\documentclass[11pt]{amsart} \usepackage{amsmath,amssymb,amsthm} \usepackage{url} \usepackage{tikz} \usepackage{longtable} \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*{theoremC}{Theorem 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{\A}{\mathbb A} \newcommand{\End}{\operatorname{End}} \newcommand{\Hom}{\operatorname{Hom}} \newcommand{\Nm}{\operatorname{Nm}} \newcommand{\rk}{\operatorname{rk}} \newcommand{\Tr}{\operatorname{Tr}} \newcommand{\Pic}{\operatorname{Pic}} \newcommand{\Hg}{\operatorname{Hg}} \newcommand{\disc}{\operatorname{disc}} \newcommand{\age}{\operatorname{age}} \newcommand{\Hur}{\mathrm{Hur}} \newcommand{\cHn}{\mathcal H_n} \newcommand{\cM}{\mathcal M} \newcommand{\cU}{\mathcal U} \newcommand{\cV}{\mathcal V} \newcommand{\cD}{\mathcal D} \newcommand{\cO}{\mathcal O} \newcommand{\fd}{\mathfrak d} \newcommand{\bn}{\mathbf n} \newcommand{\bpsi}{{\boldsymbol\psi}} \newcommand{\htot}{h_{\rm tot}} \newcommand{\qtot}{q_{\rm tot}} \newcommand{\Vp}{V_{\rm par}} \title[Weil classes for $\Q(i)$ with discriminant $-3$]{Weil classes on abelian sixfolds of Weil type for $\Q(i)$ with discriminant $-3$} \author{Benjamin Severn} \email{benzsevern@gmail.com} \subjclass[2020]{Primary 14C30; Secondary 14K20, 14D07, 14H30} \keywords{Hodge conjecture, Weil classes, abelian varieties of Weil type, Prym--Tyurin varieties, abelian covers, Fermat varieties} \date{21 October 2026 (draft version 0.7)} \begin{document} \begin{abstract} Let $K=\Q(i)$. We prove that the Weil classes of every polarized complex abelian sixfold of Weil type for $K$ whose Hermitian discriminant is the non-split class $[-3]$ are algebraic, and that every Hodge class on every power of such a sixfold is algebraic when its Hodge group is the full special unitary group. The proof starts from Galois covers of the projective line with monodromy group $W(E_8)\cdot\mu_4$, branched over twelve points: the multiplicity space of the natural representation in the cohomology of such a cover is the first cohomology of an abelian sixfold of Weil type, and a certified rank computation at one explicit cover shows that these sixfolds fill an open set of the nine-dimensional moduli space. Algebraic cycles reach these sixfolds through Milne's theorem on Lefschetz classes, from two sources: Schoen's cycles on cyclic covers, which a computer search shows do not suffice here, and products of determinant lines of characters on abelian covers of curves. For the second source we prove a general theorem (Theorem C): if such a product is of Hodge type at every Galois conjugate, and each of the cyclic covers involved has at least two branch points and order dividing $12$ after removing the common factor of its branch exponents, then the product spans a space of algebraic classes. Its proof runs Schoen's symmetric-product construction on each factor, localises at a special fibre, transfers to a product of Fermat hypersurfaces, and applies Shioda's theorem on products of Fermat varieties. \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, so they are a natural test case for the Hodge conjecture \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}. For $K=\Q(i)$ the group $\Q^\times/\Nm K^\times$ is detected by the sign and by the parities of the valuations at the primes $p\equiv3\pmod4$, which are inert in $K$. For sixfolds ($n=3$) the split class is $[-1]$; the class $[-3]$ is not split, because $3$ is inert in $K$ and so is not 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{Markman19,Markman25}). For sixfolds the split case is known: Koike for $\Q(i)$ \cite{Koike}, Schoen for $\Q(\sqrt{-3})$ \cite{Schoen98}, and Markman for every $K$ \cite[Theorem~1.5.1]{Markman25}. Mostaed describes the non-split sixfold cases as open \cite[Introduction]{Mostaed}. For $K=\Q(\sqrt{-3})$ the companion paper \cite{SevernQ3} proves algebraicity for sixfolds of every discriminant, through split eightfolds and Schoen's product argument; the class $-2$ was treated earlier in \cite{SevernG32}. For $K=\Q(i)$ we know of no non-split discriminant for which the Weil classes of sixfolds were previously known to be algebraic. Voisin's Bourbaki expos\'e \cite{Voisin} surveys this subject. This paper uses no eightfolds and no product argument. The sixfolds come directly from a family of Galois covers, dominance of the family is certified at one smooth point, and, because Schoen's cycles alone do not suffice for this family (Appendix~\ref{app:schoenfails}), a second source of cycles is added: Theorem~C below. \subsection{Results} \begin{theoremA} Let $K=\Q(i)$. The Weil classes of every polarized abelian sixfold of Weil type for $K$ with Hermitian discriminant $[-3]$ 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 $A^m$ is algebraic. \end{theoremB} The hypothesis of Theorem~B is on the member $A$ itself. It holds outside a countable union of proper closed subvarieties of the moduli space \cite[Theorem~6.11]{vanGeemen}. We make no claim about the further Hodge classes on powers of members with smaller Hodge group. The new ingredient is a theorem about abelian covers of curves. Let $D\to X$ be a connected Galois cover of a smooth projective curve $X$ of genus $g_X$ with finite abelian group $\mathsf A$. A character $\psi$ of $\mathsf A$ of order $e\ge3$ defines a cyclic cover $D/\ker\psi\to X$ of degree $e$. At each of its branch points $b$ the local monodromy is $\psi$ of a generator of inertia, say $e^{2\pi i\alpha_b/e}$; let $s_\psi$ be the number of branch points, $d_\psi=\gcd(e,\alpha_b:b\text{ a branch point})$, and call $e/d_\psi$ the \emph{reduced order}. For characters $\psi_1,\dots,\psi_R$ of orders $e_k\ge3$ (repetitions allowed), consider the line \[ \ell=\det H^1(D,\C)_{\psi_1}\otimes\dots\otimes\det H^1(D,\C)_{\psi_R}\ \subset\ H^*\big(J(D)^R,\C\big), \] an external product with one factor $J(D)$ for each $k$, and let $\ell^{(t)}$ be the same line with every $\psi_k$ replaced by $\psi_k^t$. \begin{theoremC}[Theorem~\ref{thm:C}] Suppose $g_X\ge1$, and that for every $k$ the cyclic cover $D/\ker\psi_k\to X$ has at least two branch points and reduced order dividing $12$. If every line $\ell^{(t)}$, $t$ prime to the orders $e_k$, is of Hodge type, then the rational subspace whose complexification is the sum of the distinct lines $\ell^{(t)}$ is spanned by classes of algebraic cycles. \end{theoremC} In the application no single factor $\det H^1(D,\C)_{\psi_k}$ is of Hodge type; only the product is. Section~\ref{sec:AB} states Theorem~C on the product $\prod_kJ(D/\ker\psi_k)$ (Theorem~\ref{thm:C}); the form above is Corollary~\ref{cor:ABJ}. \subsection{The construction in one page}\label{subsec:map} \[ \begin{array}{c} \text{$G$-covers $C\to\PP^1$, $G=W(E_8)\cdot\mu_4$,}\\ \text{branched at $12$ points} \end{array} \ \Longrightarrow\ \begin{array}{c} \text{$\sigma$-factor}\\ A_\sigma(C)\subset J(C),\ \dim 6 \end{array} \ \Longrightarrow \] \[ \Longrightarrow\ \begin{array}{c} \text{every sixfold for $\Q(i)$}\\ \text{with discriminant $[-3]$} \end{array} \] \emph{The sixfolds.} Let $W=W(E_8)\subset GL_8(\Q)$ and $G=W\cup iW\subset GL_8(\Q(i))$, of order $1\,393\,459\,200$, with $\sigma$ its natural representation on $K^8$, whose character field is $K=\Q(i)$. Take Galois covers $C\to\PP^1$ with group $G$, branched over twelve points, with local monodromy $i x_1$ at one point ($x_1\in W$, $x_1^2=-1$), $i x_2$ at another ($x_2\in W$ of order $12$), and a reflection of $W$ at each of the other ten. The representation $\sigma$ occurs in $H^1(C)$ with multiplicity $6$, and the multiplicity space is the first cohomology of an abelian sixfold $A_\sigma(C)$, the image in $J(C)$ of an endomorphism given by an element of $\Q[G]$ (Definition~\ref{def:pt}); the centre of $\Q[G]$ acting on $\sigma$ is $K$, and it makes $A_\sigma(C)$ a sixfold of Weil type for $K$ (Section~\ref{sec:covers}). It is not of CM type (Remark~\ref{rem:notCM}). \emph{Cycles.} For a subgroup of $G$ and a character of it, the corresponding quotients of $C$ form a cyclic or abelian cover of a quotient curve, whose Jacobian is a factor of $J(C)$. Some of these factors carry known algebraic classes. The Weil classes of $A_\sigma(C)$ are produced from them by a correspondence, which is algebraic by Milne's theorem on Lefschetz classes \cite{Milne} as soon as a certain class is invariant under the group of symplectic automorphisms of $H^1(C)$ commuting with $G$; that invariance is an identity between integer vectors of length $45$ (Section~\ref{sec:supply}). Two kinds of auxiliary classes enter: Weil classes of Prym varieties of cyclic covers, algebraic by Schoen's theorem \cite{Schoen88} (a computer search shows that these alone do not suffice; Appendix~\ref{app:schoenfails}), and products of determinant lines of several characters on abelian covers, algebraic by Theorem~C. An explicit identity with $95$ terms of the first kind and $3$ of the second (Proposition~\ref{prop:witness}) makes the Weil classes of $A_\sigma(C)$ algebraic for every cover $C$. \emph{Theorem C.} For one cyclic cover of a curve, Schoen's construction \cite[\S2]{Schoen88} produces a variety fibred over a Picard variety of the base curve whose relevant cohomology is one line. That line localises at a single special fibre, where the cover becomes, up to a vector bundle, a quotient of an open Fermat hypersurface; so the line is a Tate twist of a character line on a Fermat hypersurface. For several characters the Fermat lines multiply, and the Hodge condition on the product makes their product a Hodge class on a product of Fermat hypersurfaces of degrees dividing $12$, algebraic by Shioda \cite{Shioda}. A cone over that cycle, and Schoen's Abel--Prym correspondence, carry it back (Section~\ref{sec:AB}). \emph{Dominance and spreading.} At one explicit cover $C_0$ the codifferential of the period map, a $9\times9$ matrix, has a certified nonzero determinant (Appendix~\ref{app:rank}). So the family $A_\sigma(C)$ fills an open set of the nine-dimensional moduli space of sixfolds of its discriminant, which is $[-3]$ by an exact computation from the monodromy tuple (Proposition~\ref{prop:dominance}). The set of moduli points with algebraic Weil classes then contains an open set, and a countable-union argument makes it everything (Section~\ref{sec:spread}). \emph{What is new.