $\delta$ & $\Q(\sqrt{-3})$, $e^{2\pi i/3}$, $\delta=\sqrt{-3}=1+2\omega$\\ $G$, $\rho$ & the group $G_{32}\subset GL_4(K)$ and its natural representation on $V_\rho=K^4$\\ $\mathcal R_1$, $\mathcal R_2$, $\cI$ & reflections with eigenvalue $\omega$, with eigenvalue $\omega^2$; elements with eigenvalues $(1,1,-1,-1)$\\ $\Hur$, $\Hur^\circ$ & covers with the class datum of Definition~\ref{def:datum}; the component through $C_0$\\ $A_\rho(C)$, $M_\tau$, $m_\tau$ & the Weil-type factor; multiplicity space and multiplicity of $\tau$ in $H^1(C)$\\ $U$, $T$, $X^*(T)$ & centralizer of $G$ in $Sp(H^1(C))$, its maximal abelian quotient, its characters\\ $(H',\chi)$, $a(\chi)$, $e_\rho$ & a supplier, its character in $X^*(T)$, the character of $\det M_\rho$\\ $\cD_\rho(C)$, $\cD_{\bar\rho}(C)$, $\mu_C$ & $\Hom_G(\rho,H^0(K_C))$, $\Hom_G(\bar\rho,H^0(K_C))$, the codifferential\\ $\LL$, $\varphi$, $C_0$ & the rigid local system on $\PP^1_z$, the map of degree six, the base cover\\ $\cM$, $\cHn$ & the moduli space and the period domain \end{tabular} \end{center} %% ------------------------------------------------------------------ \section{Abelian varieties of Weil type}\label{sec:weil} %% ------------------------------------------------------------------ Throughout, $K$ is an imaginary quadratic field and $\vartheta\in K$ a nonzero element with $\bar\vartheta=-\vartheta$; for $K=\Q(\sqrt{-3})$ we take $\vartheta=\delta=\sqrt{-3}$. \begin{definition}\label{def:weiltype} A \emph{polarized abelian $2n$-fold of Weil type for $K$} is a triple $(A,\iota,E)$: a complex abelian variety $A$ of dimension $2n$, an embedding $\iota\colon K\to\End(A)\otimes\Q$ such that both embeddings of $K$ occur with multiplicity $n$ on $T_0A$, and a polarization $E$ (an alternating form on $H_1(A,\Q)$) with $E(\iota(a)x,y)=E(x,\iota(\bar a)y)$. \end{definition} Following van Geemen \cite[Lemma~5.2]{vanGeemen}, the form $H(x,y)=E(x,\iota(\vartheta)y)+\vartheta E(x,y)$ is $K$-Hermitian of signature $(n,n)$ on the $2n$-dimensional $K$-space $H_1(A,\Q)$. Its discriminant $\disc H\in\Q^\times/\Nm K^\times$ does not depend on $\vartheta$, and it is unchanged by a $K$-linear isogeny that multiplies the polarization by a positive rational number, since such a change multiplies $\det H$ by a $2n$-th power of a rational number, which is a norm. By Landherr's theorem \cite[\S5.4]{vanGeemen}, a nondegenerate $K$-Hermitian space is determined up to isometry by its dimension, signature and discriminant; those of signature $(n,n)$ with an $n$-dimensional totally isotropic subspace are those of discriminant $(-1)^n$. We write $SU_H$ for the special unitary group of $(H_1(A,\Q),H)$, an algebraic group over $\Q$. \subsection{Moduli}\label{subsec:moduli} Fix a $K$-Hermitian space $(V,H)$ of dimension $2n$, signature $(n,n)$ and discriminant $\fd$, and an $\mathcal O_K$-stable lattice $\Lambda\subset V$ on which the alternating form $\Tr_{K/\Q}(H/2\vartheta)$, scaled by a positive rational, is integral. The space $\cHn$ of $n$-dimensional complex subspaces of $V_\R$ (for the complex structure $\iota(\vartheta)/|\vartheta|$) on which $H$ is positive is the Hermitian symmetric domain of $SU(n,n)$, connected of dimension $n^2$. A point of $\cHn$ determines a Hodge structure of weight one on $V$ of Weil type for $K$, polarized by $\Tr_{K/\Q}(H/2\vartheta)$, and every such polarized Hodge structure arises in this way \cite[5.10 and the proof of 6.11]{vanGeemen}. For $\ell\ge3$ the congruence subgroup $\Gamma(\ell)$ of the unitary group of $(\Lambda,H)$ is torsion free, and $\cM=\Gamma(\ell)\backslash\cHn$ is a smooth connected quasi-projective variety with a universal abelian scheme $f\colon\mathcal A\to\cM$ with $K$-action \cite[\S\S5.5--5.11]{vanGeemen}. \begin{lemma}[Isogeny invariance]\label{lem:isogeny} If $\psi\colon A'\to A$ is a $K$-linear isogeny, then $W_K(A)$ is spanned by algebraic classes if and only if $W_K(A')$ is. Every polarized abelian $2n$-fold of Weil type for $K$ with discriminant $\fd$ is $K$-isogenous to a fibre of $\mathcal A\to\cM$. \end{lemma} \begin{proof} Pullback and pushforward by $\psi$ are algebraic correspondences, and $\psi^*$ maps $W_K(A)$ isomorphically onto $W_K(A')$, since it is a $K$-linear isomorphism on $H^1(-,\Q)$. For the second statement, Landherr's theorem gives a $K$-isometry from $(H_1(A,\Q),H)$ to $(V,H)$; the image of $H_1(A,\Z)$ is commensurable with $\Lambda$, so an integer multiple of it lies in $\Lambda$, and multiplication by that integer is a $K$-linear isogeny from $A$ to the fibre at the point given by the complex structure of $A$. \end{proof} %% ------------------------------------------------------------------ \section{The Weil-type factor and the $G_{32}$ family}\label{sec:covers} %% ------------------------------------------------------------------ \subsection{Hurwitz spaces} Fix a finite group $G$ and conjugacy classes $\mathcal C_1,\dots,\mathcal C_r$ of $G$, $r\ge4$. The \emph{Nielsen class} is the set of tuples $(g_1,\dots,g_r)$ with $g_j\in\mathcal C_j$, $g_1\cdots g_r=1$ and $\langle g_1,\dots,g_r\rangle=G$. By the Riemann existence theorem, a connected Galois cover $C\to\PP^1$ with group $G$, branched over $r$ ordered points $p_1,\dots,p_r$, is the same as such a tuple up to simultaneous conjugation, once standard generators of $\pi_1(\PP^1-\{p_j\})$ are fixed: $g_j$ is the monodromy along a loop around $p_j$. More generally, a homomorphism $\pi_1(\PP^1-\{p_j\})\to G$ that is onto determines a connected Galois $G$-cover. The Hurwitz space $\Hur$ of such covers is finite and \'etale over the configuration space of $r$ distinct ordered points; its connected components are smooth, hence irreducible, of dimension $r-3$ modulo $PGL_2$ \cite{FriedVolklein}. For an irreducible complex representation $\tau$ of $G$ write $H^1(C,\C)=\bigoplus_\tau\tau\otimes M_\tau$ and $m_\tau=\dim M_\tau$, with $M_\tau=\Hom_G(\tau,H^1(C,\C))$. Since $G$ acts by morphisms of Hodge structures, $M_\tau$ is a complex vector space with a decomposition $M_\tau=\Hom_G(\tau,H^{1,0})\oplus\Hom_G(\tau,H^{0,1})$; it is not in general defined over $\Q$. For a representation realizable over $K$, as in Subsection~\ref{subsec:pt}, $M_\tau$ is the complexification of a $K$-space $M$ (proof of Lemma~\ref{lem:weiltype}) along one embedding of $K$. For $g\in G$ write $c_\tau(g)=\rk(\tau(g)-1)$, and $\age\tau(g)=\sum_k\alpha_k$ when the eigenvalues of $\tau(g)$ are $e^{2\pi i\alpha_k}$ with $0\le\alpha_k<1$. \begin{proposition}\label{prop:mult} If $\tau$ is irreducible and nontrivial of dimension $d$, then \[ m_\tau=-2d+\sum_{j=1}^rc_\tau(g_j),\qquad \dim\Hom_G(\tau,H^0(C,K_C))=\sum_{j=1}^r\age\tau(g_j^{-1})-d. \] \end{proposition} \begin{proof} Let $U_0=\PP^1-\{p_j\}$ and $\mathbb V$ the local system on $U_0$ attached to $\tau$. Then $m_\tau$ is the dimension of the parabolic cohomology $H^1(\PP^1,j_*\mathbb V)$, the image of $H^1_c(U_0,\mathbb V)\to H^1(U_0,\mathbb V)$. Since $\tau$ is irreducible and nontrivial, $H^0(U_0,\mathbb V)=0$, and $H^2(U_0,\mathbb V)=0$ because $U_0$ is an affine curve; so $\dim H^1(U_0,\mathbb V)=(r-2)d$. Likewise $H^0_c=H^2_c=0$, the latter by duality with $H^0(U_0,\mathbb V^\vee)=0$, so $\dim H^1_c(U_0,\mathbb V)=(r-2)d$. The long exact sequence for $U_0$ and small punctured discs $\Delta_j^*$ around the $p_j$ begins \[ 0=H^0(U_0,\mathbb V)\to\textstyle\bigoplus_jH^0(\Delta_j^*,\mathbb V)\to H^1_c(U_0,\mathbb V)\to H^1(U_0,\mathbb V)\to\cdots, \] so the image of $H^1_c\to H^1$ has dimension $(r-2)d-\sum_j(d-c_\tau(g_j))$. This gives the first formula. The second is the Chevalley--Weil formula \cite[Theorem~2.10]{FGP}. \end{proof} \subsection{The Weil-type factor}\label{subsec:pt} Let $\sigma$ be an irreducible representation of $G$ whose character field is $K$ and which is realizable over $K$, on $V_\sigma=K^d$. The simple factor of $\Q[G]$ attached to $\sigma$ is $\Q[G]\varepsilon\cong\End_K(V_\sigma)\cong M_d(K)$, with centre $K$. Averaging gives a $G$-invariant positive definite $K$-Hermitian form $h_V$ on $V_\sigma$; the Rosati involution $g\mapsto g^{-1}$ acts on $\Q[G]\varepsilon$ as the $h_V$-adjoint and on its centre as complex conjugation. \begin{definition}\label{def:pt} Choose $v\in V_\sigma$ with $h_V(v,v)\ne0$, and let $e\in\Q[G]\varepsilon$ be the $h_V$-orthogonal projection onto the line $Kv$, a primitive idempotent fixed by the Rosati involution. Choose an integer $N_0\ge1$ with $N_0e\in\Z[G]$; then $N_0e$ is an endomorphism of $J(C)$, not itself idempotent. Put \[ A_\sigma(C)=\operatorname{im}\big(N_0e\colon J(C)\to J(C)\big), \] with the restricted polarization $E$ and the action $\iota$ of the centre $K$. It does not depend on $N_0$, and other choices of $v$ give $K$-isogenous abelian varieties. \end{definition} The $\sigma$-isotypic part of $J(C)$ is isogenous to $A_\sigma(C)^d$. Factors of this kind are studied as Prym--Tyurin varieties in \cite[Ch.~12]{BL}. \begin{lemma}\label{lem:weiltype} $(A_\sigma(C),\iota,E)$ is a polarized abelian variety of dimension $m_\sigma$, and the two embeddings of $K$ occur on its tangent space with multiplicities the Hodge numbers $(p_\sigma,q_\sigma)$ of $M_\sigma$. If $m_\sigma=2n$ and $(p_\sigma,q_\sigma)=(n,n)$, it is a polarized abelian $2n$-fold of Weil type for $K$. \end{lemma} \begin{proof} The $\sigma$-isotypic part of $H_1(J(C),\Q)$ is $V_\sigma\otimes_KM$ with $M=\Hom_{K[G]}(V_\sigma,H_1(J(C),\Q)_\varepsilon)$, and $H_1(A_\sigma,\Q)=Kv\otimes_KM\cong M$. After extending scalars along the two embeddings of $K$, $M$ becomes $M_\sigma$ and $M_{\bar\sigma}$. The Rosati involution of $E$ restricts on $\iota(K)$ to that of $\Q[G]$ on the centre, which is complex conjugation. \end{proof} \begin{lemma}\label{lem:constancy} On a connected component $S$ of $\Hur$, the discriminant of the Hermitian form of $A_\sigma(C)$ does not depend on $C\in S$. \end{lemma} \begin{proof} The spaces $H_1(A_\sigma(C),\Q)$ with $\iota$ and $E$ form a local system over $S$, since $e$ and $\iota$ act by flat endomorphisms and $E$ is flat; parallel transport is a $K$-isometry. \end{proof} \subsection{The group $G_{32}$ and the class datum}\label{subsec:g32} Let $K=\Q(\sqrt{-3})$, and let $G\subset GL_4(K)$ be the group generated by the matrices $r_1,r_2,x,y$ of Appendix~\ref{app:tuple}. It has order $155\,520$, and it is the Shephard--Todd group $G_{32}$; we use only the following properties, which are checked in Appendix~\ref{app:tuple}. Its natural representation $\rho$ on $V_\rho=K^4$ is absolutely irreducible, so its character field is $K$ and it is realizable over $K$. The matrices preserve the standard Hermitian form, which is therefore $h_V$. The centre $Z(G)$ has order six, $G/Z(G)\cong PSp_4(3)$ is simple of order $25\,920$, the derived group of $G$ is the kernel of $\det\rho$, of index three, and every irreducible character of $G$ takes values in $K$. Let $\mathcal R_1$ and $\mathcal R_2$ be the classes of the reflections with nontrivial eigenvalue $\omega$ and $\omega^2$ respectively; they are the classes of $r_1$ and $r_2$, and they consist of mutually inverse elements. Let $\cI$ be the class of $x^2$, whose eigenvalues are $(1,1,-1,-1)$. \begin{definition}\label{def:datum} Let $\Hur$ be the Hurwitz space of connected Galois covers $C\to\PP^1$ with group $G$, branched over thirteen points, with local monodromy in $\mathcal R_1$ at six of them, in $\mathcal R_2$ at six, and in $\cI$ at one. We call this list of classes the \emph{class datum}, and write $\Hur^\circ$ for the connected component of $\Hur$ that contains the cover $C_0$ constructed in Lemma~\ref{lem:surj}. \end{definition} \begin{lemma}\label{lem:sixfold} For every $C\in\Hur$, $\rho$ occurs in $H^1(C)$ with multiplicity $6$ and Hodge numbers $(3,3)$, so $A_\rho(C)$ is a polarized abelian sixfold of Weil type for $K$. The genus of $C$ is $505\,441$. \end{lemma} \begin{proof} A reflection has $c_\rho=1$, and an element of $\cI$ has $c_\rho=2$, so Proposition~\ref{prop:mult} gives $m_\rho=-8+12+2=6$. The inverse of an element of $\mathcal R_1$ has eigenvalues $(\omega^2,1,1,1)$, of age $\tfrac23$; that of an element of $\mathcal R_2$ has age $\tfrac13$; an element of $\cI$ has age $1$. So $\dim\Hom_G(\rho,H^0(K_C))=6\cdot\tfrac23+6\cdot\tfrac13+1-4=3$, and the Hodge numbers are $(3,3)$. Conclude with Lemma~\ref{lem:weiltype}. Riemann--Hurwitz gives $2g-2=155\,520\,\big(-2+12\cdot\tfrac23+\tfrac12\big)=1\,010\,880$. \end{proof} \begin{remark}[$A_\rho(C)$ is not of CM type]\label{rem:notCM} The family $A_\rho(C)$, $C\in\Hur^\circ$, fills an open set of its moduli space (Corollary~\ref{cor:dominant}), and the Hodge group of a very general point of that space is $SU_H$ \cite[Theorem~6.11]{vanGeemen}. So for $C$ outside a countable union of proper analytic subsets, $A_\rho(C)$ is not of CM type and not isogenous to a power of an elliptic curve. \end{remark} %% ------------------------------------------------------------------ \section{The Weil classes of every member are algebraic}\label{sec:supply} %% ------------------------------------------------------------------ We first outline the argument. The centralizer $U$ of $G$ in the symplectic group of $H^1(C)$ acts on the Weil line $\det M_\rho\subset H^6(A_\rho(C),\C)$ through a character $e_\rho$ of its maximal abelian quotient $T$. A Schoen class on a factor of $J(C)$ spans, together with its Galois conjugates, eigenlines of $U$ with characters $\pm a(\chi)$. If $e_\rho=\sum_in_ia(\chi_i)$ with integers $n_i$, then the class $\Gamma$ obtained by tensoring the generator of $\det M_\rho$ with the complementary eigenlines of the Schoen factors is invariant under $U$, hence under Milne's group of the power of $J(C)$ that contains all the factors, hence algebraic by Milne's theorem; and $\Gamma$, used as a correspondence, maps a product of Schoen classes to a nonzero multiple of the Weil class. \subsection{Schoen suppliers}\label{subsec:schoen} \begin{definition}\label{def:supplier} A \emph{supplier} is a pair $(H',\chi)$ of a subgroup $H'\le G$ and a character $\chi\colon H'\to\C^\times$ of order $m\ge3$. \end{definition} For $C\in\Hur$, the curve $C'=C/\ker\chi$ is a connected cyclic cover of $C/H'$ with group $H'/\ker\chi\cong\Z/m$. Its \emph{primitive part} $P_\chi\subset J(C)$ is the abelian subvariety on which $\Z/m$ acts through primitive characters; it is the image of $N\pi_\chi$ for a rational idempotent $\pi_\chi$ of $\Q[H']$ and an integer $N$, and $F_\chi=\Q(\mu_m)$ acts on it. For $t\in(\Z/m)^\times$ let $V_t\subset H^1(P_\chi,\C)$ be the eigenspace where $H'$ acts by $\chi^t$; then \[ V_t=\textstyle\bigoplus_\tau\Hom_{H'}(\chi^t,\tau|_{H'})\otimes M_\tau . \] The points of $C/H'$ over $p_j$ correspond to the double cosets $H'\backslash G/\langle g_j\rangle$; over the double coset of $x$ the local monodromy is $xg_j^\ell x^{-1}\in H'$, with $\ell$ the least positive integer for which this lies in $H'$. Let $q$ be the genus of $C/H'$, $s$ the number of branch points of $C'\to C/H'$, and $\alpha\in(\Z/m-\{0\})^s$ their exponents with respect to a generator, defined up to $(\Z/m)^\times$ and order. All $V_t$ have the same dimension $h=2q-2+s$ \cite[Lemma~1.5]{Schoen88}, and the Weil classes $W_{F_\chi}(P_\chi)=\bigwedge^h_{F_\chi}H^1(P_\chi,\Q)$ satisfy $W_{F_\chi}(P_\chi)\otimes\C=\bigoplus_t\det V_t$. Following Schoen \cite[\S1]{Schoen88}, $\alpha$ is \emph{simple} if, after reordering, $\alpha_{2i}=-\alpha_{2i-1}$ for all $i$; then $s$ and $h$ are even. A supplier is \emph{admissible} if $h>0$ and $\alpha$ is simple; for $s=0$ simplicity is automatic. These data depend only on $G$ and the classes $\mathcal C_j$, since replacing $g_j$ by a conjugate $yg_jy^{-1}$ replaces the double coset of $x$ by that of $xy^{-1}$, with the same local monodromy. \begin{lemma}[Schoen]\label{lem:schoen} If $(H',\chi)$ is admissible, then $W_{F_\chi}(P_\chi)$ is spanned by classes of algebraic cycles, for every $C\in\Hur$. \end{lemma} \begin{proof} We quote Schoen \cite[\S2]{Schoen88}. Let $C'$ be a smooth projective irreducible curve over $\C$ with a faithful action $\Z/m\hookrightarrow\operatorname{Aut}(C')$, $m>1$, whose invariant $h$ is even, so that the quotient map has an even number $2r$ of branch points, with exponents $1\le\alpha_j\le m-1$. \emph{Theorem~2.0} (p.~11): if the $(\Z/m)^\times$-orbit of $(\alpha_1,\dots,\alpha_{2r})$ is simple, then the relevant subspace $U$ of $H^h(C'^h)$ is generated by fundamental classes of algebraic cycles. \emph{Corollary~3.1} (p.~24), for $m>2$: under the same hypotheses, the corresponding subspace of $H^h(B,\Q)$ is generated by classes of algebraic cycles of codimension $h/2$, where $B\subseteq\operatorname{Alb}(C')$ is the primitive part, the image of $\prod_{1\ne Z'\le\Z/m}\sum_{t\in Z'}(\mathrm{Id}-t)$. The statements have no coprimality hypothesis: Schoen's \S1 treats a branch point with $\gcd(\alpha_j,m)=d_j>1$ through the local model $y^{m/d_j}=w^{\alpha_j/d_j}$. The unramified case $r=0$ is part of Theorem~2.0, where simplicity is automatic; Schoen proves it separately in \S2, where $\beta_0$ is \'etale and $P_0$ splits into $m$ connected components, using the first case of his Lemma~2.6. His Lemma~2.8 is not used when $r=0$, since there are no blow-ups. Six rows of Table~\ref{tab:witness} are unramified. Apply this to $C'=C/\ker\chi\to C/H'$, which is smooth, projective and irreducible as a quotient of $C$, with $H'/\ker\chi\cong\Z/m$ acting faithfully. The subspace of Corollary~3.1 is $W_{F_\chi}$ of the primitive part of $J(C')$, which is isogenous to $P_\chi$ compatibly with $F_\chi$. When every exponent is $m/2$ and $4\mid m$, one step of Schoen's proof is replaced by Lemma~\ref{lem:schoenR}. \end{proof} Three suppliers of the identity of Proposition~\ref{prop:lattice} (rows 13, 14 and 15 of Table~\ref{tab:witness}) have $m=12$ and every exponent equal to $6=m/2$. For such covers one step of Schoen's proof needs a replacement. Recall Schoen's construction \cite[\S2, pp.~11--19]{Schoen88}, for $C'\to X$ as in Lemma~\ref{lem:schoen} with branch points ordered so that $\alpha_{2i}=-\alpha_{2i-1}$, $1\le i\le r$. Let $W_0=C'^h/G_0$, where $G_0\subset(\Z/m)^h\rtimes S_h$ is the kernel of the sum map extended by $S_h$; the residual group $\Z/m$ acts on $W_0$, and $\beta_0\colon W_0\to S^hX$ is a $\Z/m$-cover branched along the divisors $S_j=b_j+S^{h-1}X$. Schoen chooses a linear subspace $P_0\subset S^hX$ of dimension $h/2$ (his Lemma~2.3) and, for $i=1,\dots,r$, blows up $(S^hX)_{i-1}$ along $K_{i-1}=S_{2i-1,i-1}\cap S_{2i,i-1}$, the intersection of the strict transforms of $S_{2i-1}$ and $S_{2i}$; the exceptional divisor $E_i$ is a $\PP^1$-bundle over $K_{i-1}$. Let $W_i$ be the normalization of the pulled-back cover, $\beta_i\colon W_i\to(S^hX)_i$, and $F_i=\beta_i^{-1}(E_i)$. To see that his cycles have nonzero classes, Schoen's Lemma~2.8 reduces to the statement that no primitive character of $\Z/m$ occurs in $H^*(F_i,\C)$ \cite[p.~18]{Schoen88}, which he proves (Sublemma~2.9, p.~19) by choosing a prime $p\mid m$ such that the $\Z/p$-quotient of $C'\to X$ is ramified over $b_{2i-1}$. When every exponent is $m/2$ and $4\mid m$, no such prime exists: for $m=12$ and exponent $6$, the images of $6$ in $\Z/2$ and in $\Z/3$ are zero. \begin{lemma}[Schoen's Theorem~2.0 without the choice of a prime]\label{lem:schoenR} In the situation of Schoen's Theorem~2.0, for every $i\le r$ and every primitive character $\chi$ of $\Z/m$, the $\chi$-part of $H^*(F_i,\C)$ vanishes. Hence Theorem~2.0 and Corollary~3.1 of \cite{Schoen88} hold under their stated hypotheses, without the choice of a prime in Sublemma~2.9. \end{lemma} \begin{proof} Since $\beta_i$ is finite, $H^*(F_i,\C)_\chi=H^*(E_i,(\beta_{i*}\C)_\chi|_{E_i})$. The stalk of $\beta_{i*}\C$ at a point is the permutation module on the points above it, and its $\chi$-part vanishes at every point whose stabilizer contains nontrivial inertia, because a primitive character is faithful. By Schoen's Lemma~2.5, $E_i$ is not a branch divisor of $\beta_i$: its multiplicity in the branch locus is $\alpha_{2i-1}+\alpha_{2i}\equiv0\pmod m$. So on $E_i$ the sheaf $(\beta_{i*}\C)_\chi$ is $j_!\mathcal L$, where $\mathcal L$ is a rank-one local system on the complement in $E_i$ of the strict transforms $S_{j,i}$, with monodromy $\chi(\alpha_j)\ne1$ around $S_{j,i}$. Here $j\colon U_i\to E_i$ is the inclusion of the complement $U_i$ of all the $S_{j,i}$. The strict transforms $S_{2i-1,i}$ and $S_{2i,i}$ meet $E_i$ in two disjoint sections of the $\PP^1$-bundle $p\colon E_i\to K_{i-1}$, and every other $S_{j,i}$ meets $E_i$ in a union of whole fibres \cite[\S2]{Schoen88}. Fix $k\in K_{i-1}$, let $\iota_k$ be the inclusion of the fibre $\Pi_k=p^{-1}(k)$, and $j_k$ that of $\Pi_k\cap U_i$ in $\Pi_k$. Extension by zero commutes with restriction to a closed subvariety (both sides have the same stalks), so $\iota_k^*j_!\mathcal L=(j_k)_!(\mathcal L|_{\Pi_k\cap U_i})$. There are two cases. If $\Pi_k$ lies in some $S_{j,i}$ with $j\ne2i-1,2i$, then $\Pi_k\cap U_i$ is empty and $\iota_k^*j_!\mathcal L=0$. Otherwise $\Pi_k$ meets no $S_{j,i}$ with $j\ne2i-1,2i$, and it meets $S_{2i-1,i}$ and $S_{2i,i}$ transversally in one point each, so $\Pi_k\cap U_i\cong\C^\times$, and a small loop in $\Pi_k$ around the first point is a meridian of $S_{2i-1,i}$. So $\mathcal L|_{\Pi_k\cap U_i}$ is a rank-one local system on $\C^\times$ with monodromy $\chi(\alpha_{2i-1})\ne1$, and $H^*_c(\C^\times,\mathcal L|_{\Pi_k\cap U_i})=0$: $H^0_c$ vanishes since $\C^\times$ is not compact, $H^2_c$ is dual to the invariants of the dual system, which vanish, and the Euler characteristic of $\C^\times$ is $0$. In both cases $H^*(\Pi_k,\iota_k^*j_!\mathcal L)=0$. Since $p$ is proper, proper base change \cite[Theorem~11.3.6]{Schapira} identifies the stalk of $R^bp_*(j_!\mathcal L)$ at $k$ with $H^b(\Pi_k,\iota_k^*j_!\mathcal L)=0$. So $Rp_*j_!\mathcal L=0$ and $H^*(E_i,j_!\mathcal L)=H^*(K_{i-1},Rp_*j_!\mathcal L)=0$. This uses only $\alpha_{2i-1}\not\equiv0\pmod m$. The rest of Schoen's proof is unchanged. \end{proof} \subsection{The character group}\label{subsec:characters} Let $U$ be the centralizer of $G$ in the symplectic group of $(H^1(C,\Q),E)$; we only need its complex points. Recall that an irreducible complex representation $\tau$ of a finite group is \emph{orthogonal} or \emph{quaternionic} if it is self-dual and carries a nonzero invariant symmetric, respectively alternating, bilinear form, and that otherwise $\tau\not\cong\bar\tau$. In $H^1(C,\C)=\bigoplus_\tau\tau\otimes M_\tau$ the $G$-invariant cup-product form pairs $\tau\otimes M_\tau$ with $\bar\tau\otimes M_{\bar\tau}$ and vanishes between other summands. By Schur's lemma its restriction is the tensor product of the invariant form on $\tau$, unique up to scalar, with a nondegenerate form on the multiplicity spaces: a duality between $M_\tau$ and $M_{\bar\tau}$ if $\tau\not\cong\bar\tau$, a symmetric form on $M_\tau$ if $\tau$ is quaternionic, an alternating one if $\tau$ is orthogonal. Hence $U(\C)$ is the product of the groups $GL(M_\tau)$ over the pairs $\{\tau,\bar\tau\}$ of non-self-dual representations with $m_\tau>0$, of $O(M_\tau)$ over the quaternionic $\tau$ with $m_\tau>0$, and of $Sp(M_\tau)$ over the orthogonal $\tau$ with $m_\tau>0$. Let $T=U(\C)/[U(\C),U(\C)]$. Its character group is \[ X^*(T)=\Z^{\rm pairs}\oplus(\Z/2)^{\rm quat}, \] with one free coordinate for each pair (the determinant on $M_\tau$, for a chosen member $\tau$ of the pair; for the pair of $\rho$ we choose $\rho$), and one coordinate of order two for each quaternionic $\tau$ with $m_\tau>0$ (the determinant of $O(M_\tau)$). For a supplier $(H',\chi)$ the line $\det V_\chi$, $V_\chi=\bigoplus_\tau\Hom_{H'}(\chi,\tau|_{H'})\otimes M_\tau$, is a $U$-eigenline, with character \[ a(\chi)=\big(\langle\tau|_{H'},\chi\rangle-\langle\bar\tau|_{H'},\chi\rangle\big)_{\rm pairs}\oplus\big(\langle\tau|_{H'},\chi\rangle\bmod2\big)_{\rm quat}. \] The Weil line $\det M_\rho$ has character $e_\rho$, the coordinate vector of the pair of $\rho$. \begin{lemma}\label{lem:sign} For $G=G_{32}$ and every supplier $(H',\chi)$ and $t\in(\Z/m)^\times$, the character of $\det V_{\chi^t}$ is $a(\chi)$ on the free coordinates if the automorphism $\zeta_m\mapsto\zeta_m^t$ fixes $\omega$ (extended to fix $\omega$ when $3\nmid m$), and $-a(\chi)$ otherwise; on the coordinates of order two it is $a(\chi)$. \end{lemma} \begin{proof} Let $\gamma$ be an automorphism of $\Q(\zeta_m,\omega)$ with $\gamma(\zeta_m)=\zeta_m^t$. Then $\langle\tau|_{H'},\chi^t\rangle=\langle\tau^\gamma|_{H'},\chi\rangle$, where $\tau^\gamma$ has character $\gamma\circ\operatorname{tr}\tau$. Since every character of $G$ is $K$-valued, $\tau^\gamma=\tau$ if $\gamma$ fixes $\omega$ and $\tau^\gamma=\bar\tau$ otherwise; the second case exchanges the two terms of each free coordinate, and a quaternionic $\tau$ is self-conjugate. \end{proof} If $3\nmid m$, then $\Q(\zeta_m)$ and $K$ are linearly disjoint, so every $t$ extends both to an automorphism fixing $\omega$ and to one moving it; the lemma then gives $a(\chi)=-a(\chi)$, so $a(\chi)$ vanishes on the free coordinates. Every supplier of Table~\ref{tab:witness} has $3\mid m$ ($m\in\{3,6,12,15\}$). Then $\omega=\zeta_m^{m/3}$, and $t=-1$ moves $\omega$, as used in the proof of Proposition~\ref{prop:supply}. \begin{proposition}[The character identity]\label{prop:lattice} In $X^*(T)=\Z^{44}\oplus(\Z/2)^4$, \[ e_\rho=\textstyle\sum_{i=1}^{51}n_i\,a(\chi_i) \] for the fifty-one admissible suppliers $(H'_i,\chi_i)$ and the integers $n_i$ of Appendix~\ref{app:witness}, with $\sum_i|n_i|=56$. \end{proposition} \begin{proof} Appendix~\ref{app:witness} prints, for each supplier, its branch data (which show that it is admissible) and the nonzero coordinates of $a(\chi_i)$, so that the identity can be checked by adding $51$ sparse integer vectors. The appendix also describes how these data are computed from the group. \end{proof} \subsection{The contraction} \begin{proposition}\label{prop:supply} For every $C\in\Hur$, $W_K(A_\rho(C))$ is spanned by classes of algebraic cycles. \end{proposition} \begin{proof} Write $e_\rho=\sum_in_ia(\chi_i)$ as in Proposition~\ref{prop:lattice}, put $t_i=\operatorname{sign}n_i\in\{\pm1\}\subset(\Z/m_i)^\times$, $P_i=P_{\chi_i}$, $F_i=F_{\chi_i}$, and let $V_{i,t}$ be the eigenspaces of $P_i$. By Lemma~\ref{lem:schoen} choose a nonzero rational algebraic class $y_i\in W_{F_i}(P_i)$. Put $B=\prod_iP_i^{|n_i|}$ and $Y=\boxtimes_iy_i^{\boxtimes|n_i|}\in H^*(B,\Q)$, an algebraic class. \emph{The correspondence.} Choose generators $z_{i,t}$ of the lines $\det V_{i,t}$ and put $Z_i=\prod_{t\ne t_i}z_{i,t}$. The product $\prod_tz_{i,t}$ spans the top cohomology of $P_i$, on which $U$ acts trivially, so $Z_i$ has the character opposite to that of $\det V_{i,t_i}$, which is $-t_ia(\chi_i)$ on the free coordinates (Lemma~\ref{lem:sign}, $t=-1$ moving $\omega$) and $a(\chi_i)$ on the others. Let $w$ generate $\det M_\rho\subset H^6(A_\rho,\C)$ and put \[ \Gamma=\Big(\textstyle\boxtimes_iZ_i^{\boxtimes|n_i|}\Big)\boxtimes w\in H^*(B\times A_\rho,\C). \] \emph{Degrees.} Write $\phi$ for Euler's function. $P_i$ has dimension $\phi(m_i)h_i/2$, each $z_{i,t}$ and $y_i$ has degree $h_i$, $Z_i$ has degree $(\phi(m_i)-1)h_i$, and $w$ has degree $6$. So $\Gamma$ has degree $\sum_i|n_i|(\phi(m_i)-1)h_i+6$, and for the algebraic class $Y$ of degree $\sum_i|n_i|h_i$ the class $\operatorname{pr}_{2*}(\operatorname{pr}_1^*Y\cup\Gamma)$ below has degree $\sum_i|n_i|h_i+\deg\Gamma-2\dim B=6$. Its character is $e_\rho-\sum_in_ia(\chi_i)=0$ on the free coordinates and $\sum_i|n_i|a(\chi_i)$ on the coordinates of order two, which vanishes because $n_i\equiv|n_i|$ and the identity holds there too. Characters of $U(\C)$ factor through $T$, so $\Gamma$ is fixed by all of $U(\C)$. \emph{$\Gamma$ is algebraic.} Each factor $X'$ of $B\times A_\rho$ (a copy of some $P_i$, or $A_\rho$) is the image of an endomorphism $N_0e_{X'}$ of $J(C)$ with $e_{X'}$ an idempotent of $\Q[G]$; choose one integer $N_0$ for all of them. The inclusions $X'\hookrightarrow J(C)$ and the homomorphisms $N_0e_{X'}\colon J(C)\to X'$ (the endomorphism $N_0e_{X'}$ with its target restricted to its image) give \[ B\times A_\rho\xrightarrow{\ \iota_0\ }J(C)^{R}\xrightarrow{\ \pi\ }B\times A_\rho,\qquad \pi\circ\iota_0=[N_0],\qquad R=1+\textstyle\sum_i|n_i| . \] They are built from elements of $\Q[G]$, so they commute with $U$, and $\pi^*$ is $U$-equivariant for the diagonal action of $U$ on $H^1(J(C)^R)=H^1(C)^{\oplus R}$; hence $\pi^*\Gamma$ is fixed by $U(\C)$. Milne defines $S(A)$, for an abelian variety $A$, as the algebraic group whose points are the elements $\gamma$ of the centralizer of $\End^0(A)$ with $\gamma^\dagger\gamma=1$ for the Rosati involution, ``the largest algebraic subgroup of $Sp$ whose elements commute with the endomorphisms of $A$'' \cite[\S1]{Milne}; it acts on $H^*(A^r)\cong\bigwedge(H^1(A)^{\oplus r})$ diagonally. Since $\Q[G]\subseteq\End^0(J(C))$, the elements of $S(J(C))$ commute with $G$ and preserve $E$, so $S(J(C))\subseteq U$, and $\pi^*\Gamma$ is $S(J(C))$-invariant. Milne's Theorem~3.2 \cite{Milne} states: \emph{For any abelian variety $A$ over $\Omega$ and integer $r\ge0$, the $k$-algebra $H^*(A^r)^{S(A)}$ is generated by divisor classes}; here $\Omega=\C$ and $k=\Q$, and by his Lemma~3.1 the invariants of the complexification are the complexification of the rational invariants. So $\pi^*\Gamma$ is a complex linear combination of products of divisor classes on $J(C)^R$. Pullback by $\iota_0$ preserves algebraic classes, and $\iota_0^*\pi^*\Gamma=[N_0]^*\Gamma=N_0^{\deg\Gamma}\,\Gamma$. So $\Gamma$ is a linear combination, with complex coefficients, of classes of algebraic cycles. \emph{The contraction is nonzero.} Write $y_i=\sum_tc_{i,t}v_{i,t}$ with $v_{i,t}\in\det V_{i,t}$ normalized by $\int_{P_i}v_{i,t_i}\cup Z_i=1$. For $t\ne t_i$ the product $v_{i,t}\cup Z_i$ contains $\det V_{i,t}$ twice and vanishes. Each $c_{i,t}$ is nonzero, because $W_{F_i}(P_i)$ is a one-dimensional $F_i$-vector space and a nonzero vector has a nonzero component under every embedding of $F_i$. Hence \[ \Gamma_*(Y)=\operatorname{pr}_{2*}(\operatorname{pr}_1^*Y\cup\Gamma)=\pm\textstyle\prod_ic_{i,t_i}^{|n_i|}\,w\ne0, \] and $w$ lies in the complex span of algebraic classes. The same argument with every $t_i$ replaced by $-t_i$ gives $\bar w$. \emph{Rational descent.} Let $\mathcal Z\subseteq H^6(A_\rho,\Q)$ be the span of algebraic classes. Then $W_K(A_\rho)\otimes\C=\C w\oplus\C\bar w\subseteq\mathcal Z\otimes\C$, so $(W_K(A_\rho)+\mathcal Z)\otimes\C=\mathcal Z\otimes\C$, and $W_K(A_\rho)\subseteq\mathcal Z$ by comparing dimensions. \end{proof} %% ------------------------------------------------------------------ \section{Dominance}\label{sec:dominance} %% ------------------------------------------------------------------ \subsection{The codifferential} Let $C\in\Hur$ with branch set $B_C$, $|B_C|=13$, $\infty\notin B_C$, and put $\cD_\rho(C)=\Hom_G(\rho,H^0(C,K_C))$ and $\cD_{\bar\rho}(C)=\Hom_G(\bar\rho,H^0(C,K_C))$, each of dimension three. The tangent space to $\Hur$ at $C$ modulo $PGL_2$ is Serre dual to $H^0(\PP^1,K_{\PP^1}^{\otimes2}(B_C))$, of dimension $10$. Let \[ \mu_C\colon\cD_\rho(C)\otimes\cD_{\bar\rho}(C)\to H^0\big(\PP^1,K_{\PP^1}^{\otimes2}(B_C)\big) \] be multiplication of differentials followed by contraction of $\rho\otimes\bar\rho$ against its invariant line, with values in $H^0(C,K_C^{\otimes2})^G=H^0(\PP^1,K_{\PP^1}^{\otimes2}(B_C))$. \begin{lemma}[Codifferential]\label{lem:codiff} The cotangent space of $\mathcal H_3$ at the period point of $A_\rho(C)$ is $\cD_\rho(C)\otimes\cD_{\bar\rho}(C)$, of dimension $9$, and $\mu_C$ is, up to a nonzero normalization, the codifferential at $C$ of the period map of the family $A_\rho$. If $\rk\mu_{C_1}=9$ for one $C_1$ in a connected component $S$ of $\Hur$, the period map is a submersion at $C_1$ and has rank $9$ on a nonempty Zariski open subset of $S$. \end{lemma} \begin{proof} The codifferential of the period map of $J(C)$ along the Hurwitz family is multiplication $H^0(K_C)\otimes H^0(K_C)\to H^0(K_C^{\otimes2})$ followed by restriction to $G$-invariants \cite[\S\S2--3]{FGP}. Under $H^0(C,K_C)=\bigoplus_\tau\tau\otimes\Hom_G(\tau,H^0(K_C))$, the part that pairs the $\rho$- and $\bar\rho$-components and lands in the invariants is $\mu_C$. The domain $\mathcal H_3$ parametrizes the $\rho$-eigenspace of $H^{1,0}(A_\rho)$, which is $\cD_\rho(C)$, and its tangent space is $\Hom(H^{1,0}_\rho,H^{0,1}_\rho)$; the polarization identifies $H^{0,1}_\rho$ with the dual of $H^{1,0}_{\bar\rho}=\cD_{\bar\rho}(C)$, so the cotangent space is $\cD_\rho(C)\otimes\cD_{\bar\rho}(C)$. The rank is lower semicontinuous and $9$ is its maximum, and $S$ is irreducible. \end{proof} \subsection{An explicit cover}\label{subsec:explicit} Let $\LL$ be the rank-four local system on $\PP^1_z-\{0,5,-7,\infty\}$ of the Fuchsian system $dY=\big(\frac{B_0}{z}+\frac{B_5}{z-5}+\frac{B_{-7}}{z+7}\big)Y\,dz$ with \[ B_0=\operatorname{diag}\big(-\tfrac34,-\tfrac12,-\tfrac12,\tfrac34\big),\qquad B_5=u_1v_1^{T},\qquad B_{-7}=u_2v_2^{T}, \] with $B_\infty=-(B_0+B_5+B_{-7})$ and the rational vectors $u_i,v_i$ of Appendix~\ref{app:tuple}. One checks there that $\operatorname{tr}B_5=\tfrac13$, $\operatorname{tr}B_{-7}=\tfrac23$, and that $B_\infty$ is diagonalizable with eigenvalues $\pm\tfrac23$, each twice. No two eigenvalues of a residue differ by a nonzero integer, so each local monodromy is conjugate to $\exp(2\pi iB)$ for the residue $B$: it lies in $\mathcal R_1$ at $5$, in $\mathcal R_2$ at $-7$, in the class of $x$ (eigenvalues $(i,-1,-1,-i)$) at $0$, and in the class of $y$ (eigenvalues $(\omega,\omega,\omega^2,\omega^2)$, order three) at $\infty$. \emph{The monodromy of $\LL$.