} Schoen's cycles, Milne's theorem, dominance followed by spreading \cite{Koike}, the passage from a single cyclic cover to a Fermat character, and Shioda's theorem are known. New here are Theorem~C, in which factors that are not of Hodge type are combined into a product that is, and its use as a second supply of cycles alongside Schoen's. \emph{Inputs.} Published results: Schoen's paper \cite{Schoen88}, with a repair of one step (Lemma~\ref{lem:schoenR}); Shioda's theorem on products of Fermat varieties \cite[Theorem~IV]{Shioda}, which for degree $12$ rests on a finite verification reported by Shioda and recomputed here (Subsection~\ref{subsec:shioda}); Lieberman's theorem on the Lefschetz operator \cite[Theorem~3]{Lieberman}; Milne's theorem on Lefschetz classes \cite[Theorem~3.2]{Milne}; Landherr's theorem and van Geemen's description of the moduli \cite{vanGeemen}; the Chevalley--Weil formula and the codifferential of the period map \cite{FGP}; standard sheaf theory and Hodge theory (proper base change, the K\"unneth formula, the Thom isomorphism, mixed Hodge structures). Levelt's normal form for a regular singular point with semisimple local monodromy \cite{Levelt}, used in Appendix~\ref{app:rank}. Finite certificates, described in the appendices: that the tuple generates $G$ (Appendix~\ref{app:tuple}); the character table data and the integer identity in $\Z^{45}$, with the branch data of the suppliers (Appendix~\ref{app:witness}); the discriminant, by exact linear algebra over $K$ (Proposition~\ref{prop:dominance}); and the rank of the codifferential at $C_0$ (Appendix~\ref{app:rank}). \subsection{Relation to the companion paper} Sections~\ref{sec:weil}, \ref{sec:covers}, \ref{sec:herm}, \ref{sec:spread} and \ref{sec:powers}, the Schoen part of Section~\ref{sec:supply}, and Lemma~\ref{lem:schoenR} adapt the corresponding parts of \cite{SevernQ3}, which treats $\Q(\sqrt{-3})$. Apart from the published results listed above, the proof depends on the finite certificates. Their data are printed or summarised in the appendices, and their programs and outputs are in the supplementary files, the certificate archive \cite{QiCert}; a reader who wants to verify them needs those files, not only this text. Every other step is proved here; in particular nothing is taken from \cite{SevernQ3} or \cite{SevernG32} without proof. \subsection{Organization} Section~\ref{sec:weil} recalls abelian varieties of Weil type and their moduli. Section~\ref{sec:covers} constructs the $\sigma$-factor. Section~\ref{sec:supply} defines the two kinds of suppliers and proves that an integer identity among their characters gives algebraic Weil classes. Section~\ref{sec:AB} proves Theorem~C. Section~\ref{sec:herm} computes the Hermitian discriminant from the monodromy tuple and states the dominance criterion. Section~\ref{sec:spread} spreads algebraicity from a dominant family to the moduli space. Section~\ref{sec:datum} applies all of this to $W(E_8)\cdot\mu_4$ and proves Theorem~A, and Section~\ref{sec:powers} proves Theorem~B. The appendices give the data: the monodromy tuple (Appendix~\ref{app:tuple}), the rank certificate (Appendix~\ref{app:rank}), the supplier certificates (Appendix~\ref{app:witness}), and the search showing that Schoen suppliers alone do not suffice (Appendix~\ref{app:schoenfails}). %% ------------------------------------------------------------------ \section{Abelian varieties of Weil type}\label{sec:weil} %% ------------------------------------------------------------------ Throughout, $K$ is an imaginary quadratic field and $\vartheta\in K$ is a nonzero element with $\bar\vartheta=-\vartheta$; for $K=\Q(i)$ we take $\vartheta=i$. \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; the Hermitian spaces of signature $(n,n)$ with an $n$-dimensional totally isotropic subspace are those of discriminant $(-1)^n$, called \emph{split}. 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 $V_+\in\cHn$ determines a complex structure on $V_\R$ commuting with $K$, hence a Hodge structure of weight one on $V$ of Weil type for $K$, polarized by $\Tr_{K/\Q}(H/2\vartheta)$; 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 $\phi\colon A'\to A$ 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 $\phi$ are algebraic correspondences, and $\phi^*$ 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 a multiple of it lies in $\Lambda$, and multiplication by that integer is a $K$-linear isogeny from $A$ to the fibre of $\mathcal A$ at the point given by the complex structure of $A$. \end{proof} %% ------------------------------------------------------------------ \section{Isotypic factors of Galois covers}\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{class datum}). 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 $\langle g_1,\dots,g_r\rangle=G$. A connected Galois cover $C\to\PP^1$ with group $G$, branched over $r$ ordered points $p_1,\dots,p_r$, determines such a tuple up to simultaneous conjugation, once standard generators of $\pi_1(\PP^1-\{p_j\})$ are chosen: $g_j$ is the monodromy along a loop around $p_j$. 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 $PGL_2$, and correspond to orbits of the pure braid group on the Nielsen class modulo conjugation \cite{FriedVolklein}. For an irreducible complex representation $\tau$ of $G$ write $H^1(C,\C)=\bigoplus_\tau\tau\otimes M_\tau$ (Chevalley--Weil), and $m_\tau=\dim M_\tau$. The space $M_\tau$ carries a Hodge structure of weight one. 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 $\dim H^1_c(U_0,\mathbb V)=-\chi_c(U_0)d=(r-2)d$. The long exact sequence of the pair formed by $U_0$ 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 $(r-2)d-\sum_j\dim H^0(\Delta_j^*,\mathbb V)$, and $\dim H^0(\Delta_j^*,\mathbb V)=d-c_\tau(g_j)$ is the dimension of the fixed space of $\tau(g_j)$. This gives the first formula. The second is the Chevalley--Weil formula \cite[Theorem~2.10]{FGP}. \end{proof} \subsection{The $\sigma$-factor}\label{subsec:pt} Let $\sigma$ be an irreducible representation of $G$ whose character field (the field generated by its values) is $K$ and which is realizable over $K$, on $V_\sigma=K^d$. Since $K$ is not real, the character takes non-real values and $\sigma\not\cong\bar\sigma$. 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 $K$ 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 $Kv$; it is a primitive idempotent of $\Q[G]$ 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). For a cover $C$, the \emph{$\sigma$-factor} is the abelian subvariety \[ A_\sigma(C)=\operatorname{im}\big(N_0e\colon J(C)\to J(C)\big), \] with the restricted polarization $E$ and the action $\iota$ of the centre $K$. It does not depend on $N_0$, and different choices of $v$ give $K$-isogenous factors. \end{definition} The $\sigma$-isotypic part of $J(C)$ is isogenous to $A_\sigma(C)^d$. Such factors are often called Prym--Tyurin varieties \cite[Ch.~12]{BL}; we do not use the exponent condition of that notion, and use the neutral name. \begin{lemma}\label{lem:weiltype} $(A_\sigma(C),\iota,E)$ is a polarized abelian variety of dimension $m_\sigma$, and both 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 $C\in S$; we write $\fd(S)$ for it. \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, and the discriminant is an isometry invariant. \end{proof} \begin{remark}[$A_\sigma(C)$ is not of CM type]\label{rem:notCM} In our application $(p_\sigma,q_\sigma)=(3,3)$ (Lemma~\ref{lem:sixfold}), and, as Section~\ref{sec:datum} shows by the certified rank computation of Appendix~\ref{app:rank}, the family $A_\sigma(C)$ fills an open set of its moduli space. The Hodge group of a very general point of that moduli space is $SU_H$ \cite[Theorem~6.11]{vanGeemen}, so the same holds for $A_\sigma(C)$ with $C$ outside a countable union of proper analytic subsets of the component; in particular $A_\sigma(C)$ is then not of CM type and not isogenous to a power of an elliptic curve. The situation is different when $p_\sigma q_\sigma=0$, for instance when $m_\sigma=1$: then the factor does not vary. \end{remark} %% ------------------------------------------------------------------ \section{Algebraic Weil classes from suppliers}\label{sec:supply} %% ------------------------------------------------------------------ Keep the notation of Section~\ref{sec:covers}, and assume $m_\sigma=2n$ and $(p_\sigma,q_\sigma)=(n,n)$. We produce the Weil classes of $A_\sigma(C)$ by an algebraic correspondence from a product of auxiliary abelian varieties, each a factor of $J(C)$ carrying a known algebraic class. Milne's theorem on Lefschetz classes makes a class on a product of factors of $J(C)$ algebraic as soon as it is invariant under the centralizer $U$ of $G$ (Proposition~\ref{prop:milnetransfer}); invariance under $U$ is an identity between characters of the abelianization of $U$ (Subsection~\ref{subsec:characters}). The auxiliary classes come from two sources, Schoen suppliers (Subsection~\ref{subsec:schoen}) and product suppliers (Subsection~\ref{subsec:gale}), and Subsection~\ref{subsec:contraction} combines them. \begin{center} \small \begin{tabular}{lp{0.66\textwidth}} \multicolumn{2}{l}{\emph{Notation for this section}}\\\hline $U$, $T$, $X^*(T)$ & centralizer of $G$ in $Sp(H^1(C),E)$; its maximal abelian quotient; the characters of $T$\\ $V_\chi$, $a(\chi)$ & the $\chi$-eigenspace of $H'\le G$ on $H^1(C,\C)$; the character of $U$ on $\det V_\chi$\\ $e_\sigma$ & the character of $U$ on the Weil line $\det M_\sigma$\\ $(H',\chi)$, $P_\chi$, $h_\chi$ & a Schoen supplier; the primitive part of $J(C/\ker\chi)$; $h_\chi=\dim V_\chi$\\ $(H',\bn)$, $a_{H'}(\bn)$ & a product supplier, an integer vector on the characters of $H'$; its character\\ $\psi_k$, $\ell_\bn$, $W_\bn$ & the slot characters; the product of their determinant