} The tuple $(r_1,r_2,x,y)$ of Appendix~\ref{app:tuple} has product one, lies in the same four conjugacy classes of $GL_4(\C)$, generates $G$, and is absolutely irreducible (the $K$-algebra generated by its matrices is $M_4(K)$). The dimensions of the four classes are $6+6+10+8=30=2(4^2-1)$. Let $\mathbb F$ be the local system of the tuple $(r_1,r_2,x,y)$ on $\PP^1$ minus $\{5,-7,0,\infty\}$, and recall that an irreducible local system is \emph{physically rigid} if every local system with the same local monodromy classes at the same points is isomorphic to it. By Katz's theorem \cite[Theorem~1.1.2]{Katz}, an irreducible local system whose local classes have total dimension $2(n^2-1)$, $n$ the rank, is physically rigid. This applies to $\mathbb F$, and only $\mathbb F$ needs to be irreducible; we recall the argument, since $\LL$ is not yet known to be irreducible. Let $\mathcal H=\mathcal Hom(\mathbb F,\LL)$ and $j$ the inclusion of the complement of the four points. Since the local monodromies of $\mathbb F$ and $\LL$ are conjugate, the invariants of $\mathcal H$ at each point have the dimension of the centralizer of the class, $16-\dim(\text{class})$, so \[ \chi(\PP^1,j_*\mathcal H)=(2-4)\cdot16+\textstyle\sum(16-\dim)=-32+64-30=2. \] Hence $h^0(j_*\mathcal H)+h^2(j_*\mathcal H)\ge2$, and Poincar\'e--Verdier duality gives $H^2(\PP^1,j_*\mathcal H)^\vee\cong H^0(\PP^1,j_*\mathcal H^\vee)$ with $\mathcal H^\vee=\mathcal Hom(\LL,\mathbb F)$. So there is a nonzero homomorphism $\mathbb F\to\LL$ or $\LL\to\mathbb F$. Since $\mathbb F$ is irreducible, the first is injective and the second surjective, and both are isomorphisms because the ranks are equal. So $\LL\cong\mathbb F$: its monodromy tuple with respect to standard loops around $5,-7,0,\infty$ is simultaneously conjugate to $(r_1,r_2,x,y)$, it is irreducible, and its monodromy group is $\rho(G)$. \emph{The pullback.} Let \[ \varphi(s)=\frac{s^2(s-3)^4}{(s-1)^3(s+2)^3}, \] of degree six. Its fibre over $0$ has type $[2,4]$, over $\infty$ type $[3,3]$, its fibres over $5$ and $-7$ are reduced, and its two other critical values are the roots of $729z^2-3674z+5120$, which are not real. At a point of ramification index $e$ over a branch value with local monodromy $g$, the local monodromy of $\varphi^*\LL$ is conjugate to $g^e$. So $\varphi^*\LL$ has local monodromy in $\mathcal R_1$ at the six roots of $P_5=s^2(s-3)^4-5(s-1)^3(s+2)^3$, in $\mathcal R_2$ at the six roots of $P_{-7}=s^2(s-3)^4+7(s-1)^3(s+2)^3$, in $\cI$ at $s=0$ (class of $x^2$), and trivial local monodromy at $s=3$ (since $x^4=1$) and at $s=1,-2$ (since $y^3=1$). \begin{lemma}\label{lem:surj} The monodromy group of $\varphi^*\LL$ is $\rho(G)$, so $\varphi^*\LL$ is the $\rho$-local system of a cover $C_0\in\Hur$, branched over the thirteen points $0$ and the roots of $P_5P_{-7}$. \end{lemma} \begin{proof} Over the complement of the branch values, $\varphi$ is an unramified cover of degree six, so the monodromy group $\Pi$ of $\varphi^*\LL$ is the image of a subgroup of index six of $\pi_1$ of the complement downstairs, and $[\rho(G):\Pi]\le6$. The image of $\Pi$ in $G/Z(G)\cong PSp_4(3)$ has index at most $6$, and every proper subgroup of $PSp_4(3)$ has index at least $27$ \cite{ATLAS}; so $G=\Pi Z(G)$ (identifying $G$ with $\rho(G)$), and since $Z(G)$ is central, $[G,G]=[\Pi,\Pi]\subseteq\Pi$. The derived group has index three and is the kernel of $\det\rho$, and $\Pi$ contains a reflection in $\mathcal R_1$, of determinant $\omega\ne1$; so $\Pi=G$. The local system $\varphi^*\LL$ thus defines a surjection from $\pi_1$ of the complement of the thirteen points onto $G$, hence a connected Galois $G$-cover, with the local classes listed above. \end{proof} \subsection{Differentials from a Fuchsian system}\label{subsec:dictionary} Let $C\in\Hur$, and suppose that the $\rho$-local system of $C$ is the local system of a Fuchsian system $dY=\Omega Y$, $\Omega=\sum_{p\in P}A_p\,ds/(s-p)$, on the trivial bundle of rank four over $\PP^1_s$, whose set of poles $P\subset\C$ contains the branch set $B_C$; $\infty$ is not a pole, so $\sum_pA_p=0$. At the poles not in $B_C$ (\emph{apparent singularities}) the local monodromy is trivial. Let $Y$ be a fundamental solution near a base point. \emph{Local form.} At $p\in P$ let $\Lambda_p=\operatorname{diag}(\lambda_{p,1},\dots,\lambda_{p,4})$ be the eigenvalues of $A_p$. We say that $Y$ has \emph{normal form} at $p$ if near $p$ there is a fundamental solution \[ Y_p=P_pH_p(s)\,(s-p)^{\Lambda_p},\qquad H_p\ \text{holomorphic},\quad H_p(p)=1, \] with $P_p$ an eigenbasis of $A_p$. Substituting gives the recursion $kH_k+H_k\Lambda_p-\Lambda_pH_k=R_k$ for the Taylor coefficients $H_k$ of $P_p^{-1}\cdot(P_pH_p)$, with $R_k$ determined by $H_0,\dots,H_{k-1}$ and the Taylor coefficients of $\sum_{q\ne p}A_q/(s-q)$ at $p$; entrywise $(k+\lambda_{p,j}-\lambda_{p,i})(H_k)_{ij}=(R_k)_{ij}$. If no two eigenvalues of $A_p$ differ by a positive integer, the recursion has a unique solution and normal form holds. At a \emph{resonance} $k=\lambda_{p,i}-\lambda_{p,j}$ the entry $(H_k)_{ij}$ is free, and normal form holds if and only if the corresponding entries $(R_k)_{ij}$, the \emph{logarithmic obstructions}, vanish; then, if $Y_p$ is one solution in normal form, so is $Y_p(1+cE_{ij})=P_pH_p(1+cE_{ij}(s-p)^k)(s-p)^{\Lambda_p}$, because $(s-p)^{\Lambda_p}$ conjugates $1+cE_{ij}$ to $1+cE_{ij}(s-p)^k$; this changes $(H_k)_{ij}$ by $c$ and no coefficient of lower order. So every choice of the free entries gives a solution in normal form. \begin{lemma}[The dictionary]\label{lem:dictionary} Suppose that $Y$ has normal form at every $p\in P$. Up to exchanging $\rho$ and $\bar\rho$ (which does not affect $\mu_C$), $\cD_\rho(C)$ is identified with the space of $r\in\C(s)^4$ such that the vector of forms $Y^{-1}r\,ds$ is holomorphic on $C$, and $\cD_{\bar\rho}(C)$ with the space of $q\in\C(s)^4$ such that $Y^{T}q\,ds$ is holomorphic on $C$. Concretely, $r$ belongs to the first space if and only if \[ \operatorname{ord}_p\big[(P_pH_p)^{-1}r\big]_j\ge\lfloor\lambda_{p,j}\rfloor\quad(p\in P,\ j=1,\dots,4), \] $r$ is holomorphic on $\C-P$, and $\operatorname{ord}_\infty r\ge2$; for $q$ the same holds with $(P_pH_p)^T$ and $\lfloor-\lambda_{p,j}\rfloor$. Under these identifications, $\mu_C(r\otimes q)=q^Tr\,ds^2$. \end{lemma} \begin{proof} Pulled back to the complement of the ramification points in $C$, $Y$ is single-valued, because its monodromy factors through $\rho$ and the loops that lift to closed loops on $C$ map to $1$ in $G$; and $g^*Y=Y\rho'(g)$ for $g\in G$, with $\rho'$ equal to $\rho$ or $\bar\rho$ depending on conventions. For a rational vector $r$, the vector $\omega=Y^{-1}r\,ds$ of meromorphic forms then transforms by $g^*\omega=\rho'(g)^{-1}\omega$, so its components span the image of a $G$-map from $\rho'^\vee$ to the meromorphic forms on $C$. Conversely, a $G$-map from $\rho'^\vee$ to $H^0(K_C)$ is a vector $\omega$ of holomorphic forms with this transformation law, and $r=Y\omega/ds$ is $G$-invariant, hence descends to $\PP^1_s-P$; it is meromorphic at $P$ because $Y$ and $Y^{-1}$ have moderate growth at the poles of a Fuchsian system, and at $\infty$, which is neither a pole nor a branch point, $Y$ is holomorphic and invertible, so $r$ vanishes to order at least two there, $ds$ having a double pole. So $r$ is rational. Since $\rho$ is unitary, $\rho'^\vee\cong\bar\rho'$; the same argument with $Y^T$ gives the second space, and $q^TY\cdot Y^{-1}r=q^Tr$ is the invariant contraction. The order condition: at $p\in P$ the local monodromy has finite order $e$, so $e\lambda_{p,j}\in\Z$; in a local coordinate $y$ on $C$ with $s-p=y^e$ the $j$-th component of $Y_p^{-1}r\,ds$ is $[(P_pH_p)^{-1}r]_j\,(s-p)^{-\lambda_{p,j}}ds$, of order $e\operatorname{ord}_p[(P_pH_p)^{-1}r]_j-e\lambda_{p,j}+e-1$ in $y$, and this is $\ge0$ exactly when $\operatorname{ord}_p[(P_pH_p)^{-1}r]_j\ge\lfloor\lambda_{p,j}\rfloor$. Holomorphy of the vector $Y^{-1}r\,ds$ does not depend on the choice of fundamental solution, since two of them differ by a constant matrix. \end{proof} \begin{lemma}\label{lem:kernel} In the situation of Lemma~\ref{lem:dictionary}, put $M_p=\max(0,-\min_j\lfloor\lambda_{p,j}\rfloor)$. \begin{enumerate} \item Every $r\in\cD_\rho(C)$ is uniquely of the form $r=\sum_{p\in P}\sum_{k=1}^{M_p}c_{p,k}(s-p)^{-k}$ with $c_{p,k}\in\C^4$; likewise for $\cD_{\bar\rho}(C)$ with $-\lambda_{p,j}$ in place of $\lambda_{p,j}$. \item $\cD_\rho(C)$ is the kernel of the linear map $\Phi$ from the space of these coefficients to the Laurent coefficients that the order conditions require to vanish, together with the coefficient of $s^{-1}$ at $\infty$; it has dimension $3$. \item If a square submatrix $\Phi_{R,F'}$ of size $n-3$ ($n$ the number of unknowns), on a row set $R$ and the complement $F'$ of three columns $F$, is invertible, then the vectors $x_f$, $f\in F$, with coordinate $1$ at $f$, $0$ at the other columns of $F$, and $-\Phi_{R,F'}^{-1}\Phi_{R,f}$ on $F'$, form a basis of $\ker\Phi$. \end{enumerate} \end{lemma} \begin{proof} (1) Away from $P$, $Y$ is holomorphic and invertible, so $r$ is holomorphic there; at $p\in P$, $P_pH_p$ is holomorphic and invertible, so $\operatorname{ord}_pr\ge-M_p$; and $r$ vanishes at $\infty$, so it is the sum of its principal parts, which are independent. (2) Each condition is the vanishing of finitely many Laurent coefficients, linear in the unknowns; the dimension is $3$ by Lemma~\ref{lem:sixfold}. (3) The $x_f$ lie in the kernel of the rows $R$, which has dimension $3$ since those rows have rank $n-3$, and they are independent; this kernel contains $\ker\Phi$, of dimension $3$. \end{proof} \begin{lemma}\label{lem:numerators} Let $B\subset\C$ be a set of $13$ points and $D_B$ the monic polynomial with roots $B$. Then $H^0(\PP^1,K_{\PP^1}^{\otimes2}(B))=\{N(s)\,ds^2/D_B(s):\deg N\le9\}$. \end{lemma} \begin{proof} A section is $g(s)\,ds^2$ with $g$ holomorphic on $\C-B$, with at