lines; its rational span\\ $\varsigma$ & the Hodge circle \end{tabular} \end{center} \subsection{The character group}\label{subsec:characters} Let $U$ be the centralizer of $G$ in the symplectic group of $(H^1(C,\Q),E)$, an algebraic group over $\Q$; 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 (Frobenius--Schur indicator $+1$ or $-1$), and that otherwise $\tau\not\cong\bar\tau$. In $H^1(C,\C)=\bigoplus_\tau\tau\otimes M_\tau$ the $G$-invariant alternating cup-product form pairs $\tau\otimes M_\tau$ with $\bar\tau\otimes M_{\bar\tau}$ and is zero 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. An automorphism commuting with $G$ acts through $\prod_\tau GL(M_\tau)$, and preserving the cup-product form means preserving these forms. Hence $U(\C)$ is the product of the groups $GL(M_\tau)$ over the pairs $\{\tau,\bar\tau\}$ of non-self-dual representations with $m_\tau>0$ (acting on $M_\tau$ and contragrediently on $M_{\bar\tau}$), 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 $T=U(\C)/[U(\C),U(\C)]$ be its maximal abelian quotient. Its character group is \[ X^*(T)=\Z^{\rm pairs}\oplus(\Z/2)^{\rm quat}, \] where ``pairs'' and ``quat'' run over the pairs and the quaternionic $\tau$ with $m_\tau>0$: one free coordinate for each pair (the determinant on $M_\tau$, for a chosen member $\tau$ of the pair; for the pair of $\sigma$ we choose $\sigma$), and one coordinate of order two (the determinant of $O(M_\tau)$) for each quaternionic $\tau$. For a subgroup $H'\le G$ and a linear character $\chi$ of $H'$ let \[ V_\chi=\textstyle\bigoplus_\tau\Hom_{H'}(\chi,\tau|_{H'})\otimes M_\tau\subseteq H^1(C,\C), \] the $\chi$-eigenspace of $H'$, with Hodge numbers $(p_\chi,q_\chi)$. The line $\det V_\chi$ 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 line $\det M_\sigma\subset H^{2n}(A_\sigma(C),\C)$, one of the two lines of $W_K(A_\sigma)\otimes\C$, has character $e_\sigma$: coordinate $+1$ at the pair of $\sigma$ and $0$ elsewhere. \begin{lemma}\label{lem:conj} $a(\chi^{-1})=-a(\chi)$ on the free coordinates, and $a(\chi^{-1})=a(\chi)$ on the others. Moreover $p_{\chi^{-1}}=q_\chi$. \end{lemma} \begin{proof} Complex conjugation of $H^1(C,\C)$ maps $V_\chi$ to $V_{\chi^{-1}}$ and $H^{1,0}$ to $H^{0,1}$. It also gives $\langle\tau|_{H'},\chi^{-1}\rangle=\langle\bar\tau|_{H'},\chi\rangle$, which exchanges the two terms of each free coordinate; a quaternionic $\tau$ is self-conjugate. \end{proof} The \emph{Hodge circle} $\varsigma\colon S^1\to GL(H^1(C,\R))$, acting by $z^p\bar z^q$ on $H^{p,q}$, commutes with $G$ and preserves the polarization, so it lies in $U$; it acts on $\det V_\chi$ through $z^{p_\chi}\bar z^{q_\chi}$. Restricting characters of $T$ to it gives a homomorphism $\delta_T\colon X^*(T)\to\Z$ with $\delta_T(a(\chi))=p_\chi-q_\chi$. \subsection{The transfer principle} \begin{proposition}\label{prop:milnetransfer} Let $C\in\Hur$, let $\mathfrak X=\prod_jX_j$ be a finite product of abelian varieties each isogenous to the image $\operatorname{im}(N_je_j)\subseteq J(C)$ of an endomorphism with $e_j$ an idempotent of $\Q[G]$, and let $\Xi\in H^*(\mathfrak X,\C)$ be fixed by $U(\C)$, acting diagonally through the identifications $H^1(X_j)\subseteq H^1(C)$. Then $\Xi$ is a complex linear combination of classes of algebraic cycles on $\mathfrak X$. \end{proposition} \begin{proof} Choose one integer $N_0$ for all factors. The inclusions and the corestrictions of the $N_0e_j$ give homomorphisms \[ \mathfrak X\xrightarrow{\ i_{\mathfrak X}\ }J(C)^{R'}\xrightarrow{\ p_{\mathfrak X}\ }\mathfrak X,\qquad p_{\mathfrak X}\circ i_{\mathfrak X}=[N_0], \] up to isogenies of the factors. They are built from elements of $\Q[G]$, so they commute with $U$, and $p_{\mathfrak X}^*$ is $U$-equivariant for the diagonal action of $U$ on $H^1(J(C)^{R'})=H^1(C)^{\oplus R'}$; hence $p_{\mathfrak X}^*\Xi$ 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 $S(J(C))\subseteq U$, compatibly with the diagonal actions, and $p_{\mathfrak X}^*\Xi$ is $S(J(C))$-invariant. Milne's Theorem~3.2 \cite{Milne} states that for any abelian variety $A$ over an algebraically closed field and any $r\ge0$, the $\Q$-algebra $H^*(A^r)^{S(A)}$ is generated by divisor classes; by his Lemma~3.1 the invariants of the complexification are the complexification of the rational invariants. With $A=J(C)$ and $r=R'$, $p_{\mathfrak X}^*\Xi$ is a complex linear combination of products of divisor classes. Since $i_{\mathfrak X}^*p_{\mathfrak X}^*\Xi=[N_0]^*\Xi$ is a nonzero multiple of each homogeneous component of $\Xi$, the class $\Xi$ is a complex combination of algebraic classes. \end{proof} A class of the form (generator of an eigenline)$\,\boxtimes\,$(generators of complementary eigenlines) is fixed by $U(\C)$ exactly when its character in $X^*(T)$ is $0$, since characters of $U(\C)$ factor through $T$. This is how the proposition will be applied. \subsection{Schoen suppliers}\label{subsec:schoen} \begin{definition}\label{def:supplier} A \emph{Schoen 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 primitive part $P_\chi\subset J(C)$, the part on which $\Z/m$ acts through primitive characters, is cut out by a rational idempotent of $\Q[H']$, 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 \[ V_t=\textstyle\bigoplus_\tau\Hom_{H'}(\chi^t,\tau|_{H'})\otimes M_\tau, \] so $V_t=V_{\chi^t}$ and the character of $\det V_t$ is $a(\chi^t)$; all $V_t$ have the same dimension $h_\chi$, and the Weil classes $W_{F_\chi}(P_\chi)=\bigwedge^{h_\chi}_{F_\chi}H^1(P_\chi,\Q)$, a one-dimensional $F_\chi$-vector space, satisfy $W_{F_\chi}(P_\chi)\otimes\C=\bigoplus_t\det V_t$. The branch data of $C'\to C/H'$ are read off from the tuple. 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; then $h_\chi=2q-2+s$ \cite[Lemma~1.5]{Schoen88}. Following Schoen \cite[\S1]{Schoen88}, $\alpha$ is \emph{simple} if, after reordering, $\alpha_{2i}=-\alpha_{2i-1}$ for all $i$. A Schoen supplier is \emph{Schoen-admissible} if $h_\chi$ is even and positive and $\alpha$ is simple (automatic when $s=0$). \begin{example}\label{ex:schoenearly} Let $H'=\langle i\rangle N_W(P_3)$, of order $7776$, for $P_3$ a Sylow $3$-subgroup of $W=W(E_8)$, in the group $G=W(E_8)\cdot\mu_4$ of Section~\ref{sec:datum}. It has a character $\chi$ of order $4$ for which $C/H'$ has genus $q=384\,479$ and $C/\ker\chi\to C/H'$ has $s=88$ branch points, all with exponent $2$; so $h_\chi=769\,044$, the exponents are simple ($2\equiv-2$), and the supplier is Schoen-admissible. It is one of the terms of the identity of Proposition~\ref{prop:witness} (Example~\ref{ex:schoen3215}). \end{example} \begin{lemma}\label{lem:classdatum} For a subgroup $H'\le G$ and a character $\chi$ of $H'$, the numbers $q$, $s$, $h$, the exponents $\alpha$ up to $(\Z/m)^\times$ and order, and the Hodge numbers of the eigenspaces $V_t$, depend only on $G$ and the classes $\mathcal C_j$; in particular Schoen-admissibility does. \end{lemma} \begin{proof} 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; Riemann--Hurwitz for $C\to C/H'$ gives $q$. The Hodge numbers of $M_\tau$ are given by Proposition~\ref{prop:mult} in terms of the classes. \end{proof} \begin{lemma}[Schoen]\label{lem:schoen} If $(H',\chi)$ is Schoen-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 $\prod_{1\ne Z'\le\Z/m}\sum_{t\in Z'}(\mathrm{Id}-t)$. Neither statement asks the exponents to be prime to $m$: Schoen's \S1 treats a branch point with $\gcd(\alpha_j,m)=d_j>1$ through the local model $y^{m/d_j}=w^{\alpha_j/d_j}$. Simplicity is automatic for $r\le1$, and the unramified case $r=0$ is included, with its own proof. 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, which is isogenous to $P_\chi$ compatibly with $F_\chi$. Admissibility does not depend on $C$ by Lemma~\ref{lem:classdatum}. 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} In Section~\ref{sec:datum} one Schoen supplier of the identity has every exponent equal to $m/2$ with $m=4$ (Example~\ref{ex:schoen3215}). For such covers one step of Schoen's proof needs a replacement, which we now give. 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 $F_i=\beta_i^{-1}(E_i)$. After $r$ steps the strict transform of $P_0$ avoids the branch locus, its inverse image splits into $m$ components, and these give the cycles of Theorem~2.0 (his Lemmas~2.4--2.6). To see that their classes on $W_0$ are nonzero, his Lemma~2.8 shows that no primitive character of $\Z/m$ occurs in $H^*(W_i,\C)/H^*(W_{i-1},\C)$, and this reduces to the statement that no primitive character occurs in $H^*(F_i,\C)$ \cite[p.~18]{Schoen88}. Schoen proves the latter (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 equals $m/2$ and $4\mid m$, no such prime exists. \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; in particular they hold when every exponent is $m/2$ and $4\mid m$. \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 $S_{j,i}$. The strict transforms $S_{2i-1,i}$ and $S_{2i,i}$ meet $E_i$ in two disjoint sections of $p\colon E_i\to K_{i-1}$, and every other $S_{j,i}$ meets $E_i$ in a union of whole fibres. Such a fibre lies in the closed set $S_{j,i}$, so the restriction of $j_!\mathcal L$ to it is the zero sheaf. On any other fibre $\PP^1$ it is the extension by zero of a rank-one local system on $\C^\times=\PP^1-\{0,\infty\}$ with monodromy $\chi(\alpha_{2i-1})\ne1$, whose compactly supported cohomology vanishes: $H^0_c=0$, $H^2_c$ is dual to the invariants of the dual system, which are $0$, and the Euler characteristic is $0$. Since $p$ is proper and $j_!\mathcal L$ is constructible, proper base change identifies the stalk of $R^bp_*(j_!\mathcal L)$ at $k\in K_{i-1}$ with $H^b(p^{-1}(k),j_!\mathcal L|_{p^{-1}(k)})$ \cite[Theorem~11.3.6]{Schapira}, which we have just shown to vanish. Hence $Rp_*(j_!\mathcal L)=0$ and $H^*(E_i,j_!\mathcal L)=H^*(K_{i-1},Rp_*j_!