most simple poles on $B$, and, since $ds^2$ has a pole of order four at $\infty$, $g=O(s^{-4})$ there. So $gD_B$ is an entire function of growth $O(s^{9})$, a polynomial of degree at most $9$; conversely every such quotient qualifies. \end{proof} We call the coefficients of $N$ the \emph{numerator coordinates}. In these coordinates and in bases of $\cD_\rho(C)$ and $\cD_{\bar\rho}(C)$, $\mu_C$ is a $9\times10$ matrix. \subsection{The isomonodromic deformation}\label{subsec:schlesinger} Let $P=\{t_q\}$ be the poles of a Fuchsian system with residues $A_q$, $\sum_qA_q=0$. Let $t_q(\theta)$ be analytic functions of a complex parameter $\theta$ near $0$, with $t_q(0)=t_q$, pairwise distinct, and consider the \emph{Schlesinger equations} \begin{equation}\label{eq:schlesinger} \frac{dA_p}{d\theta}=\sum_{q\ne p}[A_q,A_p]\,\frac{d}{d\theta}\log\big(t_p(\theta)-t_q(\theta)\big),\qquad A_p(0)=A_p . \end{equation} \begin{lemma}\label{lem:schlesinger} The solution of \eqref{eq:schlesinger} is analytic near $\theta=0$, and its Taylor series is the unique formal power series solution. For $\theta$ near $0$: \begin{enumerate} \item $\sum_qA_q(\theta)=0$, and each $A_q(\theta)$ is conjugate to $A_q$; \item the connection $\Omega=\sum_qA_q(\theta)\,d\log(s-t_q(\theta))$ on $\{(s,\theta):s\ne t_q(\theta)\}$ is flat, so the monodromy representation of the system at $\theta$, transported along the fibration, does not depend on $\theta$; in particular local monodromies stay trivial at the apparent singularities; \item if a pole $t_p$ does not move ($t_p(\theta)=t_p$) and the system at $\theta=0$ has normal form at $t_p$, then the system at $\theta$ has normal form at $t_p$, with $P_{t_p}(\theta)$ and $H_{t_p}(s,\theta)$ analytic in $\theta$. \end{enumerate} \end{lemma} \begin{proof} Equation \eqref{eq:schlesinger} is an analytic ordinary differential equation, so the initial value problem has a unique analytic solution, whose Taylor coefficients satisfy the recursion obtained by expanding \eqref{eq:schlesinger}; that recursion determines them uniquely. (1) The derivative of $\sum_pA_p$ is $\sum_p\sum_{q\ne p}[A_q,A_p]\,m_{pq}$ with $m_{pq}=\frac{d}{d\theta}\log(t_p-t_q)$ symmetric in $p,q$, which vanishes by antisymmetry. Each equation has the form $A_p'=[S_p,A_p]$, so $A_p(\theta)=G_p(\theta)A_pG_p(\theta)^{-1}$ with $G_p'=S_pG_p$, $G_p(0)=1$. (2) Put $\ell_q=d\log(s-t_q(\theta))$. Then $d\Omega=\sum_qdA_q\wedge\ell_q$ and $\Omega\wedge\Omega=\sum_{p=stealth] \foreach \a/\n in {90/0,0/1,-90/2,180/3} { \draw[->] (0,0) -- (\a:2.9); \node at (\a:3.3) {$e_{\n}$}; } \foreach \a/\n in {45/1,-45/2,-135/3,135/4} { \fill (\a:1.4) circle (2pt); \node at ($(\a:1.4)+(\a+90:0.32)$) {$p_{\n}$}; \draw[->] (0,0) -- (\a:1.33); \node at ($(\a:0.75)+(\a-90:0.2)$) {$f_{\n}$}; \draw[->,dashed] (\a:1.47) -- (\a:2.9); \node at ($(\a:2.25)+(\a-90:0.22)$) {$g_{\n}$}; } \fill (0,0) circle (2pt); \node at (-67.5:0.42) {$b_0$}; \node at (3.6,-2.6) {$b_\infty=\infty$}; \end{tikzpicture} \caption{The $\Delta$-complex of Lemma~\ref{lem:cocycle} for $r=4$, drawn in $\PP^1-\{\infty\}$. The edges $e_i$ and $g_i$ end at $b_\infty=\infty$. The sector between $e_{i-1}$ and $e_i$ contains $p_i$ and is cut by the path $f_i\cup g_i$ into the two triangles at $p_i$. The loop $\ell_i$, out along $e_{i-1}$ and back along $e_i$, runs clockwise around $p_i$.}\label{fig:delta} \end{figure} \begin{proof} We use the following $\Delta$-complex structure on $\PP^1$ (Figure~\ref{fig:delta}). Its vertices are a base point $b_0$, a second point $b_\infty$, and the punctures $p_1,\dots,p_r$. Its edges are $e_0,\dots,e_{r-1}$ from $b_0$ to $b_\infty$, cutting the sphere into $r$ sectors, with the sector between $e_{i-1}$ and $e_i$ containing $p_i$ (indices of $e$ modulo $r$); and, for each $i$, an edge $f_i$ from $b_0$ to $p_i$ and an edge $g_i$ from $p_i$ to $b_\infty$ inside that sector. Its faces are the $2r$ triangles with vertices $(b_0,p_i,b_\infty)$: for each $i$ one with edges $f_i,g_i,e_{i-1}$ and one with edges $f_i,g_i,e_i$. The loop $\ell_i$ goes out along $e_{i-1}$ and back along $e_i$, and we orient $\PP^1$ so that it runs clockwise around $p_i$. The monodromy group is finite, so there is a finite Galois cover branched over the punctures on which $\mathsf W$ becomes trivial; lift the complex to it. There the cup product of $1$-cochains followed by $h_{\mathsf W}$, evaluated on the fundamental class, is the simplicial cup product $(f\cup f')[v_0,v_1,v_2]=h_{\mathsf W}(f[v_0,v_1],f'[v_1,v_2])$ \cite[\S3.2]{Hatcher}. Equivariant cochains contribute the same term on every lift of a triangle, so dividing by the degree gives a sum over the triangles of $\PP^1$. Choose the lifts of the two triangles at $p_i$ so that they share their edge $g_i$; their edges $f_i$ are then two lifts that differ by the local monodromy at $p_i$, and the two triangles enter the fundamental class with opposite signs. The terms of the two triangles combine, by the cocycle condition on each triangle, into \[ \cS_{\rm simp}(f,f')=\sum_ih_{\mathsf W}\big(v^f_{e_i}-v^f_{e_{i-1}},\ v^{f'}_{g_i}\big), \] with $v^f_\epsilon$ the value of $f$ on the chosen lift of $\epsilon$. Write $a,c,F$ for the data of $f$ and $a',c',F'$ for those of $f'$. The cocycle conditions on the two triangles at the $i$-th puncture give $a_i=Q_i^{-1}v_{f_i}$ and $v_{e_{i-1}}=v_{e_0}-F_i$. Substituting these into $\cS_{\rm simp}$ and summing gives \[ \cS_{\rm simp}(f,f')=-\sum_i\bigl[h_{\mathsf W}(c_i,a'_i)+h_{\mathsf W}(Q_ic_i,F'_i)\bigr]+h_{\mathsf W}\Bigl(\sum_iQ_ic_i,\ v^{f'}_{e_0}\Bigr), \] and the last term vanishes by the cocycle relation $\sum_iQ_ic_i=0$. By unitarity of $Q_i$, $h_{\mathsf W}(c_i,Q_i^{-1}F'_i)=h_{\mathsf W}(Q_ic_i,F'_i)$, so \[ \overline{\cS(b,a)}=\sum_ih_{\mathsf W}\bigl(c_i,\ Q_i^{-1}F'_i+a'_i\bigr)=\sum_i\bigl[h_{\mathsf W}(c_i,a'_i)+h_{\mathsf W}(Q_ic_i,F'_i)\bigr], \] and $\cS_{\rm simp}(f,f')=-\overline{\cS(b,a)}$. It remains to show $\cS(a,b)+\overline{\cS(b,a)}=0$. Since $T_i$ is unitary and $c_i=(1-T_i)a_i$, one has $h_{\mathsf W}(c_i,c'_i)=h_{\mathsf W}(a_i,c'_i)+h_{\mathsf W}(c_i,a'_i)$, and since $F_{i+1}=F_i+Q_ic_i$, \[ h_{\mathsf W}(F_{i+1},F'_{i+1})-h_{\mathsf W}(F_i,F'_i)=h_{\mathsf W}(F_i,Q_ic'_i)+h_{\mathsf W}(Q_ic_i,F'_i)+h_{\mathsf W}(c_i,c'_i). \] Hence \[ \begin{aligned} \cS(a,b)+\overline{\cS(b,a)}&=\sum_i\bigl[h_{\mathsf W}(F_{i+1},F'_{i+1})-h_{\mathsf W}(F_i,F'_i)\bigr]\\ &=h_{\mathsf W}(F_{r+1},F'_{r+1})-h_{\mathsf W}(F_1,F'_1)=0, \end{aligned} \] because $F_1=0$ and $F_{r+1}=\sum_iQ_ic_i=0$. So $\cS_{\rm simp}=\cS$, and $\cS$ is skew-Hermitian. A coboundary pairs to zero by the same telescoping. Finally, replacing $a_i$ by $a_i+k$ with $T_ik=k$ changes neither $c_i$ nor any $F_j$, and changes $\cS(a,b)$ by $h_{\mathsf W}(k,c^b_i)=h_{\mathsf W}(k,a^b_i)-h_{\mathsf W}(T_i^{-1}k,a^b_i)=0$, by unitarity; the second variable enters only through the $c^b_i$. \end{proof} \begin{proposition}\label{prop:disc} For every $C\in\Hur^\circ$, $A_\rho(C)$ has Hermitian signature $(3,3)$ and discriminant $-2$ in $\Q^\times/\Nm K^\times$. \end{proposition} \begin{proof} By Lemma~\ref{lem:constancy} it suffices to treat $C_0$. By Proposition~\ref{prop:herm} we need $\det\hpar$ for $\LL'$, which is $\varphi^*\LL$ or its conjugate; the conjugate has the conjugate form, with the same determinant. Since $\varphi$ is finite, $H^1_{\rm par}(\PP^1_s,\varphi^*\LL)=H^1_{\rm par}(\PP^1_z,\LL\otimes\varphi_*K)$ with the same cup product, where $\varphi_*K$ is the local system of the permutation representation of $\varphi$, with its standard form. This is a parabolic cohomology on $\PP^1_z$ minus six points: $5,-7,0$, the two other critical values $c_1,c_2$ of $\varphi$, and $\infty$. Choose standard loops around them in this order. With respect to them the monodromy of $\LL$ is $(M_5,M_{-7},M_0,1,1,M_\infty)$, and $(M_5,M_{-7},M_0,M_\infty)$ is a product-one tuple in the classes of $r_1,r_2,x,y$ with respect to standard loops of $\PP^1_z-\{5,-7,0,\infty\}$; by rigidity (Section~\ref{subsec:explicit}) it is conjugate to $(r_1,r_2,x,y)$, by an element of $GL_4(K)$ since both tuples are defined over $K$ and absolutely irreducible, and the conjugation carries $h_{\LL}$ to a positive rational multiple of the standard form, which does not change the discriminant. The monodromy of $\varphi_*K$ is a product-one tuple $(1,1,\pi_0,\pi_1,\pi_2,\pi_\infty)$ of permutations of cycle types $[2,4]$, $[2,1^4]$, $[2,1^4]$, $[3,3]$ generating a transitive group. The transitive product-one tuples of these cycle types form exactly four classes under simultaneous conjugation by $S_6$, with representatives $(\pi_0,\pi_1,\pi_2,\pi_\infty)$: \[ \begin{array}{l} \big((1,5,2,4)(3,6),\ (1,6),\ (2,3),\ (1,4,3)(2,5,6)\big),\\ \big((1,5,2,4)(3,6),\ (1,6),\ (1,5),\ (1,4,2)(3,5,6)\big),\\ \big((1,5,2,4)(3,6),\ (1,6),\ (4,6),\ (1,6,3)(2,5,4)\big),\\ \big((1,5,2,4)(3,6),\ (1,4),\ (1,6),\ (1,6,3)(2,5,4)\big). \end{array} \] The computation below gives the same matrix for each, so it is not necessary to determine which class $\varphi$ realizes. For each of the four, Lemma~\ref{lem:cocycle} applied to the tuple $(T_1,\dots,T_6)=(r_1\otimes1,\,r_2\otimes1,\,x\otimes\pi_0,\,1\otimes\pi_1,\,1\otimes\pi_2,\,y\otimes\pi_\infty)$ of rank $24$, with $h_{\mathsf W}$ the tensor product of the standard forms, computes $\cS$ exactly over $K$. The parameters are the tuples $(a_1,\dots,a_6)\in(K^{24})^6$ with $\sum_iQ_i(1-T_i)a_i=0$; they form a space $Z$ of dimension $120$. On $Z$ the form has rank $6$; its radical, of dimension $114$, is spanned by the tuples coming from coboundaries and the tuples with every $a_i\in\ker(1-T_i)$. Let $z_1,\dots,z_{120}$ be the reduced row echelon basis of $Z$, in the coordinates $(a_1,\dots,a_6)$, and $z_{j_1},\dots,z_{j_6}$ the first six of them that are independent modulo the radical. On them $\hpar=\delta\cS$ has the Gram matrix \[ h_0=\begin{pmatrix} \tfrac23&-\tfrac13&0&-\omega-\tfrac13&-\tfrac23\omega&0\\[2pt] -\tfrac13&\tfrac23&0&\omega+\tfrac23&\tfrac13\omega&0\\[2pt] 0&0&\tfrac13&0&0&-\tfrac13\omega\\[2pt] \omega+\tfrac23&-\omega-\tfrac13&0&\tfrac23&-\tfrac23\omega&0\\[2pt] \tfrac23\omega+\tfrac23&-\tfrac13\omega-\tfrac13&0&\tfrac23\omega+\tfrac23&-\tfrac23&0\\[2pt] 0&0&\tfrac13\omega+\tfrac13&0&0&-\tfrac13 \end{pmatrix}, \] the same matrix for all four classes. Its characteristic polynomial is \[ x^6-\tfrac43x^5-\tfrac{26}9x^4+\tfrac{41}{27}x^3+\tfrac{19}{27}x^2-\tfrac{22}{81}x-\tfrac2{81}=\Big(x^2-\tfrac29\Big)\Big(x^4-\tfrac43x^3-\tfrac83x^2+\tfrac{11}9x+\tfrac19\Big), \] with six real roots, approximately $-1.326$, $-0.471$, $-0.078$, $0.471$, $0.474$ and $2.263$. So the signature is $(3,3)$ and $\det\hpar=-\tfrac2{81}=-2\cdot\Nm(\tfrac19)$, whose class is $-2$. \end{proof} %% ------------------------------------------------------------------ \section{From an open set to the whole moduli space}\label{sec:spread} %% ------------------------------------------------------------------ Let $\cM=\cM(K,3,-2)$ be the moduli space of Subsection~\ref{subsec:moduli}. \begin{lemma}\label{lem:locus} Let $\mathcal W=\bigwedge^{6}_KR^1f_*\Q\subset R^{6}f_*\Q$ on $\cM$. The set $\cM^{\rm alg}$ of points $t$ at which $\mathcal W_t$ is spanned by classes of algebraic cycles is a countable union of closed algebraic subsets of $\cM$. \end{lemma} \begin{proof} This is the standard argument with relative Hilbert schemes \cite[\S3.3.2, pp.