\mathcal L)=0$. The rest of Schoen's proof is unchanged. \end{proof} \subsection{Product suppliers}\label{subsec:gale} A single line $\det V_\chi$ is of Hodge type only when $p_\chi=q_\chi$. Several such lines, none of Hodge type, can have a product that is of Hodge type, and Section~\ref{sec:AB} shows that, under conditions on the branching, such a product spans algebraic classes. \begin{definition}\label{def:gale} A \emph{product supplier} is a pair $(H',\bn)$ of a subgroup $H'\le G$ and a finitely supported integer vector $\bn=(n_\chi)$ on the linear characters $\chi$ of $H'$ with $\dim V_\chi>0$, such that every $\chi$ with $n_\chi\ne0$ has order at least $3$. Let $M_\bn$ be the least common multiple of the orders of these $\chi$. Since $\chi^t$ depends only on $t$ modulo the order of $\chi$, the numbers \[ \delta_t(\bn)=\textstyle\sum_\chi n_\chi\,\big(p_{\chi^t}-q_{\chi^t}\big),\qquad t\in(\Z/M_\bn)^\times, \] are well defined. The supplier is \emph{balanced} if $\delta_t(\bn)=0$ for every $t\in(\Z/M_\bn)^\times$. Its character is $a_{H'}(\bn)=\sum_\chi n_\chi a(\chi)\in X^*(T)$. An \emph{integral lift} of a vector $v\in X^*(T)$ is an integer vector $\bn$ with $a_{H'}(\bn)=v$; different lifts of one vector give different products of lines. \end{definition} By the last subsection, $\delta_1(\bn)=\delta_T(a_{H'}(\bn))$ depends only on the character of the supplier. Fix a balanced product supplier and $C\in\Hur$. Put $\kappa_\bn=\bigcap_{n_\chi\ne0}\ker\chi$, a normal subgroup of $H'$ with abelian quotient $\mathsf A=H'/\kappa_\bn$, and $D=C/\kappa_\bn$, a connected Galois $\mathsf A$-cover of $X=C/H'$; the characters $\chi$ with $n_\chi\ne0$ factor through $\mathsf A$, and $V_\chi$ is identified with $H^1(D,\C)_\chi$ by pullback. Let $R=\sum_\chi|n_\chi|$. A \emph{slot} is one of the $R$ occurrences: the character $\chi$, used $n_\chi$ times, if $n_\chi>0$, and the character $\chi^{-1}$, used $|n_\chi|$ times, if $n_\chi<0$. List the slot characters as $\psi_1,\dots,\psi_R$ and put \[ \ell_\bn=\textstyle\bigotimes_{k=1}^R\det H^1(D,\C)_{\psi_k}\ \subset\ H^{d_\bn}\big(J(D)^R,\C\big), \] with $d_\bn=\sum_k\dim H^1(D,\C)_{\psi_k}$, an external product with one factor $J(D)$ for each slot. A negative coefficient thus contributes the determinant of the opposite character, in positive degree. For $t\in(\Z/M_\bn)^\times$ let $\ell_\bn^{(t)}$ be the line obtained by replacing each $\psi_k$ by $\psi_k^t$. \begin{lemma}\label{lem:Wn} The distinct lines among the $\ell_\bn^{(t)}$ are independent, and their sum is $W_\bn\otimes\C$ for a $\Q$-subspace $W_\bn\subseteq H^{d_\bn}(J(D)^R,\Q)$ consisting of Hodge classes of type $(d_\bn/2,d_\bn/2)$. The line $\ell_\bn$ is a $U$-eigenline with character $a_{H'}(\bn)$ on the free coordinates and $\sum_\chi|n_\chi|a(\chi)$ on the others. \end{lemma} \begin{proof} Two distinct lines $\ell^{(t)}$, $\ell^{(t')}$ differ in some slot, where they lie in the determinants of two different character eigenspaces of $\mathsf A$ on $H^1(D)$, each of positive dimension; so they lie in different $\mathsf A^R$-isotypic summands of $H^*(J(D)^R,\C)$, and the sum is direct. The eigenspaces are defined over $\Q(\mu_{M_\bn})$, and the automorphism $\zeta\mapsto\zeta^t$ of that field maps $\ell_\bn$ to $\ell_\bn^{(t)}$, because $H^1(D)$ is defined over $\Q$; so the sum of the distinct lines is Galois stable and descends to $\Q$. The line $\ell^{(t)}$ has Hodge type $(P,d_\bn-P)$ with $P-(d_\bn-P)=\sum_k(p_{\psi_k^t}-q_{\psi_k^t})=\delta_t(\bn)$, by Lemma~\ref{lem:conj} for the negative slots; balance makes it $(d_\bn/2,d_\bn/2)$. The character statement is Lemma~\ref{lem:conj}. \end{proof} \begin{definition}\label{def:galeadm} A balanced product supplier $(H',\bn)$ is \emph{product-admissible} if $X=C/H'$ has genus $g_X\ge1$ and every slot character $\psi_k$ satisfies hypotheses (H2) and (H3) of Theorem~\ref{thm:C}: the cyclic cover $D/\ker\psi_k\to X$ has at least two branch points, and $e_k/d_k$ divides $12$, with $e_k$ the order of $\psi_k$ and $d_k$ the greatest common divisor of $e_k$ and its branch exponents. \end{definition} \begin{example}\label{ex:productearly} In the group $G$ of Section~\ref{sec:datum}, let $w_1\in W$ have order $3$ with centralizer of order $311\,040$, and $H'=\langle i\rangle C_W(w_1)$. Then $X=C/H'$ has genus $3259$, and there are characters $\chi_2,\chi_7$ of $H'$ such that $\bn=-\chi_2-\chi_7$ is a balanced product supplier with two slots $\psi_1=\bar\chi_2$ (order $4$, $6732$ branch points) and $\psi_2=\bar\chi_7$ (order $12$, $20$ branch points), both with $d_k=1$. Neither $\det H^1(D)_{\psi_k}$ is of Hodge type, while their product is, at every $t$. Example~\ref{ex:gale3231} follows it through the proof of Theorem~C. \end{example} By Lemma~\ref{lem:classdatum} product-admissibility depends only on the class datum. Balance is hypothesis (H4) of Theorem~\ref{thm:C}, since $2p_{\psi_k^t}-h_k=p_{\psi_k^t}-q_{\psi_k^t}$. Hence: \begin{corollary}\label{cor:galesupply} If $(H',\bn)$ is a product-admissible supplier, then for every $C\in\Hur$ the space $W_\bn$ is spanned by classes of algebraic cycles on $J(D)^R$. \end{corollary} \begin{proof} Theorem~\ref{thm:C} and Corollary~\ref{cor:ABJ}, applied to $D=C/\kappa_\bn\to X=C/H'$ and the slot characters. \end{proof} \subsection{The contraction}\label{subsec:contraction} \begin{definition}\label{def:lattice} Let $L(G;\mathcal C_1,\dots,\mathcal C_r)\subseteq X^*(T)$ be the subgroup generated by the $a(\chi)$ of the Schoen-admissible suppliers and the $a_{H'}(\bn)$ of the product-admissible suppliers. \emph{Condition (L)} is that $e_\sigma\in L$. \end{definition} \begin{proposition}\label{prop:supply} Let $C\in\Hur$ and let $X'=\prod_jX'_j$ be a finite product of abelian subvarieties $X'_j=\operatorname{im}(N_je_j)\subseteq J(C)$, with $e_j$ idempotents of $\Q[G]$. Let $\ell\subset H^p(X',\C)$ be an eigenline of $U$ whose character $\xi$ lies in $L$. Then $\ell$ lies in the complex span of classes of algebraic cycles on $X'$. \end{proposition} We first record the complements of the supplier lines. \begin{lemma}[Complements]\label{lem:complements} \textup{(a)} Let $(H',\chi)$ be a Schoen supplier with eigenspaces $V_t$, $t\in(\Z/m)^\times$, and $t_0=\pm1$. Choose generators $z_t$ of the lines $\det V_t$ and put $Z=\prod_{t\ne t_0}z_t$. Then $Z$ is a $U$-eigenline with character $-t_0a(\chi)$ on the free coordinates and $a(\chi)$ on the others, and for $v\in\det V_t$, $v\cup Z=0$ unless $t=t_0$, while $\cup Z$ is nonzero on $\det V_{t_0}$. \textup{(b)} Let $(H',\bn)$ be a balanced product supplier. On the copy of $J(D)$ belonging to a slot with character $\psi$, choose a generator of the top exterior power of the sum of all $\mathsf A$-eigenspaces of $H^1(D,\C)$ other than $H^1(D,\C)_\psi$, and let $Z$ be the external product of these over the slots. Then $Z$ has the character opposite to that of $\ell_\bn$, $\cup Z$ is nonzero on $\ell_\bn$, and $v_L\cup Z=0$ for every line $L\ne\ell_\bn$ among the $\ell_\bn^{(t)}$. \end{lemma} \begin{proof} (a) The product $\prod_tz_t$ spans the top cohomology of $P_\chi$, on which $U$ acts trivially, so $Z$ has the character opposite to $a(\chi^{t_0})$, which Lemma~\ref{lem:conj} computes. For $t\ne t_0$ the product $v\cup Z$ contains $\det V_t$ twice. (b) The product of $Z$ with $\ell_\bn$ spans the top cohomology of $J(D)^R$, on which $U$ acts trivially. A line $L\ne\ell_\bn$ differs from $\ell_\bn$ in some slot, where its factor already occurs in the complement. \end{proof} \begin{proof}[Proof of Proposition~\ref{prop:supply}] Write $\xi=\sum_in_ia(\chi_i)+\sum_la_{H'_l}(\bn_l)$ with Schoen-admissible suppliers $(H'_i,\chi_i)$, $n_i\ne0$, and product-admissible suppliers $(H'_l,\bn_l)$; a product term with a coefficient $c$ is absorbed by replacing $\bn_l$ by $c\,\bn_l$. This is again product-admissible: $\delta_t(c\,\bn_l)=c\,\delta_t(\bn_l)=0$; its slots are those of $\bn_l$ repeated $|c|$ times, inverted when $c<0$; and $\psi^{-1}$ has the same branch points, order and gcd $d$ as $\psi$ (its exponents are the negatives), so (H2) and (H3) still hold. Its character is $a_{H'_l}(c\,\bn_l)=c\,a_{H'_l}(\bn_l)$. Put $t_i=\operatorname{sign}n_i$, $P_i=P_{\chi_i}$, $F_i=F_{\chi_i}$. By Lemma~\ref{lem:schoen} choose a nonzero rational algebraic class $y_i\in W_{F_i}(P_i)$, and by Corollary~\ref{cor:galesupply} a nonzero rational algebraic class $y_l\in W_{\bn_l}$ on $J(D_l)^{R_l}$, $D_l=C/\kappa_{\bn_l}$. Put $\mathfrak B=\prod_iP_i^{|n_i|}\times\prod_lJ(D_l)^{R_l}$ and $\mathbf y=\boxtimes_iy_i^{\boxtimes|n_i|}\boxtimes\boxtimes_ly_l\in H^*(\mathfrak B,\Q)$, an algebraic class. \emph{The correspondence.} With the complements $Z_i$ (for $t_0=t_i$) and $Z_l$ of Lemma~\ref{lem:complements} and a generator $x$ of $\ell$, put \[ \Xi=\Big(\textstyle\boxtimes_iZ_i^{\boxtimes|n_i|}\boxtimes\boxtimes_lZ_l\Big)\boxtimes x\in H^*(\mathfrak B\times X',\C). \] By Lemmas~\ref{lem:complements} and~\ref{lem:Wn} its character is $\xi-\sum_in_ia(\chi_i)-\sum_la_{H'_l}(\bn_l)=0$ on the free coordinates. On the coordinates of order two, $Z_i$ has character $a(\chi_i)$ whatever the sign of $n_i$, and $|n_i|$ copies give $|n_i|a(\chi_i)\equiv n_ia(\chi_i)\pmod2$. Likewise $Z_l$ has the character of $\ell_{\bn_l}$ there, $\sum_\chi|n_{l,\chi}|a(\chi)\equiv a_{H'_l}(\bn_l)\pmod2$, because $a(\chi^{-1})=a(\chi)$ on these coordinates (Lemma~\ref{lem:conj}). So the character of $\Xi$ is $\xi-\sum_in_ia(\chi_i)-\sum_la_{H'_l}(\bn_l)=0$ there too. (For the group of Section~\ref{sec:datum} there are no coordinates of order two.) Every factor of $\mathfrak B\times X'$ is isogenous to the image of an endomorphism $N_0e'$ of $J(C)$ with $e'$ an idempotent of $\Q[G]$: for $P_i$ and $X'_j$ by construction, and for $J(D_l)$ with $e'$ the averaging idempotent of $\kappa_{\bn_l}$, since pullback identifies $H^1(D_l)$ with $H^1(C)^{\kappa_{\bn_l}}$. So Proposition~\ref{prop:milnetransfer} makes $\Xi$ a complex combination of algebraic classes. \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 $\int_{P_i}v_{i,t_i}\cup Z_i=1$. 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 $F_i$. Similarly write $y_l=\sum c_{l,L}v_L$ over the distinct lines $L$ of Lemma~\ref{lem:Wn}, with $v_{\ell_{\bn_l}}$ normalized against $Z_l$; the coefficient $c_{l,\ell_{\bn_l}}$ is nonzero because $y_l$ is rational and nonzero and the Galois action on coefficients permutes the distinct lines transitively. By Lemma~\ref{lem:complements}, \[ \Xi_*(\mathbf y)=\operatorname{pr}_{2*}(\operatorname{pr}_1^*\mathbf y\cup\Xi)=\pm\textstyle\prod_ic_{i,t_i}^{|n_i|}\prod_lc_{l,\ell_{\bn_l}}\,x\ne0, \] and $x$ lies in the complex span of algebraic classes. \end{proof} \begin{corollary}\label{cor:everymember} If condition (L) holds, then $W_K(A_\sigma(C))$ is spanned by classes of algebraic cycles for every $C\in\Hur$, on every component. \end{corollary} \begin{proof} $W_K(A_\sigma)\otimes\C=\det M_\sigma\oplus\det M_{\bar\sigma}$, two eigenlines of $U$ with characters $\pm e_\sigma$ on the free coordinates and $0$ on the others, so both characters lie in $L$. Proposition~\ref{prop:supply} puts both in the complex span of algebraic classes. If $\mathcal Z$ is the rational span of algebraic classes and $W\otimes\C\subseteq\mathcal Z\otimes\C$ for a rational subspace $W$, then $W\subseteq\mathcal Z$. \end{proof} \begin{remark}\label{rem:integer} The identity $e_\sigma\in L$ must hold with integer coefficients. An identity for a multiple $ke_\sigma$ would give a $k$-fold tensor power of the target line, and no $k$-th root of it. Large coefficients only require finitely many copies of each factor. \end{remark} %% ------------------------------------------------------------------ \section{Products of determinant lines on abelian covers: Theorem C}\label{sec:AB} %% ------------------------------------------------------------------ This section is independent of the rest of the paper: it uses only the definitions of Subsection~\ref{subsec:ABsetting}, and Corollary~\ref{cor:ABJ} connects it with Section~\ref{sec:supply}. The proof runs as follows. For a single cyclic cover $D'\to X$ with a character $\psi$, Schoen constructs a variety $W_0$ fibred over $\Pic^h(X)$ whose $\psi$-cohomology is the single line $\det H^1(D')_\psi$. That line is supported over a single point $L_0$ of $\Pic^h(X)$, and over $L_0$ the cover is, up to a vector bundle, an open piece of a Fermat hypersurface divided by a finite group. So the line is, up to a Tate twist, a character line on a Fermat hypersurface. For a product of factors only the total Hodge type matters, so a product of lines that is of Hodge type corresponds to a Hodge class on a product of Fermat hypersurfaces. That class is algebraic by Shioda's theorem, and a cone over its cycle, followed by Schoen's Abel--Prym correspondence, gives algebraic classes in the product line. For a single factor, Schoen observed that the Hodge condition is exactly the condition for the corresponding Fermat character to be of Hodge type \cite[Corollary~1.9 and Remark~1.10, p.~10]{Schoen88}; the construction below turns this correspondence of Hodge structures into one of cycles. \begin{center} \small \begin{tabular}{lp{0.6\textwidth}} \multicolumn{2}{l}{\emph{Notation for this section}}\\\hline $X$, $g_X$ & the base curve and its genus\\ $\mathsf A$, $D\to X$ & finite abelian group, connected Galois $\mathsf A$-cover\\ $\psi_k$, $k=1,\dots,R$ & the slots: characters of $\mathsf A$, of orders $e_k\ge3$\\ $D_k$, $\Gamma_k$ & the cyclic cover $D/\ker\psi_k\to X$ and its group\\ $s_k$, $h_k$, $d_k$, $e'_k$ & branch points, $h_k=2g_X-2+s_k$, $\gcd$ of $e_k$ and the exponents, $e'_k=e_k/d_k$\\ $M_0$ & $\operatorname{lcm}_ke_k$\\ $\htot$, $\qtot$, $\nu$ & $\sum_kh_k$, $Rg_X$, $\tfrac12\sum_k(s_k-2)$\\ $L_0$, $\Pi_0$ & the special point $K_X+\sum_jb_j$ of $\Pic^h(X)$; the common base locus $\sum_jb_j+|K_X|$ of the hyperplanes $H_j$ in $|L_0|$\\ $\Omega$ & the complement of $s$ general hyperplanes in $\PP^{s-2}$\\ $\bpsi$ & the character $\boxtimes_k\psi_k$ of $\Gamma_{\rm tot}=\prod_k\Gamma_k$ \end{tabular} \end{center} \subsection{Setting and statement}\label{subsec:ABsetting} Let $X$ be a smooth projective curve of genus $g_X$, $\mathsf A$ a finite abelian group, and $D\to X$ a connected Galois cover with group $\mathsf A$. Fix characters $\psi_1,\dots,\psi_R$ of $\mathsf A$ (repetitions allowed), the \emph{slots}, of orders $e_k\ge3$. For each slot: \begin{itemize} \item $D_k=D/\ker\psi_k\to X$ is a connected cyclic cover with group $\Gamma_k=\mathsf A/\ker\psi_k$, cyclic of order $e_k$, on which $\psi_k$ is faithful; \item its branch points are the points $b\in X$ whose inertia group in $\mathsf A$ is not contained in $\ker\psi_k$; there are $s_k$ of them, and $h_k=2g_X-2+s_k$; \item the \emph{exponent} $\alpha_{k,b}\in\Z/e_k-\{0\}$ at $b$ is defined by $\psi_k(g_b)=e^{2\pi i\alpha_{k,b}/e_k}$, for the generator $g_b$ of inertia given by a small positive loop around $b$; and $d_k=\gcd(e_k,\alpha_{k,b}:b\text{ a branch point})$, $e'_k=e_k/d_k$, the \emph{reduced order}. \end{itemize} By Chevalley--Weil, $H^1(D_k,\C)_{\psi_k}$ has dimension $h_k$ \cite[Lemma~1.5]{Schoen88}, and pullback identifies it with $H^1(D,\C)_{\psi_k}$. For $t$ prime to $M_0$ put $p_{k,t}=\dim H^{1,0}(D_k)_{\psi_k^t}$. Define the lines \[ \ell^{(t)}=\textstyle\bigotimes_k\det H^1(D_k,\C)_{\psi_k^t}\subset H^{\htot}\big(\textstyle\prod_kJ(D_k),\C\big),\qquad \htot=\sum_kh_k . \] The distinct lines among them are independent, their sum is stable under the Galois action on coefficients, and so it is $W\otimes\C$ for a rational subspace $W\subseteq H^{\htot}(\prod_kJ(D_k),\Q)$; the proof is that of Lemma~\ref{lem:Wn}. \begin{theorem}[Theorem C]\label{thm:C} Assume \begin{enumerate} \item[(H1)] $g_X\ge1$; \item[(H2)] $s_k\ge2$ for every $k$; \item[(H3)] $e'_k$ divides $12$ for every $k$; \item[(H4)] $\sum_k(2p_{k,t}-h_k)=0$ for every $t\in(\Z/M_0)^\times$. \end{enumerate} Then $W$ is spanned by classes of algebraic cycles on $\prod_kJ(D_k)$. \end{theorem} \begin{corollary}\label{cor:ABJ} Under the same hypotheses, the space $W_\bn$ of Lemma~\ref{lem:Wn}, formed on $J(D)^R$ with one copy of $J(D)$ for each slot, is spanned by classes of algebraic cycles. \end{corollary} \begin{proof} For each slot, the norm homomorphism $J(D)\to J(D_k)$ induces on $H^1$ the pullback $H^1(D_k)\to H^1(D)$, which identifies $H^1(D_k,\C)_{\psi_k^t}$ with $H^1(D,\C)_{\psi_k^t}$ for every $t$. The product of these homomorphisms $J(D)^R\to\prod_kJ(D_k)$ pulls $\ell^{(t)}$ back to $\ell_\bn^{(t)}$, so it maps $W$ onto $W_\bn$, and pullback by a homomorphism of abelian varieties preserves algebraic classes. \end{proof} (H4) says that every $\ell^{(t)}$ is of Hodge type: $\ell^{(t)}$ has type $(P_t,\htot-P_t)$ with $P_t=\sum_kp_{k,t}$. Under (H1) and (H2) we have $h_k\ge2g_X\ge2$, and $\htot$ is even by (H4). Hypothesis (H1) is used in Lemma~\ref{lem:L3}, where the special fibre is a single point of $\Pic^h(X)$ with base locus $\Pi_0\cong\PP^{g_X-1}$; when $g_X=0$, $\Pic^h(X)$ is a point and no localisation is needed, but we do not treat that case. In the application $g_X\ge3259$. Hypothesis (H3) is used only in Lemma~\ref{lem:fermatinput}, to map Fermat varieties of the single degree $12$ onto the Fermat models of all slots; the same proof works whenever all $e'_k$ divide one integer $m$ for which Shioda's theorem gives the Hodge conjecture for products of Fermat varieties of degree $m$ ($m$ prime or $m\le20$, Theorem~\ref{thm:shioda}). We state the case $m=12$, which the application needs. \begin{example}\label{ex:Csmall} For the supplier of Example~\ref{ex:productearly}, $g_X=3259$, and the two slots have $(e_k,s_k,d_k)=(4,6732,1)$ and $(12,20,1)$, so $h_1=13\,248$, $h_2=6536$, $e'_1=4$ and $e'_2=12$. The Fermat models of Subsection~\ref{subsec:fermat} are the Fermat hypersurfaces of degree $4$ and dimension $6730$ and of degree $12$ and dimension $18$. Example~\ref{ex:gale3231} computes their Hodge types and the cone. \end{example} \emph{Conventions.} Fix one slot and drop the index $k$ until Subsection~\ref{subsec:product}. Write $D'=D_k$, $\Gamma=\Z/e$, $\psi$ faithful, branch points $b_1,\dots,b_s$ with exponents $\alpha_1,\dots,\alpha_s\ne0$ as above, and $h=2g_X-2+s$. Loops around points of $X$, and meridians of divisors below, are oriented positively for the complex orientation. With these conventions, the Chevalley--Weil formula reads $p=\dim H^{1,0}(D')_\psi=g_X-1+\sum_j\{-\alpha_j/e\}$, with $\{\cdot\}$ the fractional part; the opposite convention replaces $\psi$ by $\bar\psi$ throughout, which does not affect the statements. \emph{Three standard facts.} (F1) $\sum_j\alpha_j\equiv0\pmod e$: in $\pi_1$ of $X$ minus the branch points, the product of the loops around the branch points is a product of commutators, whose image in the abelian group $\Gamma$ is trivial. (F2) For $L\in\Pic^h(X)$ the divisor $S_j$ meets the fibre $|L|$ of $\pi$ transversally, in the hyperplane $H_j=|L-b_j|$: the restriction of $\cO(S_j)$ to $|L|$ is $\cO(1)$. So a meridian of $H_j$ in $|L|$ is a meridian of $S_j$ in $S^hX$. (F3) The $\psi$-parts below are defined over $\Q(\mu_e)$, not over $\Q$; ``mixed Hodge structure'' on a $\psi$-part means complex mixed Hodge structure. \subsection{Schoen's construction}\label{subsec:schoenW0} Following \cite[\S2]{Schoen88}, let $\hat G=\Gamma^h\rtimes S_h$ act on $D'^h$, let $\mathsf N\subset\Gamma^h$ be the kernel of the sum map $\Gamma^h\to\Gamma$, and $G_0=\mathsf N\rtimes S_h$. Put $W_0=D'^h/G_0$. The quotient $\hat G/G_0\cong\Gamma$ acts on $W_0$ with quotient $S^hX$, giving a $\Gamma$-cover $\beta\colon W_0\to S^hX$, and the Abel--Jacobi map $\pi\colon S^hX\to\Pic^h(X)$ is a projective bundle with fibres $|L|\cong\PP^{h-g_X}$, since $h\ge2g_X-1$. Let $S_j=b_j+S^{h-1}X\subset S^hX$, the divisors of the branch points. \begin{lemma}[{\cite[Lemmas~1.1--1.2, pp.