~90--94, and the proofs of Theorems~7.19 and 8.26]{VoisinII}. Fix a relatively ample line bundle on $f\colon\mathcal A\to\cM$. For each Hilbert polynomial, the relative Hilbert scheme of three-dimensional closed subschemes of the fibres is projective over $\cM$; there are countably many Hilbert polynomials, and each relative Hilbert scheme has finitely many irreducible components. A rational algebraic class in $H^{6}(\mathcal A_t,\Q)$ is a rational combination of classes of such subschemes. Fix a finite tuple $\mathcal H_1,\dots,\mathcal H_k$ of components, let $T$ be an irreducible component of $\mathcal H_1\times_\cM\dots\times_\cM\mathcal H_k$, $\widetilde T\to T$ a resolution, and $g\colon\widetilde T\to\cM$ the composite; $g$ is proper, so $g(\widetilde T)$ is closed. The universal subschemes pull back to subschemes $\mathcal Z_l$ of $\mathcal A\times_\cM\widetilde T$, flat over $\widetilde T$, and their fibre classes are global flat sections of $g^*R^{6}f_*\Q$ \cite[proof of Theorem~3.33, p.~94]{VoisinII}. Let $\mathcal F_u$ be the span of $[\mathcal Z_{1,u}],\dots,[\mathcal Z_{k,u}]$. Parallel transport carries each of these global sections to itself, so it carries $\mathcal F_u$ onto $\mathcal F_{u'}$ and $g^*\mathcal W_u$ onto $g^*\mathcal W_{u'}$; so the condition $g^*\mathcal W_u\subseteq\mathcal F_u$ holds on all of the connected $\widetilde T$ or nowhere. Call $T$ \emph{good} in the first case; then $\mathcal W_t$ is algebraic for every $t\in g(\widetilde T)$. Conversely, if $\mathcal W_t$ is spanned by algebraic classes, choose subschemes $Z_1,\dots,Z_k$ of $\mathcal A_t$ whose classes span a space containing $\mathcal W_t$; the point $(Z_1,\dots,Z_k)$ lies on a component $T$, which is good. So $\cM^{\rm alg}$ is the union of the $g(\widetilde T)$ over the countably many good $T$. \end{proof} \begin{theorem}\label{thm:transfer} Let $S$ be a connected component of $\Hur$ on which $A_\rho(C)$ has discriminant $-2$, such that $W_K(A_\rho(C))$ is algebraic for every $C\in S$ and $\rk\mu_{C_1}=9$ at some $C_1\in S$. Then the Weil classes of every polarized abelian sixfold of Weil type for $K$ with discriminant $-2$ are algebraic. \end{theorem} \begin{proof} Choose a small polydisc $\Delta\ni C_1$ in the configuration space over which the covers form an analytic family. Over the simply connected $\Delta$ the spaces $H_1(A_\rho(C),\Q)$ with $\iota$ and $H$ form a trivial flat bundle of Hermitian spaces of discriminant $-2$, so Landherr's theorem gives a $K$-isometry $m\colon H_1(A_\rho(C_1),\Q)\to(V,H)$, which we transport over $\Delta$. Sending $C$ to the complex structure of $H_1(A_\rho(C),\R)$, transported by $m$, gives the period map $\Phi\colon\Delta\to\mathcal H_3$, a submersion at $C_1$ by Lemma~\ref{lem:codiff}. Shrinking $\Delta$, $\Phi(\Delta)$ contains a ball $\mathcal B$, whose image in $\cM$ is open, since $\Gamma(\ell)$ acts properly and freely. Let $C\in\Delta$ with $z=\Phi(C)\in\mathcal B$, and let $A_z$ be the fibre of $\mathcal A$ over the image of $z$. The lattice $m(H_1(A_\rho(C),\Z))$ is commensurable with $\Lambda$; if $N_1m(H_1(A_\rho(C),\Z))\subseteq\Lambda$, then $N_1m$ is a $K$-linear map of lattices compatible with the complex structures, hence a $K$-linear isogeny $A_\rho(C)\to A_z$. By hypothesis and Lemma~\ref{lem:isogeny}, the Weil classes of $A_z$ are algebraic. So $\cM^{\rm alg}$ contains a nonempty open set. A nonempty open subset of $\cM$ is a Baire space, so by Lemma~\ref{lem:locus} one of the closed sets has nonempty interior; since $\cM$ is irreducible, it is all of $\cM$. Every polarized sixfold of Weil type for $K$ with discriminant $-2$ is $K$-isogenous to a fibre of $\mathcal A$ (Lemma~\ref{lem:isogeny}). \end{proof} \begin{proof}[Proof of Theorem~A] By Lemma~\ref{lem:sixfold}, $A_\rho(C)$ is a polarized sixfold of Weil type for $K$ for every $C\in\Hur$; by Proposition~\ref{prop:supply} its Weil classes are algebraic; by Proposition~\ref{prop:disc} its discriminant is $-2$ on $\Hur^\circ$; and by Corollary~\ref{cor:dominant} $\rk\mu_{C_\theta}=9$ at points of $\Hur^\circ$. Theorem~\ref{thm:transfer} with $S=\Hur^\circ$ concludes. \end{proof} %% ------------------------------------------------------------------ \section{Type III sixfolds}\label{sec:typeIII} %% ------------------------------------------------------------------ Let $A$ be a simple complex abelian sixfold with $\End^0(A)=D$, a totally definite quaternion algebra over $\Q$, with canonical involution $b\mapsto\bar b$. Then $H_1(A,\Q)\cong D^3$ as a left $D$-module, and the polarization is $E(x,y)=\operatorname{Trd}_{D/\Q}T(x,y)$ for a nondegenerate form $T$ on $D^3$ with $T(ax,by)=a\,T(x,y)\,\bar b$ and $\overline{T(y,x)}=-T(x,y)$ \cite[\S9.6]{BL}. Such a $T$ is diagonal in a suitable $D$-basis, with entries pure quaternions $\lambda_1,\lambda_2,\lambda_3$ (since $T(x,x)\ne0$ for some $x$, a Gram--Schmidt argument applies), and $\disc(T)=\lambda_1^2\lambda_2^2\lambda_3^2\in\Q^\times/\Q^{\times2}$ does not depend on the basis, since a change of basis by $P$ multiplies it by $\operatorname{Nrd}(P)^2$. Suppose $K=\Q(u)\subset D$ with $u^2=-3$; choose a pure quaternion $j$ with $ju=-uj$, and put $b=j^2<0$, so that $D=(-3,b)_\Q$ and $D=K\oplus Kj$. \begin{lemma}\label{lem:typeIII} With $K$ acting by left multiplication, $(A,K,E)$ is a polarized sixfold of Weil type for $K$, and its Hermitian discriminant is $-b\,\disc(T)$ in $\Q^\times/\Nm K^\times$. \end{lemma} \begin{proof} The Rosati involution of $E$ is the canonical involution of $D$, which restricts to complex conjugation on $K$. Since $j$ anticommutes with $u$ and commutes with the complex structure, it exchanges the two eigenspaces of $u$ on $H^{1,0}(A)$, so both embeddings of $K$ occur three times. For $q\in D$ write $q=q_1+q_2j$ with $q_1,q_2\in K$ and put $\operatorname{pr}(q)=q_1$. Then $T_K=\operatorname{pr}\circ T$ is $K$-sesquilinear (linear in the first variable) and skew-Hermitian, and $\Tr_{K/\Q}T_K=\operatorname{Trd}\circ T=E$, because $q_2j$ is a pure quaternion. Van Geemen's form $H(x,y)=E(x,\iota(u)y)+uE(x,y)$ is linear in the second variable; with $t=T_K(x,y)$ one has $T_K(x,uy)=-ut$, so $H(x,y)=\Tr_{K/\Q}(-ut)+u\Tr_{K/\Q}(t)=2u\,\bar t$, that is $H=2u\,\overline{T_K}$. Since ${}^t\overline{T_K}=-T_K$ and the size is six, $\det T_K$ is rational, and $\det H=(2u)^6\det T_K=-27\cdot64\det T_K$, which is $-\det T_K$ in $\Q^\times/\Nm K^\times$ because $27=\Nm(3u)$ and $64=\Nm(8)$. On a block $De$ with $T(e,e)=\lambda=\ell_1u+\ell_2j+\ell_3uj$, take the $K$-basis $(e,je)$; using $jz=\bar zj$ for $z\in K$ one finds \[ T_K=\begin{pmatrix}\ell_1u&-b(\ell_2+\ell_3u)\\ b(\ell_2-\ell_3u)&b\,\ell_1u\end{pmatrix}, \] with $\det T_K=-3b\ell_1^2+b^2\ell_2^2+3b^2\ell_3^2=b\lambda^2$ on this block. Taking the three blocks, $\det T_K=b^3\disc(T)$ for the whole form, and $\disc H=-b^3\disc(T)=-b\,\disc(T)$, since $b^2=\Nm(b)$. \end{proof} \begin{corollary}\label{cor:typeIII} If $-b\,\disc(T)=-2$ in $\Q^\times/\Nm K^\times$, then $W_K(A)$ is spanned by algebraic classes. If moreover the PEL family of type III defined by $(D,T)$ is effective, then the Hodge conjecture holds for every power of every general member of that family. \end{corollary} \begin{proof} The first statement is Theorem~A with Lemma~\ref{lem:typeIII}. For the second, Abdulali calls a PEL type \emph{effective} if its arithmetic group is torsion free and the generic member has endomorphism algebra $D$ \cite[\S2.5, p.~670]{AbdulaliIII}, and a member \emph{general} if its Hodge group is that of the generic member \cite[\S2, p.~670]{AbdulaliIII}. His Theorem~4.1 \cite[p.~673]{AbdulaliIII} states: \emph{Let $A$ be a general member of an effective PEL-family of type III. The Hodge ring of $A$ is generated by divisors, together with the Weil cycles associated to a maximal subfield of $\End_\Q(A)$. The algebraicity of the Weil cycles on $A$ implies the usual Hodge conjecture for all powers of $A$.} The field $K$ is a maximal subfield of $D$. \end{proof} \begin{remark}\label{rem:typeIII} Take $D=(-1,-1)_\Q$, ramified exactly at $2$ and $\infty$. It contains $K$, since $2$ is inert in $K$, and $D\cong(-3,-2)_\Q$, so we may take $b=-2$. Let $i,j$ be the standard generators of $(-3,-2)_\Q$, $k=ij$, and $T=\operatorname{diag}(i,\,j,\,3j+2k)$. The squares of the entries are $-3$, $-2$ and $-2\cdot9-6\cdot4=-42$, so $\disc(T)=-252=-7\cdot6^2$ and $-b\,\disc(T)=-504=-2\cdot252$, with $252=\Nm(3+9u)$; the class is $-2$. No imaginary quadratic subfield $K'=\Q(u')\subset D$, $u'^2=-d'$, gives a split class: writing $D=(-d',b')_\Q$, the class for $K'$ is $-b'\disc(T)$, which is split exactly when $b'\disc(T)\in\Nm K'^\times$, that is, when the quaternion algebras $(-d',\disc(T))_\Q$ and $(-d',b')_\Q=D$ coincide. The first contains $\Q(\sqrt{\disc T})=\Q(\sqrt{-7})$, in which $2$ splits; a quaternion algebra containing a quadratic field is split at every prime that splits in that field, since a division algebra over $\Q_2$ contains no copy of $\Q_2\times\Q_2$. So the first algebra is split at $2$, while $D$ is ramified at $2$, and they differ. So neither Markman's theorem nor Schoen's split case applies to this family. \end{remark} %% ------------------------------------------------------------------ \section{Powers: proof of Theorem~B}\label{sec:powers} %% ------------------------------------------------------------------ Let $\mathsf W$ be a complex vector space of dimension $N$, and $\mathsf U_1,\mathsf U_2$ complex vector spaces of dimension $m$. Put \[ \Lambda=\textstyle\bigwedge\big(\mathsf W\otimes\mathsf U_1\big)\otimes\bigwedge\big(\mathsf W^*\otimes\mathsf U_2\big)=\bigwedge\big((\mathsf W\otimes\mathsf U_1)\oplus(\mathsf W^*\otimes\mathsf U_2)\big), \] with the commuting actions of $GL(\mathsf W)$ and $GL(\mathsf U_1)\times GL(\mathsf U_2)$. For $u\in\mathsf U_1$, $u'\in\mathsf U_2$ let $\Theta(u,u')=\sum_i(w_i\otimes u)\wedge(w_i^*\otimes u')$ for dual bases, and for $x\in\mathsf U_1$ let $\det(\mathsf W\otimes x)\in\bigwedge^N(\mathsf W\otimes\mathsf U_1)$ be the image of a generator of $\bigwedge^N\mathsf W$ under $w\mapsto w\otimes x$; define $\det(\mathsf W^*\otimes x')$ similarly. \begin{lemma}\label{lem:sl} $\Lambda^{SL(\mathsf W)}$ is generated by the $\Theta(u,u')$, the $\det(\mathsf W\otimes x)$ and the $\det(\mathsf W^*\otimes x')$. \end{lemma} \begin{proof} We deduce it from the first fundamental theorem for $SL(\mathsf W)$, in the form of Kraft and Procesi \cite[\S8.4, Theorem (FFT for $SL_n$), p.