~5--6]{Schoen88}}]\label{lem:L1} The $\psi$-isotypic part of $H^*(W_0,\C)$ for the action of $\Gamma$ is one-dimensional, concentrated in degree $h$, and its pullback to $D'^h$ is the line $U_\psi=\bigwedge^hH^1(D',\C)_\psi$, embedded in $H^h(D'^h,\C)$ by the K\"unneth formula. \end{lemma} \begin{proof} $H^*(W_0,\C)=H^*(D'^h,\C)^{G_0}$. Decompose $H^*(D'^h)=H^*(D')^{\otimes h}$ into the tensor products $\bigotimes_iH^*(D')_{\chi_i}$ of eigenspaces of $\Gamma$. An element $(\gamma_i)\in\mathsf N$ acts on such a product by $\prod_i\chi_i(\gamma_i)$, and this is $1$ for all $(\gamma_i)$ with $\sum_i\gamma_i=0$ exactly when all $\chi_i$ are equal; the residual $\Gamma$ then acts by the common character. For $\chi=\psi$ faithful, $H^0(D')$ and $H^2(D')$ have no $\psi$-part, so the $\psi$-part is $H^1(D')_\psi^{\otimes h}$, and invariance under $S_h$, with the Koszul sign, gives its top exterior power $\bigwedge^hH^1(D')_\psi$, which is one-dimensional because $\dim H^1(D')_\psi=h$. \end{proof} By Schoen's Remark~1.4 \cite[pp.~6--7]{Schoen88}, $U_\psi$ is the pullback, under the sum map $D'^h\to J(D')$, of the line $\det H^1(D',\C)_\psi\subset H^h(J(D'),\C)$. \subsection{Localisation}\label{subsec:localisation} \begin{lemma}\label{lem:L2} Let $j$ be the inclusion of $S^hX-\bigcup_jS_j$. The $\psi$-isotypic part of $\beta_*\C$ is $j_!\mathcal L$ for a rank-one local system $\mathcal L$ whose monodromy around $S_j$ is $\psi(g_{b_j})\ne1$. \end{lemma} \begin{proof} The stalk of $\beta_*\C$ at a point is $\C[\Gamma/\Gamma_y]$, with $\Gamma_y$ the stabilizer of a point $y$ above it, and its $\psi$-part is nonzero exactly when $\psi$ is trivial on $\Gamma_y$. At a divisor containing some $b_j$, with multiplicity $\mu\ge1$, a point of $D'^h$ above it has $\mu$ coordinates over $b_j$, each fixed by the inertia group $\langle g_{b_j}\rangle$; the element of $\hat G$ that applies $g_{b_j}$ to one such coordinate fixes the point, and its image in $\Gamma$ is $g_{b_j}$. So $\Gamma_y\ni g_{b_j}$, on which the faithful $\psi$ is nontrivial, and the $\psi$-part vanishes there. Off $\bigcup_jS_j$, coincident points are permuted by $S_h\subset G_0$, so $\beta$ is \'etale for the residual action, and $(\beta_*\C)_\psi$ is a rank-one local system $\mathcal L$. Near a general point of $S_j$, write the divisor as $x_1+x_2+\dots+x_h$ with $x_1$ near $b_j$ and the other points distinct and unbranched. Moving $x_1$ once around $b_j$ lifts, in $D'^h$, to a path from a point $(y_1,\dots,y_h)$ to $(g_{b_j}y_1,y_2,\dots,y_h)$, which differs from the starting point by the element $(g_{b_j},0,\dots,0)$ of $\Gamma^h$, whose image in $\hat G/G_0=\Gamma$ is $g_{b_j}$. So the monodromy of $\mathcal L$ around $S_j$ is $\psi(g_{b_j})$. \end{proof} For $L\in\Pic^h(X)$ the divisors in $|L|$ that contain $b_j$ form a hyperplane $H_j=|L-b_j|$: since $\deg(L-b_j)=h-1\ge2g_X-1$, the point $b_j$ is not a base point. By Riemann--Roch, $h^0(L-\sum_jb_j)$, of degree $2g_X-2$, equals $g_X-1$ unless $L=L_0:=K_X+\sum_jb_j$, when it is $g_X$; and any $s-1$ of the hyperplanes are independent, because $L$ minus $s-1$ of the points has degree $2g_X-1$. \begin{lemma}\label{lem:L3} Assume (H1) and (H2). \begin{enumerate} \item[(a)] For $L\ne L_0$, $|L|-\bigcup H_j\cong(\C^\times)^{s-1}\times\A^{g_X-1}$, and $H^*_c(|L|-\bigcup H_j,\mathcal L)=0$. \item[(b)] If $s\ge3$, all $H_j$ in $|L_0|$ contain $\Pi_0=\sum_jb_j+|K_X|\cong\PP^{g_X-1}$, and projection from $\Pi_0$ makes $|L_0|-\bigcup H_j$ the total space of a vector bundle of rank $g_X$ over $\Omega=\PP^{s-2}$ minus $s$ hyperplanes in general position. \item[(c)] If $s=2$, then $H_1=H_2$ in $|L_0|$, and the complement $|L_0|-H_1$ is $\A^{g_X}$, on which $\mathcal L$ is trivial. \end{enumerate} \end{lemma} \begin{proof} (a) Off $L_0$ the $s$ linear forms defining the $H_j$ are linearly independent on $H^0(L)$, of dimension $h-g_X+1\ge s$. Choose homogeneous coordinates $x_0,\dots,x_{h-g_X}$ on $|L|$ with $H_j=\{x_{j-1}=0\}$ for $j=1,\dots,s$. Then $|L|-\bigcup H_j$ is the set where $x_0,\dots,x_{s-1}$ are nonzero, and in the chart $x_0=1$ it is $(\C^\times)^{s-1}\times\A^{h-g_X-s+1}$, with $h-g_X-s+1=g_X-1$. A small loop in the $j$-th factor $\C^\times$, $j=1,\dots,s-1$, is a meridian of $H_{j+1}$, around which $\mathcal L$ has monodromy $\psi(g_{b_{j+1}})\ne1$ (Lemma~\ref{lem:L2} and (F2)). The compactly supported cohomology of $\C^\times$ with a nontrivial rank-one local system vanishes, so the K\"unneth formula for compact supports \cite[Corollary~11.3.11]{Schapira} gives the vanishing. (b) At $L_0$ the intersection of all $H_j$ is $|L_0-\sum b_j|=\sum b_j+|K_X|$, of dimension $g_X-1$. Hyperplanes containing the centre of projection descend to hyperplanes of $\PP^{h-2g_X}=\PP^{s-2}$, and they are in general position because any $s-1$ of the $H_j$ are independent. (c) For $s=2$, $L_0-b_1=K_X+b_2$ has $b_2$ as a base point, so $H_1=b_1+b_2+|K_X|=H_2$, a hyperplane of $|L_0|\cong\PP^{g_X}$; its complement $\A^{g_X}$ is simply connected. \end{proof} \begin{proposition}[Localisation]\label{prop:localisation} Assume (H1) and (H2). Put $W^\circ=\beta^{-1}(|L_0|-\bigcup_jH_j)$. Restriction gives isomorphisms \[ U_\psi\cong H^h(W_0,\C)_\psi\xrightarrow{\ \sim\ }H^h\big(\beta^{-1}|L_0|,\C\big)_\psi\cong H^h_c(W^\circ,\C)_\psi \] of (complex) mixed Hodge structures on the $\psi$-parts, and both sides are one-dimensional. More generally, in every degree $b$, restriction $H^b(W_0,\C)_\psi\to H^b(\beta^{-1}|L_0|,\C)_\psi$ and extension by zero $H^b_c(W^\circ,\C)_\psi\to H^b(\beta^{-1}|L_0|,\C)_\psi$ are isomorphisms, and these spaces vanish for $b\ne h$. \end{proposition} \begin{proof} By Lemma~\ref{lem:L2}, $H^*(W_0,\C)_\psi=H^*(S^hX,j_!\mathcal L)=H^*(\Pic^h(X),R\pi_*j_!\mathcal L)$, hypercohomology of the derived direct image. Since $\pi$ is proper and $j_!\mathcal L$ is constructible, proper base change \cite[Theorem~11.3.6 and (11.3.3)]{Schapira} identifies the stalk of $R^b\pi_*j_!\mathcal L$ at $L$ with $H^b(|L|,j_!\mathcal L)=H^b_c(|L|-\bigcup H_j,\mathcal L)$, which vanishes for $L\ne L_0$ by Lemma~\ref{lem:L3}(a). So every cohomology sheaf of $R\pi_*j_!\mathcal L$ is supported at the point $L_0$, the complex is a skyscraper at $L_0$, and its hypercohomology in degree $b$ is its stalk $H^b(|L_0|,j_!\mathcal L)=H^b(\beta^{-1}|L_0|,\C)_\psi$, which is $H^b_c(|L_0|-\bigcup H_j,\mathcal L)$ through extension by zero; the restriction map to $\beta^{-1}|L_0|$ is this identification, in every degree. The same lemma identifies the stalk with $H^b_c(W^\circ,\C)_\psi$, compactly supported cohomology of the open part. Lemma~\ref{lem:L1} gives the dimension and the degree $b=h$. The maps are restrictions and the identification of a $j_!$-extension with compactly supported cohomology of the open part, so they are morphisms of mixed Hodge structures. \end{proof} \subsection{The special fibre and the Fermat hypersurface}\label{subsec:fermat} Assume first $s\ge3$, and put $r=s-2$, $d=\gcd(e,\alpha_1,\dots,\alpha_s)$, $e'=e/d$. Choose coordinates in which $\Omega=\{x\in\PP^{s-1}:\sum_jx_j=0,\ x_j\ne0\text{ for all }j\}$ (any $s$ hyperplanes in general position in $\PP^{s-2}$ can be put in this form). Let \[ F=F_{e'}^{r}=\Big\{Y\in\PP^{s-1}:\textstyle\sum_jY_j^{e'}=0\Big\}, \] the Fermat hypersurface of degree $e'$ and dimension $r$, and $F^\circ\subset F$ the open set where all $Y_j\ne0$. The map $Y\mapsto(Y_j^{e'})_j$ is a Galois cover $F^\circ\to\Omega$ with group $\Gamma_F=\mu_{e'}^s/\mu_{e'}$ (diagonal). A vector $a\in(\Z/e')^s$ with $\sum_ja_j=0$ defines a character $\zeta\mapsto\prod_j\zeta_j^{a_j}$ of $\Gamma_F$. For such a character $b$ let $V(b)=H^r(F,\C)_b$. \begin{lemma}[The special fibre]\label{lem:special} Assume (H1) and $s\ge3$, and let $a=\varepsilon\,(\alpha_j/d)_j$, where $\varepsilon\in\{\pm1\}$ is the sign relating the deck action on $F^\circ\to\Omega$ to monodromy; it depends only on the conventions, not on the slot or on $t$. By (F1), $\sum_ja_j=0$, so $a$ is a character of $\Gamma_F$. Let $H'_F$ be the kernel of the character $a$ of $\Gamma_F$ and $S'^\circ=F^\circ/H'_F$. \begin{enumerate} \item[(a)] $W^\circ$ has $d$ connected components, permuted transitively by $\Gamma$. Each is isomorphic to the pullback of the cyclic cover $S'^\circ\to\Omega$ to the rank-$g_X$ vector bundle of Lemma~\ref{lem:L3}(b). \item[(b)] There are isomorphisms of complex mixed Hodge structures \[ H^h_c(W^\circ,\C)_\psi\cong H^r_c(S'^\circ,\C)_{a}(-g_X)\cong H^r_c(F^\circ,\C)_a(-g_X)\cong V(a)(-g_X), \] and $V(a)$ is one-dimensional. \item[(c)] $V(a)$ has Hodge type $(p-g_X,\,h-p-g_X)$, where $p=\dim H^{1,0}(D')_\psi$. \end{enumerate} \end{lemma} \begin{proof} (a) Over the vector bundle, $W^\circ$ is pulled back from a $\Gamma$-torsor over $\Omega$ (the branch divisors have been removed) whose monodromy around the $j$-th hyperplane is $g_{b_j}$, of exponent $\alpha_j$ (Lemma~\ref{lem:L2}). Torsors under the abelian group $\Gamma$ over the connected $\Omega$ are classified by homomorphisms $H_1(\Omega,\Z)\to\Gamma$, and $H_1(\Omega,\Z)$ is generated by the meridians of the hyperplanes: a loop in $\Omega$ bounds a disc in the simply connected $\PP^{s-2}$, which can be moved transverse to the hyperplanes away from their intersections, and removing small discs at the intersection points writes the loop as a product of conjugates of meridians. The image of the homomorphism is $d\Z/e$, so the torsor is induced from a connected $d\Z/e$-torsor and has $d$ components, permuted transitively by $\Gamma$; each component is a connected cyclic cover of $\Omega$ of degree $e'$ whose meridian monodromies, in $d\Z/e\cong\Z/e'$, have exponents $\alpha_j/d$. The quotient $S'^\circ\to\Omega$ of $F^\circ\to\Omega$ is a connected cyclic cover with group $\Gamma_F/H'_F\cong\mu_{e'}$ whose meridian monodromies are given by the character $a$; so, after identifying $\mu_{e'}$ with $d\Z/e$ (which fixes $\varepsilon$), each component is $S'^\circ$ pulled back to the bundle. (b) $H^*_c(W^\circ)$ is induced from the stabilizer $d\Z/e$ of a component, so by Frobenius reciprocity its $\psi$-part is the $\psi|_{d\Z/e}$-part of one component; the restriction of a faithful character of $\Z/e$ to $d\Z/e\cong\Z/e'$ is faithful, and under the identification of (a) it is the character $a$. For a rank-$g_X$ algebraic vector bundle the Thom isomorphism shifts compactly supported cohomology by $2g_X$ and twists the mixed Hodge structure by $(-g_X)$. The $a$-part of $H^r_c(S'^\circ)=H^r_c(F^\circ)^{H'_F}$ is the $a$-part of $H^r_c(F^\circ)$, since $a$ is trivial on $H'_F$. Finally, $F-F^\circ$ is the union of the coordinate strata $F_J=F\cap\{Y_j=0\text{ for }j\in J\}$, $J\ne\emptyset$. Filter it by the closed sets $F_{\ge c}=\bigcup_{|J|\ge c}F_J$; the difference $F_{\ge c}-F_{\ge c+1}$ is the disjoint union of the locally closed strata $F^\circ_J$, $|J|=c$, where exactly the coordinates in $J$ vanish. Each $F^\circ_J$ is fixed pointwise by the factor $\mu_{e'}$ of $\Gamma_F$ in a coordinate $j\in J$, which therefore acts trivially on $H^*_c(F^\circ_J)$, while it acts on the $a$-part through the nontrivial character $\zeta\mapsto\zeta^{a_j}$ ($a_j\ne0$). So the $a$-part of $H^*_c(F^\circ_J)$ vanishes, and the long exact sequences of compactly supported cohomology for the decompositions $F_{\ge c}=(F_{\ge c}-F_{\ge c+1})\sqcup F_{\ge c+1}$ give $H^*_c(F-F^\circ)_a=0$; then the sequence for $F=F^\circ\sqcup(F-F^\circ)$ gives $H^r_c(F^\circ)_a\cong H^r(F)_a$. By Shioda \cite[Theorem~I, p.~176]{Shioda}, $V(a)$ is one-dimensional for $a$ with all entries nonzero and sum zero. (c) By (b) and Lemma~\ref{lem:L1}, $V(a)(-g_X)\cong U_\psi$, which has type $(p,h-p)$; the twist $(-g_X)$ raises both indices by $g_X$. As a check, Shioda's Theorem~I(ii) \cite[p.~176]{Shioda} gives $V(a)$ the type $(r+1-|a|,\,|a|-1)$ with $|a|=\sum_j\langle a_j\rangle/e'$, $\langle a_j\rangle\in\{1,\dots,e'-1\}$; for $a=(\alpha_j/d)_j$ and the convention for $p$ above, $p=g_X-1+\sum_j(1-\langle a_j\rangle/e')=g_X+r+1-|a|$, in agreement. \end{proof} \emph{Slots with $s=2$.} By Lemma~\ref{lem:L3}(c), $W^\circ$ is then a $\Gamma$-torsor over the simply connected $\A^{g_X}$, hence trivial: $W^\circ\cong\Gamma\times\A^{g_X}$, with $\Gamma$ acting by translation on the first factor. We put $r=0$, $F=F^\circ=\Gamma$ (a set of $e$ points), $\Gamma_F=\Gamma$ acting by translation, $a=\psi$, $H'_F=1$ and $S'^\circ=F$, so that $W^\circ$ is the trivial rank-$g_X$ bundle over $S'^\circ$ and $V(a)=H^0(F,\C)_\psi$ is one-dimensional. Lemma~\ref{lem:special} then holds with these conventions: (a) and (b) by the Thom isomorphism for the trivial bundle, and (c) because $p=g_X-1+\{-\alpha_1/e\}+\{\alpha_1/e\}=g_X$ and $h=2g_X$, so both sides have type $(0,0)$ after the twist. With these conventions all statements below hold for slots with $s_k=2$, including products in which some slots have $s_k=2$ and others $s_k\ge3$. In the application of Section~\ref{sec:datum} every slot has $s\ge20$. \subsection{Hodge types on the product}\label{subsec:product} Now restore the index $k$. Put $Y=\prod_kW_{0,k}$, an ordinary product over $\C$, with the action of $\Gamma_{\rm tot}=\prod_k\Gamma_k$ and the character $\bpsi=\boxtimes_k\psi_k$, and $F_{\rm tot}=\prod_kF_k$, of dimension $2\nu:=\sum_kr_k$. Put $\qtot=Rg_X$, so that $\htot=2\nu+2\qtot$. \begin{lemma}[Hodge type]\label{lem:HT} Under (H1)--(H2), the K\"unneth formula and Proposition~\ref{prop:localisation} give \[ H^{\htot}(Y,\C)_\bpsi=\textstyle\bigotimes_kH^{h_k}(W_{0,k},\C)_{\psi_k}\cong\Big(\bigotimes_kV(a_k)\Big)(-\qtot), \] a line, and the same for every simultaneous conjugate $\bpsi^t$, with $a_k$ replaced by $ta_k$. The line $\bigotimes_kV(ta_k)\subset H^{2\nu}(F_{\rm tot},\C)$ has type $(\nu,\nu)$ if and only if $\sum_k(2p_{k,t}-h_k)=0$. So (H4) holds if and only if every simultaneous conjugate is of type $(\nu,\nu)$. \end{lemma} \begin{proof} The first statement is K\"unneth for a finite product over a field, together with the slot isomorphisms; the $\bpsi$-part of a product of $\Gamma_{\rm tot}$-modules is the tensor product of the $\psi_k$-parts. By Lemma~\ref{lem:special}(c), $\bigotimes_kV(ta_k)$ has type $(\sum_kp_{k,t}-\qtot,\ \htot-\sum_kp_{k,t}-\qtot)$, which is $(\nu,\nu)$ exactly when $\sum_kp_{k,t}=\htot/2$. \end{proof} In particular, (H4) forces $\sum_kr_k$ to be even, though no single $r_k$ need be. Let $\cV\subseteq H^{2\nu}(F_{\rm tot},\Q)$ be the rational subspace whose complexification is the sum of the distinct lines $\bigotimes_kV(ta_k)$; under (H4) it consists of Hodge classes. \subsection{The Fermat input}\label{subsec:shioda} For an integer $m\ge1$ write $X^r_m\subset\PP^{r+1}$ for the Fermat variety $x_0^m+\dots+x_{r+1}^m=0$ of degree $m$ and dimension $r$. Shioda proves: \begin{theorem}[{Shioda \cite[Theorem~IV and the Corollary preceding it, p.~182]{Shioda}}]\label{thm:shioda} \emph{Fix $m\ge1$. Then the Hodge Conjecture for the product $X^{r_1}_m\times\dots\times X^{r_k}_m$ is true provided that it is true for $X^r_m$ for all even $r\le r_1+\dots+r_k+2(k-1)$. In particular, if $m$ is a prime or $m\le20$, then the Hodge Conjecture for arbitrary product $X^{r_1}_m\times\dots\times X^{r_k}_m$ is true.} \end{theorem} Here ``the Hodge Conjecture'' means that every rational Hodge class is algebraic, and the dimensions $r_1,\dots,r_k$ are arbitrary; the parity condition concerns only the auxiliary single Fermat varieties in Shioda's proof. The case $m\le20$ rests on Shioda's Theorem~III and a finite verification. Let $M_m$ be the semigroup of nonnegative integer solutions $(x_1,\dots,x_{m-1};y)$, $y>0$, of the equations $\sum_{v=1}^{m-1}\langle tv\rangle x_v=my$ for all $t\in(\Z/m)^\times$, where $\langle tv\rangle\in\{0,\dots,m-1\}$ is the representative, and $M_m(y)$ its elements of length $y$. An element is \emph{decomposable} if it is a sum of two elements of $M_m$, and \emph{quasi-decomposable} if adding some element of $M_m(1)$ makes it decomposable in a different way \cite[p.~180]{Shioda}. Shioda's condition $(P^n_m)$ is: \emph{every indecomposable element of $M_m(y)$ with $3\le y\le n/2+1$ is quasi-decomposable}; by his Theorem~III \cite[p.~180]{Shioda} it implies the Hodge conjecture for $X^n_m$. Shioda reports that he verified $(P^n_m)$ for $12\le m\le20$ and all $n$, with a table of indecomposable elements computed with Maruyama \cite[p.~182]{Shioda}. Theorem~C needs it for $m=12$, and we have recomputed this case exactly. The indecomposable elements of $M_{12}$ are its Hilbert basis, computed with Normaliz \cite{Normaliz}: $56$ elements, of lengths $y=1,2,3,4,5$ in numbers $6,16,22,8,4$. Each of the $34$ with $y\ge3$ is quasi-decomposable, with an explicit witness; for example \[ \begin{aligned} &(0,0,0,4,0,0,0,0,0,2,0;3)+(0,0,0,0,0,2,0,0,0,0,0;1)\\ &\qquad=2\,(0,0,0,2,0,1,0,0,0,1,0;2). \end{aligned} \] So $(P^n_{12})$ holds for every $n$, since the condition concerns only the finite Hilbert basis. The same program reproduces Shioda's table on p.~181 for $m=6,8,9,10$, and finds that the condition fails for $m=25$. The program and its output, with all $34$ witnesses, are in the supplementary files. In the application, slots of reduced order $12$ occur in each of the three product suppliers, so degree $12$ cannot be avoided with the identity used here. \begin{lemma}[Fermat input]\label{lem:fermatinput} Assume (H3) and (H4). Then every class in $\cV$ is algebraic. \end{lemma} \begin{proof} For each $k$ with $r_k\ge1$, the coordinate map $\phi_k\colon X^{r_k}_{12}\to F_k$, $[Y_j]\mapsto[Y_j^{12/e'_k}]$, is defined because $e'_k\mid12$; it has no base points and is finite and surjective (it is the quotient by $(\mu_{12/e'_k})^{s_k}$ modulo the diagonal). Slots with $r_k=0$ are finite sets of points and are kept as they are. Let $\Phi$ be the product map. For rational cohomology $\Phi_*\Phi^*=(\deg\Phi)\,\mathrm{id}$, so $\Phi^*$ is injective and carries Hodge classes to Hodge classes. A product of Fermat varieties of degree $12$ with a finite set is a disjoint union of copies of a product of Fermat varieties of degree $12$, so by Theorem~\ref{thm:shioda} with $m=12$ every Hodge class upstairs is algebraic. For $v\in\cV$ choose an algebraic cycle $\tilde z$ with class $\Phi^*v$; then $v=(\deg\Phi)^{-1}\Phi_*[\tilde z]$ is algebraic. \end{proof} The class in $\cV$ is not a product of classes on the factors, which are not Hodge classes on their own; this is why Shioda's theorem for products is needed. \subsection{The cone cycle}\label{subsec:cone} \emph{Notation.} In this subsection all (co)homology has complex coefficients unless stated otherwise; for a finite abelian group acting, $(\ )_\chi$ denotes the $\chi$-isotypic part, an exact functor. Put $W^\circ_{\rm tot}=\prod_kW^\circ_k$, an open subset of the compact space $Y_{L_0}=\prod_k\beta_k^{-1}|L_{0,k}|$, which is closed in $Y$; let $j\colon W^\circ_{\rm tot}\hookrightarrow Y_{L_0}$ and $i\colon Y_{L_0}\hookrightarrow Y$ be the inclusions, and $B=Y_{L_0}-W^\circ_{\rm tot}$. Similarly put $F^\circ_{\rm tot}=\prod_kF^\circ_k$ with inclusion $j'\colon F^\circ_{\rm tot}\hookrightarrow F_{\rm tot}$, and let $p\colon F_{\rm tot}\to S'_{\rm tot}=\prod_kF_k/H'_{F,k}$ and $p^\circ\colon F^\circ_{\rm tot}\to S'^\circ_{\rm tot}$ be the quotients by $H'_{F,\rm tot}=\prod_kH'_{F,k}$; since $\Gamma_{F_k}$ acts freely on $F^\circ_k$, $p^\circ$ is finite \'etale. Write $\bar\bpsi=\boxtimes_k\bar\psi_k$, and $-a=(-a_k)_k$ for the corresponding character of $\Gamma_{F,\rm tot}=\prod_k\Gamma_{F_k}$. We use three standard facts about a locally compact space $T$ that is a complex algebraic variety. (i) A closed subvariety $V\subseteq T$ of pure dimension $m$ has a fundamental class $[V]\in H^{BM}_{2m}(T)$ in Borel--Moore homology, natural under restriction to open subsets and under pushforward along closed embeddings; for compact $T$, Borel--Moore homology is singular homology. (ii) For an open inclusion $j_T\colon T^\circ\hookrightarrow T$, a class $\alpha\in H^{BM}_n(T)$ and $\beta\in H^n_c(T^\circ)$, one has $\langle\alpha,j_{T!}\beta\rangle_T=\langle\alpha|_{T^\circ},\beta\rangle_{T^\circ}$, where $j_{T!}$ is extension by zero and $\langle\ ,\ \rangle$ is the evaluation pairing (cap product followed by the degree map): extension by zero is adjoint to restriction. (iii) For a complex vector bundle $\pi_E\colon E\to S$ of rank $c$ with Thom class $\tau_E$, the map $w\mapsto\pi_E^*w\cup\tau_E$ is an isomorphism $H^b_c(S)\to H^{b+2c}_c(E)$ (Thom isomorphism), and $\langle\pi_E^{-1}[V],\pi_E^*w\cup\tau_E\rangle_E=\pm\langle[V],w\rangle_S$ for a closed subvariety $V\subseteq S$, by the projection formula and $\int_{\rm fibre}\tau_E=1$. \begin{lemma}\label{lem:boundary} Assume (H1) and (H2). For every $b$, $H^b(B)_{\bar\bpsi}=0$, and extension by zero $j_!