~90]{KraftProcesi}: \emph{the ring of invariants $K[V^p\oplus V^{*q}]^{SL(V)}$ is generated by the scalar products $\langle j\mid i\rangle$ and the determinants $[i_1,\dots,i_n]$ and $[j_1,\dots,j_n]^*$}, for a vector space $V$ of dimension $n$ over a field of characteristic zero. An $SL(\mathsf W)$-invariant element of $\mathsf W^{\otimes p}\otimes\mathsf W^{*\otimes q}$ is an invariant polynomial on $\mathsf W^{*p}\oplus\mathsf W^q$ that is homogeneous of degree one in each of the $p+q$ vector variables; apply the theorem with $V=\mathsf W^*$. A polynomial in the scalar products and determinants that is of degree one in each variable is a linear combination of products in which every variable occurs exactly once. So the invariants in $\mathsf W^{\otimes p}\otimes\mathsf W^{*\otimes q}$ are spanned by tensor products of contractions $\sum_iw_i\otimes w_i^*$ (one factor $\mathsf W$ and one factor $\mathsf W^*$), determinant tensors $\sum_{\pi\in S_N}\operatorname{sgn}(\pi)\,w_{\pi(1)}\otimes\dots\otimes w_{\pi(N)}$ (in $N$ factors $\mathsf W$), and dual determinant tensors. The homogeneous piece $\bigwedge^p(\mathsf W\otimes\mathsf U_1)\otimes\bigwedge^q(\mathsf W^*\otimes\mathsf U_2)$ is the image of $\mathsf W^{\otimes p}\otimes\mathsf W^{*\otimes q}\otimes\mathsf U_1^{\otimes p}\otimes\mathsf U_2^{\otimes q}$ under antisymmetrization, which is equivariant and surjective; since $SL(\mathsf W)$ is reductive, the invariants of the image are the image of the invariants. Antisymmetrization turns tensor products into wedge products, up to sign. The image of a contraction in the factors $(w\otimes u_k)$ and $(w^*\otimes u'_l)$ is $\Theta(u_k,u'_l)$. The image of a determinant tensor in the factors $u_{k_1},\dots,u_{k_N}$ is the full polarization of the homogeneous map $x\mapsto N!\det(\mathsf W\otimes x)$ of degree $N$, evaluated at $(u_{k_1},\dots,u_{k_N})$, which by the polarization identity is a linear combination of values $\det(\mathsf W\otimes x)$; likewise for the dual determinant tensors. \end{proof} \begin{theorem}\label{thm:powers} Let $(A,\iota,E)$ be a polarized abelian $2n$-fold of Weil type for $K$, $n\ge2$, whose Weil classes are algebraic and whose Hodge group is $SU_H$. Then every Hodge class on every power $A^m$ is algebraic. \end{theorem} \begin{proof} Let $H^1(A,\C)=\mathsf W\oplus\mathsf W'$ be the decomposition into the two eigenspaces of $K$. The polarization identifies $\mathsf W'$ with $\mathsf W^*$ as representations of $SU_H(\C)=SL(\mathsf W)$. The Hodge group of $A^m$ is $SU_H$ acting diagonally, so the Hodge classes on $A^m$, tensored with $\C$, are $\Lambda^{SL(\mathsf W)}$ with $N=2n$ and $\mathsf U_1=\mathsf U_2=\C^m$, the coordinates of $A^m$ kept separate on the two eigenspaces. The $\Theta(u,u')$ are Hodge classes of degree two, hence combinations of divisor classes. Let $\mathcal O=\{y\in K:\iota(y)\in\End(A)\}$. For $y\in\mathcal O^m$ the homomorphism $f_y\colon A^m\to A$, $(a_k)\mapsto\sum_k\iota(y_k)a_k$, pulls the two Weil lines of $A$ back to $\det(\mathsf W\otimes\tilde y)$ and $\det(\mathsf W^*\otimes\overline{\tilde y})$ separately, with $\tilde y\in\C^m$ the image of $y$ under the given embedding; these are algebraic. Since $\mathcal O$ is a lattice in $K$, the set of $\tilde y$ is Zariski dense in $\mathsf U_1$, and so is the set of $\overline{\tilde y}$ in $\mathsf U_2$; so these classes span all $\det(\mathsf W\otimes x)$ and all $\det(\mathsf W^*\otimes x')$. By Lemma~\ref{lem:sl}, products of algebraic classes span all Hodge classes over $\C$, and rational descent as at the end of the proof of Proposition~\ref{prop:supply} finishes the proof. \end{proof} \begin{proof}[Proof of Theorem~B] Theorem~A and Theorem~\ref{thm:powers} with $n=3$. \end{proof} Members of $\cM$ whose Hodge group is smaller than $SU_H$ carry further Hodge classes on their powers; we make no claim about them. %% ------------------------------------------------------------------ \appendix \section{The group, the rigid system and the base cover}\label{app:tuple} %% ------------------------------------------------------------------ With $\omega=e^{2\pi i/3}$, let \[ r_1=\begin{pmatrix}\frac{\omega+2}3&\frac{2\omega+1}3&0&-\frac{2\omega+1}3\\[2pt]-\frac{\omega+2}3&\frac{\omega+2}3&0&\frac{1-\omega}3\\[2pt]0&0&1&0\\[2pt]\frac{\omega+2}3&\frac{1-\omega}3&0&\frac{\omega+2}3\end{pmatrix},\qquad r_2=\begin{pmatrix}\frac{1-\omega}3&\frac{2\omega+1}3&\frac{2\omega+1}3&0\\[2pt]\frac{1-\omega}3&\frac{1-\omega}3&-\frac{\omega+2}3&0\\[2pt]\frac{1-\omega}3&-\frac{\omega+2}3&\frac{1-\omega}3&0\\[2pt]0&0&0&1\end{pmatrix}, \] \[ x=\begin{pmatrix}-1&0&0&0\\[2pt]0&-\frac{\omega+2}3&-\frac{2\omega+1}3&\frac{1-\omega}3\\[2pt]0&\frac{\omega+2}3&-\frac{\omega+2}3&\frac{\omega+2}3\\[2pt]0&-\frac{\omega+2}3&\frac{\omega-1}3&\frac{2\omega+1}3\end{pmatrix}, \] \[ y=\begin{pmatrix}\frac{\omega-1}3&0&-\frac{\omega+2}3&-\frac{\omega+2}3\\[2pt]0&\frac{\omega-1}3&\frac{1-\omega}3&\frac{\omega-1}3\\[2pt]\frac{1-\omega}3&-\frac{\omega+2}3&-\frac{\omega+2}3&0\\[2pt]\frac{1-\omega}3&\frac{\omega+2}3&0&-\frac{\omega+2}3\end{pmatrix}, \] matrices acting on column vectors. They are unitary for the standard Hermitian form, $r_1r_2xy=1$, $r_1$ has eigenvalues $(\omega,1,1,1)$, $r_2$ has $(\omega^2,1,1,1)$, $x$ has $(i,-1,-1,-i)$ with $x^2$ of eigenvalues $(1,1,-1,-1)$ and $x^4=1$, and $y$ has order three with eigenvalues $(\omega,\omega,\omega^2,\omega^2)$. The group $G=\langle r_1,r_2,x,y\rangle$ has order $155\,520$; the $K$-algebra generated by $r_1,r_2,x$ is $M_4(K)$; the derived group of $G$ has index three and is $\ker\det$; $G/Z(G)$ has order $25\,920$; every irreducible character of $G$ is $K$-valued; and $\det r_1=\omega$. These facts are checked by the certificate scripts in GAP~4.14 \cite{GAP} and SageMath~10.7 \cite{Sage} described in Appendix~\ref{app:files}. The residues of the rigid system of Subsection~\ref{subsec:explicit} are $B_0=\operatorname{diag}(-\tfrac34,-\tfrac12,-\tfrac12,\tfrac34)$, $B_5=u_1v_1^T$ and $B_{-7}=u_2v_2^T$ with \begin{align*} u_1&=(1,\,1,\,0,\,1),\\ v_1&=\big(\tfrac{119}{648},\,\tfrac{28}{135},\,-\tfrac{41472}{289},\,-\tfrac{187}{3240}\big),\\ u_2&=\big(-\tfrac{5598720}{14161},\,0,\,1,\,\tfrac{5598720}{22253}\big),\\ v_2&=\big(-\tfrac{2648107}{3627970560},\,-\tfrac{1090397}{755827200},\,\tfrac{56}{135},\,-\tfrac{2648107}{18139852800}\big). \end{align*} One checks directly that $\operatorname{tr}B_5=\tfrac13$, $\operatorname{tr}B_{-7}=\tfrac23$, and that $B_\infty=-(B_0+B_5+B_{-7})$ is diagonalizable with eigenvalues $\pm\tfrac23$, each twice. The derivative of $\varphi$ is \[ \varphi'(s)=\frac{3s(s-3)^3(5s^2-3s+4)}{(s-1)^4(s+2)^4}, \] so the critical points are $0$, $3$, $1$, $-2$ and the roots of $5s^2-3s+4$, whose critical values are the roots of $729z^2-3674z+5120$; $\varphi(\infty)=1$; and the fibres over $5$ and $-7$ consist of the six distinct roots of $P_5$ and of $P_{-7}$. The permutation monodromy of $\varphi$ generates $S_6$. %% ------------------------------------------------------------------ \section{The character identity}\label{app:witness} %% ------------------------------------------------------------------ Table~\ref{tab:witness} lists the fifty-one suppliers $(H'_i,\chi_i)$ of Proposition~\ref{prop:lattice}. The columns are: the coefficient $n_i$; the order of $H'_i$; the order $m$ of $\chi_i$; the genus $q$ of $C/H'_i$; the number $s$ of branch points of $C/\ker\chi_i\to C/H'_i$; $h=2q-2+s$; $d=\gcd(m,\alpha)$ for ramified rows; the exponents, written $a^k$ for the exponent $a$ occurring $k$ times; and the type: U for unramified, S for ramified with simple exponents, and S$^*$ for the three rows that use Lemma~\ref{lem:schoenR}. Every row is admissible: in each ramified row the exponents $a$ and $m-a$ occur equally often (for $a=m/2$ an even number of times), and $h$ is even and positive. {\small \setlength{\tabcolsep}{3.5pt} \begin{longtable}{rrrrrrrrlc} \caption{The suppliers.}\label{tab:witness}\\ $i$ & $n_i$ & $|H'_i|$ & $m$ & $q$ & $s$ & $h$ & $d$ & exponents & type\\ \hline \endfirsthead $i$ & $n_i$ & $|H'_i|$ & $m$ & $q$ & $s$ & $h$ & $d$ & exponents & type\\ \hline \endhead 1 & $-1$ & $155520$ & $3$ & $0$ & $12$ & $10$ & 1 & $1^{6}, 2^{6}$ & S \\ 2 & $-1$ & $3456$ & $6$ & $108$ & $12$ & $226$ & 3 & $3^{12}$ & S \\ 3 & $1$ & $4320$ & $6$ & $115$ & $12$ & $240$ & 3 & $3^{12}$ & S \\ 4 & $1$ & $1728$ & $3$ & $221$ & $108$ & $548$ & 1 & $1^{54}, 2^{54}$ & S \\ 5 & $-1$ & $486$ & $3$ & $905$ & $288$ & $2096$ & 1 & $1^{144}, 2^{144}$ & S \\ 6 & $1$ & $144$ & $6$ & $3505$ & $12$ & $7020$ & 3 & $3^{12}$ & S \\ 7 & $1$ & $1944$ & $3$ & $229$ & $96$ & $552$ & 1 & $1^{48}, 2^{48}$ & S \\ 8 & $-1$ & $648$ & $6$ & $685$ & $288$ & $1656$ & 2 & $2^{144}, 4^{144}$ & S \\ 9 & $-2$ & $486$ & $6$ & $913$ & $96$ & $1920$ & 2 & $2^{48}, 4^{48}$ & S \\ 10 & $-3$ & $486$ & $6$ & $913$ & $384$ & $2208$ & 2 & $2^{192}, 4^{192}$ & S \\ 11 & $-1$ & $486$ & $6$ & $913$ & $96$ & $1920$ & 2 & $2^{48}, 4^{48}$ & S \\ 12 & $-2$ & $486$ & $6$ & $913$ & $384$ & $2208$ & 2 & $2^{192}, 4^{192}$ & S \\ 13 & $-1$ & $24$ & $12$ & $21025$ & $72$ & $42120$ & 6 & $6^{72}$ & S$^*$ \\ 14 & $-1$ & $24$ & $12$ & $21025$ & $72$ & $42120$ & 6 & $6^{72}$ & S$^*$ \\ 15 & $-1$ & $96$ & $12$ & $5257$ & $18$ & $10530$ & 6 & $6^{18}$ & S$^*$ \\ 16 & $1$ & $15$ & $15$ & $33697$ & $0$ & $67392$ & -- & -- & U \\ 17 & $-1$ & $1296$ & $6$ & $339$ & $144$ & $820$ & 2 & $2^{72}, 4^{72}$ & S \\ 18 & $-1$ & $36$ & $6$ & $14017$ & $96$ & $28128$ & 3 & $3^{96}$ & S \\ 19 & $-1$ & $1152$ & $6$ & $436$ & $12$ & $882$ & 3 & $3^{12}$ & S \\ 20 & $1$ & $144$ & $6$ & $3397$ & $324$ & $7116$ & 2 & $2^{162}, 4^{162}$ & S \\ 21 & $1$ & $972$ & $3$ & $449$ & $192$ & $1088$ & 1 & $1^{96}, 2^{96}$ & S \\ 22 & $-1$ & $576$ & $6$ & $871$ & $0$ & $1740$ & -- & -- & U \\ 23 & $1$ & $108$ & $3$ & $4385$ & $432$ & $9200$ & 1 & $1^{216}, 2^{216}$ & S \\ 24 & $-1$ & $1152$ & $6$ & $430$ & $36$ & $894$ & 3 & $3^{36}$ & S \\ 25 & $-1$ & $576$ & $6$ & $871$ & $12$ & $1752$ & 3 & $3^{12}$ & S \\ 26 & $-1$ & $576$ & $6$ & $769$ & $324$ & $1860$ & 2 & $2^{162}, 4^{162}$ & S \\ 27 & $-1$ & $486$ & $3$ & $913$ & $96$ & $1920$ & 1 & $1^{48}, 2^{48}$ & S \\ 28 & $-1$ & $486$ & $6$ & $913$ & $288$ & $2112$ & 2 & $2^{144}, 4^{144}$ & S \\ 29 & $-1$ & $486$ & $3$ & $913$ & $384$ & $2208$ & 1 & $1^{192}, 2^{192}$ & S \\ 30 & $-1$ & $324$ & $6$ & $1537$ & $48$ & $3120$ & 3 & $3^{48}$ & S \\ 31 & $1$ & $18$ & $6$ & $28057$ & $96$ & $56208$ & 3 & $3^{96}$ & S \\ 32 & $-1$ & $648$ & $3$ & $685$ & $288$ & $1656$ & 1 & $1^{144}, 2^{144}$ & S \\ 33 & $-1$ & $432$ & $3$ & $989$ & $108$ & $2084$ & 1 & $1^{54}, 2^{54}$ & S \\ 34 & $-1$ & $480$ & $15$ & $1045$ & $0$ & $2088$ & -- & -- & U \\ 35 & $-1$ & $432$ & $3$ & $989$ & $540$ & $2516$ & 1 & $1^{270}, 2^{270}$ & S \\ 36 & $-1$ & $384$ & $6$ & $1306$ & $36$ & $2646$ & 3 & $3^{36}$ & S \\ 37 & $-1$ & $144$ & $6$ & $3073$ & $12$ & $6156$ & 3 & $3^{12}$ & S \\ 38 & $-1$ & $144$ & $3$ & $3505$ & $0$ & $7008$ & -- & -- & U \\ 39 & $1$ & $144$ & $6$ & $3505$ & $12$ & $7020$ & 3 & $3^{12}$ & S \\ 40 & $1$ & $144$ & $3$ & $3397$ & $324$ & $7116$ & 1 & $1^{162}, 2^{162}$ & S \\ 41 & $1$ & $54$ & $6$ & $9361$ & $0$ & $18720$ & -- & -- & U \\ 42 & $1$ & $576$ & $3$ & $769$ & $324$ & $1860$ & 1 & $1^{162}, 2^{162}$ & S \\ 43 & $-1$ & $432$ & $6$ & $989$ & $4$ & $1980$ & 3 & $3^{4}$ & S \\ 44 & $-1$ & $432$ & $3$ & $989$ & $108$ & $2084$ & 1 & $1^{54}, 2^{54}$ & S \\ 45 & $-1$ & $486$ & $3$ & $905$ & $384$ & $2192$ & 1 & $1^{192}, 2^{192}$ & S \\ 46 & $-2$ & $432$ & $6$ & $1159$ & $48$ & $2364$ & 3 & $3^{48}$ & S \\ 47 & $-1$ & $432$ & $3$ & $989$ & $432$ & $2408$ & 1 & $1^{216}, 2^{216}$ & S \\ 48 & $-1$ & $384$ & $6$ & $1306$ & $0$ & $2610$ & -- & -- & U \\ 49 & $-1$ & $324$ & $12$ & $1369$ & $288$ & $3024$ & 4 & $4^{144}, 8^{144}$ & S \\ 50 & $1$ & $243$ & $3$ & $1825$ & $576$ & $4224$ & 1 & $1^{288}, 2^{288}$ & S \\ 51 & $-1$ & $216$ & $3$ & $1979$ & $1080$ & $5036$ & 1 & $1^{540}, 2^{540}$ & S \\ \end{longtable} } \emph{The vectors.} The coordinates of $X^*(T)=\Z^{44}\oplus(\Z/2)^4$ are numbered $1,\dots,44$ for the pairs of non-real irreducible characters with $m_\tau>0$ and $45,\dots,48$ for the quaternionic ones, in the order in which the certificate enumerates the irreducible characters of $G$; the pair of $\rho$ is coordinate $3$, so $e_\rho$ is the third coordinate vector. Table~\ref{tab:vectors} lists the nonzero coordinates of each $a(\chi_i)$ as pairs (coordinate:value). Summing $n_i\,a(\chi_i)$ gives $1$ in coordinate $3$, $0$ in the other free coordinates, and even values in coordinates $45$ to $48$. {\small \setlength{\tabcolsep}{3pt} \begin{longtable}{rrp{0.8\textwidth}} \caption{The nonzero coordinates of $a(\chi_i)$.}\label{tab:vectors}\\ $i$ & $n_i$ & $a(\chi_i)$\\ \hline \endfirsthead $i$ & $n_i$ & $a(\chi_i)$\\ \hline \endhead 1 & $-1$ & 1:1\\ 2 & $-1$ & 6:1, 31:1\\ 3 & $1$ & 8:1, 26:1\\ 4 & $1$ & 25:1, 40:1\\ 5 & $-1$ & 2:1, 9:1, 23:$-1$, 27:1\\ 6 & $1$ & 36:1, 37:1, 39:$-1$, 46:1\\ 7 & $1$ & 1:1, 6:1, 7:1, 12:1, 22:1, 26:1\\ 8 & $-1$ & 13:1, 22:1, 26:1, 34:1, 35:1, 44:1\\ 9 & $-2$ & 19:1, 28:1, 37:1, 38:1, 42:1, 43:1\\ 10 & $-3$ & 17:1, 29:1, 38:1, 39:1, 42:1, 43:1\\ 11 & $-1$ & 3:1, 16:1, 18:1, 28:1, 37:2, 38:1, 39:1\\ 12 & $-2$ & 4:1, 16:1, 20:1, 29:1, 37:1, 38:1, 39:2\\ 13 & $-1$ & 2:$-1$, 3:1, 4:$-1$, 15:$-1$, 17:$-1$, 19:1, 27:$-3$, 28:3, 29:$-3$, 36:2, 37:2, 39:$-2$, 46:1\\ 14 & $-1$ & 2:$-1$, 3:$-1$, 15:$-1$, 16:$-1$, 17:1, 18:2, 20:2, 27:$-3$, 28:$-3$, 36:2, 37:$-1$, 39:1, 42:$-2$, 43:2, 46:1\\ 15 & $-1$ & 16:1, 17:1, 18:1, 19:1, 20:1, 28:2, 29:2, 37:4, 38:4, 39:4, 42:4, 43:5\\ 16 & $1$ & 3:1, 4:1, 6:1, 7:1, 8:1, 10:2, 11:2, 12:3, 13:3, 16:4, 17:4, 18:4, 19:4, 20:4, 21:4, 22:5, 24:6, 25:6, 26:6, 28:7, 29:7, 31:8, 32:8, 34:9, 35:9, 37:12, 38:12, 39:12, 40:12, 41:13, 42:13, 43:16, 44:16\\ 17 & $-1$ & 13:1, 22:1, 44:1\\ 18 & $-1$ & 5:$-1$, 23:1, 30:$-1$\\ 19 & $-1$ & 7:1, 25:1, 32:1, 40:1\\ 20 & $1$ & 9:1, 23:$-1$, 30:$-1$, 33:$-1$\\ 21 & $1$ & 10:1, 12:1, 22:1, 24:1, 44:1\\ 22 & $-1$ & 21:1, 22:1, 41:1, 44:2\\ 23 & $1$ & 9:1, 23:$-2$, 30:$-1$, 33:$-1$\\ 24 & $-1$ & 6:1, 7:1, 13:1, 26:1, 31:1, 32:1\\ 25 & $-1$ & 12:1, 21:1, 26:1, 40:1, 41:1, 44:1\\ 26 & $-1$ & 10:1, 24:1, 32:1, 34:1, 41:1, 44:1\\ 27 & $-1$ & 25:1, 31:1, 35:1, 40:1, 41:1, 44:1\\ 28 & $-1$ & 15:1, 27:1, 36:$-1$, 46:1, 47:1, 48:1\\ 29 & $-1$ & 24:1, 32:1, 34:1, 40:1, 41:1, 44:1\\ 30 & $-1$ & 2:1, 3:$-1$, 4:1, 27:1, 28:$-1$, 29:1\\ 31 & $1$ & 5:$-1$, 23:1, 30:$-1$, 45:1, 46:1, 48:1\\ 32 & $-1$ & 1:1, 12:1, 21:1, 22:1, 26:1, 34:1, 35:1, 40:1\\ 33 & $-1$ & 11:1, 25:2, 31:1, 35:1, 40:1, 41:1, 44:1\\ 34 & $-1$ & 12:1, 21:1, 22:1, 24:1, 25:1, 40:1, 41:1, 44:1\\ 35 & $-1$ & 13:1, 22:1, 24:1, 34:1, 35:1, 40:2, 44:1\\ 36 & $-1$ & 26:1, 31:1, 32:1, 34:1, 35:1, 40:1, 41:1, 44:1\\ 37 & $-1$ & 3:$-1$, 19:$-1$, 36:$-1$, 37:$-1$, 38:$-1$, 42:1, 43:1, 46:1\\ 38 & $-1$ & 7:1, 8:$-1$, 12:1, 21:$-1$, 23:$-1$, 24:1, 32:1, 41:$-1$\\ 39 & $1$ & 4:1, 17:1, 36:$-1$, 38:1, 39:1, 42:$-1$, 43:$-1$, 46:1\\ 40 & $1$ & 1:1, 13:1, 21:1, 22:1, 23:$-1$, 24:$-1$, 26:1, 40:$-1$\\ 41 & $1$ & 4:1, 16:1, 17:$-1$, 20:1, 37:1, 39:1, 42:$-1$, 43:$-1$\\ 42 & $1$ & 7:1, 8:1, 12:1, 21:1, 24:2, 32:1, 40:1, 41:1\\ 43 & $-1$ & 3:1, 16:1, 19:1, 28:2, 37:2, 38:1, 42:1\\ 44 & $-1$ & 8:1, 11:1, 21:1, 25:1, 31:1, 35:1, 41:2, 44:1\\ 45 & $-1$ & 4:1, 10:1, 12:1, 22:1, 24:1, 29:1, 37:1, 39:1, 44:1\\ 46 & $-2$ & 8:1, 26:2, 31:1, 32:1, 34:1, 35:1, 40:1, 41:1\\ 47 & $-1$ & 8:1, 10:1, 21:1, 24:1, 32:1, 34:1, 41:2, 44:1\\ 48 & $-1$ & 12:1, 21:1, 22:1, 24:1, 25:1, 40:1, 41:1, 44:2\\ 49 & $-1$ & 16:1, 28:1, 29:1, 37:1, 38:1, 39:1, 42:2, 43:1\\ 50 & $1$ & 15:$-1$, 23:1, 27:$-1$, 30:1, 33:1, 36:1, 46:1, 47:1, 48:1\\ 51 & $-1$ & 9:1, 23:$-1$, 27:2, 30:$-1$, 33:$-1$, 36:$-1$, 45:1, 48:1\\ \end{longtable} } \emph{How the data are computed.} The group is built in GAP from the four matrices of Appendix~\ref{app:tuple}, with its character table. A product-one generating tuple in the class datum is fixed. For each supplier, $H'_i$ is given by words in the generators and $\chi_i$ by its values on them; the branch data come from the double cosets $H'_i\backslash G/\langle g_j\rangle$, and $a(\chi_i)$ from the restrictions $\tau|_{H'_i}$ by Frobenius reciprocity. The checking script reads these fixed data, recomputes every row, and checks the identity without reference to the order of the characters: for every non-real $\tau$ with $m_\tau>0$ it computes $\sum_in_i(\langle\tau|_{H'_i},\chi_i\rangle-\langle\bar\tau|_{H'_i},\chi_i\rangle)$ and requires $+1$ at $\tau=\rho$, $-1$ at $\tau=\bar\rho$ and $0$ otherwise, and for every quaternionic $\tau$ with $m_\tau>0$ it requires $\sum_in_i\langle\tau|_{H'_i},\chi_i\rangle$ to be even. A second script, not using GAP, rechecks the identity and the simplicity of the exponents from the exported data. The suppliers were found among the linear characters, of orders $3$ to $18$, of representatives of the conjugacy classes of subgroups of $G$. %% ------------------------------------------------------------------ \section{The rank certificate}\label{app:rank} %% ------------------------------------------------------------------ \emph{The computation.} The program computes, exactly over $\Q$ and modulo $\theta^{13}$, the quantities (1)--(7) of Subsection~\ref{subsec:certificate} along the direction $(V_5,V_{-7})$ of Proposition~\ref{prop:rank}: the residues, with each residue at a root of $P_\epsilon$ represented by one $4\times4$ matrix over $E_\epsilon[\theta]/(\theta^{13})$ and the sums of item (1) computed as traces; the local data at $s=0,3,1,-2$ and the eigenvectors at the moving poles; the matrices $\Phi$ for $\cD_\rho$ ($36$ unknowns, $45$ rows) and for $\cD_{\bar\rho}$ ($48$ unknowns, $57$ rows); the kernel bases for the free columns of item (6); and the matrix $\mathbf M(\theta)$ of item (7). \emph{The result.} $\rk\mathbf M(0)=5$, and each of the ten $9\times9$ minors $m_j(\theta)$ of $\mathbf M(\theta)$, omitting the column of $s^j$, has its first nonzero Taylor coefficient at $\theta^7$. The coefficients $c_j$ are rational numbers with denominator $2^{72}$ and numerators of $64$ to $66$ digits; to four digits they are \[ \begin{array}{c|ccccc} j&0&1&2&3&4\\\hline c_j/10^{42}&-1.031&1.609&-1.701&1.946&-2.572\\\hline\hline j&5&6&7&8&9\\\hline c_j/10^{42}&4.150&-7.929&16.51&-33.78&60.57 \end{array} \] The exact value of $c_0$ is printed in Proposition~\ref{prop:rank}. In the Schur form of Remark~\ref{rem:order7}, $\mathbf S_1$ has rank one, with nonzero entry $16770025629/32000$ in adapted bases, and the $3\times4$ block of $\mathbf S_2$ has rank three. \emph{Identities verified along the way.} The following consequences of the lemmas of Section~\ref{sec:dominance} hold exactly modulo $\theta^{13}$: the sum of all residues is $0$; the characteristic polynomial of every residue is constant; the logarithmic obstructions at $s=0,3,1,-2$ vanish; the vectors $w_t$ are left eigenvectors; $\Phi\,x_f=0$ on all rows, not only on the rows $R$; the polar coefficients of $q^Tr$ of order at most $-2$ at $s=0$ and at most $-1$ at the apparent singularities vanish; and each numerator has degree at most $9$. \emph{Independent computations.} Two computations in interval arithmetic, written separately from the exact program, integrate the Schlesinger equations numerically with rigorous error bounds. Along the direction of Proposition~\ref{prop:rank}, at $\theta=10^{-13}$ and $2\cdot10^{-13}$, the second reproduces ratios of the minors $m_j$ predicted by the exact jets, and its minors grow by the factor $2^7$ when $\theta$ is doubled. The first, along another direction, also finds a $9\times9$ minor vanishing to order $7$. %% ------------------------------------------------------------------ \section{Supplementary files}\label{app:files} %% ------------------------------------------------------------------ \begin{sloppypar} The programs and their outputs are in the certificate archive \cite{G32Cert}, \texttt{doi:10.5281/zenodo.23549518}, whose README lists every file named below with its path in the archive. The rank certificate of Appendix~\ref{app:rank} is in the directory \[ \texttt{dominance-exact}: \] the program \texttt{g32exact.sage} (SageMath~10.7 \cite{Sage}; run as \texttt{sage g32exact.sage 12}, one process, under $4$\,GB), its output, the exact coefficients $c_j$ in \texttt{minor\_coeffs.json}, and the two interval-arithmetic computations. The other computations are in the directory \texttt{certificate}, in four subdirectories: \texttt{A}, the character identity (GAP); \texttt{B}, the facts about $G$ and the tuple of Appendix~\ref{app:tuple} (Sage, exact); \texttt{D}, the discriminant computation of Proposition~\ref{prop:disc} (\texttt{disc\_cert.sage}); \texttt{E}, a separately written computation of the pairing of Lemma~\ref{lem:cocycle} on the same local system and on small covers. (Subdirectory \texttt{C} holds an earlier numerical rank certificate, superseded by Appendix~\ref{app:rank}.) A README in each directory lists the commands and the SHA-256 hashes of the outputs. \end{sloppypar} %% ------------------------------------------------------------------ \begin{thebibliography}{99} \bibitem{AbdulaliIII} S. Abdulali, \emph{Abelian varieties of type III and the Hodge conjecture}, Internat. J. Math. 10 (1999), 667--675. \bibitem{ATLAS} J. H. Conway, R. T. Curtis, S. P. Norton, R. A. Parker and R. A. Wilson, \emph{Atlas of Finite Groups}, Oxford University Press, 1985 (the group $U_4(2)\cong PSp_4(3)$, whose maximal subgroups have indices $27$, $36$, $40$, $40$ and $45$). \bibitem{BL} C. Birkenhake and H. Lange, \emph{Complex Abelian Varieties}, second edition, Grundlehren 302, Springer, 2004. \bibitem{FGP} P. Frediani, A. Ghigi and M. Penegini, \emph{Shimura varieties in the Torelli locus via Galois coverings}, Int. Math. Res. Not. IMRN 2015, 10595--10623. \bibitem{FriedVolklein} M. Fried and H. 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