\colon H^b_c(W^\circ_{\rm tot})_{\bar\bpsi}\to H^b(Y_{L_0})_{\bar\bpsi}$ is an isomorphism; restriction $H^b(Y)_{\bar\bpsi}\to H^b(Y_{L_0})_{\bar\bpsi}$ is an isomorphism. Likewise $H^b_c(F_{\rm tot}-F^\circ_{\rm tot})_{-a}=0$, and $j'_!\colon H^b_c(F^\circ_{\rm tot})_{-a}\to H^b(F_{\rm tot})_{-a}$ is an isomorphism. \end{lemma} \begin{proof} For one slot, Proposition~\ref{prop:localisation}, applied to the faithful character $\bar\psi_k$, which satisfies the same hypotheses, gives the two isomorphisms in every degree. The K\"unneth formula for the compact products $Y$ and $Y_{L_0}$ and for compactly supported cohomology of $W^\circ_{\rm tot}$ \cite[Corollary~11.3.11]{Schapira}, together with the fact that the $\bar\bpsi$-part of a tensor product of $\Gamma_{\rm tot}$-modules is the tensor product of the $\bar\psi_k$-parts, gives them for the products. The long exact sequence $\cdots\to H^b_c(W^\circ_{\rm tot})\to H^b(Y_{L_0})\to H^b(B)\to H^{b+1}_c(W^\circ_{\rm tot})\to\cdots$ is $\Gamma_{\rm tot}$-equivariant, and taking $\bar\bpsi$-parts is exact; since $j_!$ is an isomorphism on $\bar\bpsi$-parts in every degree, $H^*(B)_{\bar\bpsi}=0$. For $F_{\rm tot}$, stratify $F_{\rm tot}-F^\circ_{\rm tot}$ by the locally closed sets $\prod_kF^\circ_{k,J_k}$, where $F^\circ_{k,J}\subseteq F_k$ is the locus on which exactly the coordinates in $J$ vanish, and $J_k\ne\emptyset$ for at least one $k$. On such a stratum the factor $\mu_{e'_k}\subseteq\Gamma_{F_k}$ in a coordinate $j\in J_k$ acts trivially, since it rescales $Y_j=0$, while the character $-a$ is nontrivial on it because $a_{k,j}\ne0$. So the $(-a)$-part of the compactly supported cohomology of every stratum vanishes. Filtering by the closed unions of strata of dimension at most $c$ and using the open--closed long exact sequences gives $H^*_c(F_{\rm tot}-F^\circ_{\rm tot})_{-a}=0$, and the open--closed sequence for $F^\circ_{\rm tot}\subseteq F_{\rm tot}$ gives the last claim. \end{proof} \emph{The cycle.} Choose a nonzero $v\in\cV$ and, by Lemma~\ref{lem:fermatinput}, a rational algebraic cycle $z=\sum_\lambda n_\lambda z_\lambda$ on $F_{\rm tot}$ with $[z]=v$, the $z_\lambda$ irreducible of dimension $\nu$. Every component of $v$ on a distinct conjugate line $\bigotimes_kV(ta_k)$ is nonzero: the Galois action on coefficients permutes these lines transitively and fixes $v$. Put \[ Z=p_*z=\textstyle\sum_\lambda n_\lambda\deg\big(z_\lambda\to p(z_\lambda)\big)\,p(z_\lambda), \] a rational cycle of dimension $\nu$ on $S'_{\rm tot}$, and $Z^\circ$ its restriction to $S'^\circ_{\rm tot}$, so that $Z^\circ=p^\circ_*(z^\circ)$ with $z^\circ$ the restriction of $z$ to $F^\circ_{\rm tot}$. In each slot with $s_k\ge3$ fix one connected component $\mathcal E_k$ of $W^\circ_k$, the total space of a rank-$g_X$ vector bundle over $S'^\circ_k$ (Lemma~\ref{lem:special}(a)); in a slot with $s_k=2$ put $\mathcal E_k=W^\circ_k$, the trivial bundle over $S'^\circ_k=F_k$ (Subsection~\ref{subsec:fermat}). Let $\pi_{\mathcal E}\colon\mathcal E=\prod_k\mathcal E_k\to S'^\circ_{\rm tot}$, a vector bundle of rank $\qtot$. It is open and closed in $W^\circ_{\rm tot}$. For each $\lambda$ with $p(z_\lambda)$ meeting $S'^\circ_{\rm tot}$, the set $\pi_{\mathcal E}^{-1}(p(z_\lambda)\cap S'^\circ_{\rm tot})$ is irreducible of dimension $\nu+\qtot$ and closed in $W^\circ_{\rm tot}$; let $C_\lambda$ be its Zariski closure in $Y$, which is irreducible of the same dimension and contained in the closed set $Y_{L_0}$. Define \[ C(Z)=\textstyle\sum_\lambda n_\lambda\deg\big(z_\lambda\to p(z_\lambda)\big)\,C_\lambda, \] the sum over those $\lambda$. It is a rational algebraic cycle on $Y$ of pure dimension $\nu+\qtot=\htot/2$, and its restriction to $W^\circ_{\rm tot}$ is the cycle $\pi_{\mathcal E}^{-1}(Z^\circ)$. The closure of an irreducible locally closed set is irreducible of the same dimension.We write $[C(Z)]\in H_{\htot}(Y,\Q)$ for its fundamental class in homology, and $\langle[C(Z)],u\rangle$ for the evaluation of a class $u\in H^{\htot}(Y)$ on it. \emph{Map of the argument.} The spaces involved in Lemma~\ref{lem:cone}, Corollary~\ref{cor:coneup} and Proposition~\ref{prop:Ualg} are \[ P\xrightarrow{\ \varpi\ }Y\ \supseteq\ Y_{L_0}\ \supseteq\ W^\circ_{\rm tot}\ \supseteq\ \mathcal E\xrightarrow{\ \pi_{\mathcal E}\ }S'^\circ_{\rm tot}\xleftarrow{\ p^\circ\ }F^\circ_{\rm tot}\ \subseteq\ F_{\rm tot}. \] Here $\varpi$ is the quotient by $G_{0,\rm tot}$, $\mathcal E$ is open and closed in $W^\circ_{\rm tot}$, $\pi_{\mathcal E}$ is a vector bundle of rank $\qtot$, and $p^\circ$ is finite \'etale. The group $\Gamma_{\rm tot}$ acts on $Y$, $Y_{L_0}$ and $W^\circ_{\rm tot}$, and $\Gamma_{F,\rm tot}$ acts on $F_{\rm tot}$ and $F^\circ_{\rm tot}$. The cycles move from right to left: $z$ on $F_{\rm tot}$, then $Z=p_*z$, then $C(Z)$, the closure of $\pi_{\mathcal E}^{-1}Z^\circ$ in $Y$, then $\varpi^!C(Z)$ on $P$, then its projection by $e_{[\bpsi]}$. The cohomology classes move from left to right, as follows. \begin{center} \small \begin{tabular}{lp{0.6\textwidth}ll} step & class & group & character\\\hline 1 & $u\in H^{\htot}(Y)$, $i^*u=j_!u_c$, $u_c\in H^{\htot}_c(W^\circ_{\rm tot})$ & $\Gamma_{\rm tot}$ & $\bar\bpsi$\\ 2--3 & $u_c|_{\mathcal E}\in H^{\htot}_c(\mathcal E)$, nonzero & & \\ 4 & $u_c|_{\mathcal E}=\pi_{\mathcal E}^*w\cup\tau_{\mathcal E}$, $w\in H^{2\nu}_c(S'^\circ_{\rm tot})$ & & \\ 5 & $(p^\circ)^*w\in H^{2\nu}_c(F^\circ_{\rm tot})$ & $\Gamma_{F,\rm tot}$ & $-a$\\ 6 & $\xi=j'_!(p^\circ)^*w\in H^{2\nu}(F_{\rm tot})$ & $\Gamma_{F,\rm tot}$ & $-a$\\ 7 & paired with $[z]=v$, whose component of character $a$ is nonzero & $\Gamma_{F,\rm tot}$ & $a$ \end{tabular} \end{center} Here $2\nu=\dim F_{\rm tot}$ and $\htot=2\nu+2\qtot$; the pairing in step 7 is Poincar\'e duality on $F_{\rm tot}$, and steps 1--6 identify it with $\langle[C(Z)],u\rangle$ up to sign. In Corollary~\ref{cor:coneup} the nonvanishing is pulled back to $P$, and in Proposition~\ref{prop:Ualg} the $\Q[\Gamma_{\rm tot}]$-module generated by the projection gives all of $\cU$. \begin{lemma}[The cone pairing]\label{lem:cone} Assume (H1)--(H4). Let $u$ generate the line $H^{\htot}(Y)_{\bar\bpsi}$ (Lemma~\ref{lem:HT}). Then there is a generator $\xi$ of $\bigotimes_kV(-a_k)$ such that \[ \langle[C(Z)],u\rangle=\pm\int_{F_{\rm tot}}[z]\cup\xi\ne0 . \] \end{lemma} \begin{proof} \emph{Step 1.} By Lemma~\ref{lem:boundary} there is a unique $u_c\in H^{\htot}_c(W^\circ_{\rm tot})_{\bar\bpsi}$ with $j_!u_c=i^*u$, and $u_c\ne0$. Since $C(Z)$ is supported in $Y_{L_0}$, $\langle[C(Z)],u\rangle_Y=\langle[C(Z)],i^*u\rangle_{Y_{L_0}}=\langle[C(Z)],j_!u_c\rangle_{Y_{L_0}}$. \emph{Step 2 (no boundary terms).} By fact (ii), this equals $\langle[C(Z)]|_{W^\circ_{\rm tot}},u_c\rangle=\langle[\pi_{\mathcal E}^{-1}Z^\circ],u_c\rangle_{W^\circ_{\rm tot}}$, so whatever part of $C(Z)$ lies in $B$ does not enter. \emph{Step 3 (one component).} The cycle $\pi_{\mathcal E}^{-1}Z^\circ$ lies in the open and closed subset $\mathcal E$, so the pairing is $\langle[\pi_{\mathcal E}^{-1}Z^\circ],u_c|_{\mathcal E}\rangle_{\mathcal E}$. The components of $W^\circ_{\rm tot}$ are permuted transitively by $\Gamma_{\rm tot}$ (Lemma~\ref{lem:special}(a) in each slot), and $H^*_c(W^\circ_{\rm tot})=\bigoplus_{\gamma}H^*_c(\gamma\mathcal E)$ over the distinct translates. If $\gamma^*u_c=\bar\bpsi(\gamma)u_c$ for all $\gamma$, then restricting to $\mathcal E$ gives $\gamma^*(u_c|_{\gamma\mathcal E})=\bar\bpsi(\gamma)\,u_c|_{\mathcal E}$, where $\gamma\colon\mathcal E\to\gamma\mathcal E$. So if $u_c|_{\mathcal E}=0$ then $u_c$ vanishes on every component, hence $u_c=0$. Since $u_c\ne0$, $u_c|_{\mathcal E}\ne0$. \emph{Step 4 (Thom).} By fact (iii) there is a unique $w\in H^{2\nu}_c(S'^\circ_{\rm tot})$ with $\pi_{\mathcal E}^*w\cup\tau_{\mathcal E}=u_c|_{\mathcal E}$, it is nonzero, and the pairing equals $\pm\langle[Z^\circ],w\rangle_{S'^\circ_{\rm tot}}$. \emph{Step 5 (finite quotient).} Since $Z^\circ=p^\circ_*z^\circ$ and $p^\circ$ is proper, the projection formula gives $\langle[Z^\circ],w\rangle=\langle[z^\circ],(p^\circ)^*w\rangle_{F^\circ_{\rm tot}}$. The class $(p^\circ)^*w$ is nonzero: $p^\circ$ is finite \'etale of degree $|H'_{F,\rm tot}|$, and $p^\circ_*(p^\circ)^*w=|H'_{F,\rm tot}|\,w\ne0$ for the trace map $p^\circ_*$ on compactly supported cohomology. Moreover, by Lemma~\ref{lem:special}(b) for the opposite characters, slot by slot with K\"unneth, it lies in $H^{2\nu}_c(F^\circ_{\rm tot})_{-a}$. \emph{Step 6 (back to $F_{\rm tot}$).} Put $\xi=j'_!(p^\circ)^*w$. By Lemma~\ref{lem:boundary} it is a nonzero element of $H^{2\nu}(F_{\rm tot})_{-a}=\bigotimes_kV(-a_k)$, a line by Lemma~\ref{lem:special}(b), so it generates it. By fact (ii) for $j'$, $\langle[z^\circ],(p^\circ)^*w\rangle_{F^\circ_{\rm tot}}=\langle[z],\xi\rangle_{F_{\rm tot}}=\int_{F_{\rm tot}}[z]\cup\xi$; components of $z$ inside $F_{\rm tot}-F^\circ_{\rm tot}$ contribute zero to both sides. \emph{Step 7 (non-vanishing).} Poincar\'e duality on the smooth projective $F_{\rm tot}$ is invariant under $\Gamma_{F,\rm tot}$, so it pairs the $b$-part with the $b'$-part trivially unless $b+b'=0$, and it is perfect; hence it pairs $\bigotimes_kV(a_k)$ perfectly with $\bigotimes_kV(-a_k)$. The component of $[z]=v$ on $\bigotimes_kV(a_k)$ is nonzero, so $\int_{F_{\rm tot}}[z]\cup\xi\ne0$. Signs from orientation conventions affect only the sign in the statement. \end{proof} In the simplest case, one slot with $s=2$, $C(Z)$ is the closure of one copy of $\A^{g_X}$ in $W_0$, a component of the inverse image of $|L_0|$, and the lemma recovers the nonvanishing of Schoen's cycle \cite[Lemma~2.6]{Schoen88}. Let $P=\prod_kD_k^{h_k}$, a smooth projective variety, $G_{0,\rm tot}=\prod_kG_{0,k}$, and $\varpi\colon P\to Y=P/G_{0,\rm tot}$ the quotient map. For an irreducible subvariety $V\subseteq Y$, choose an irreducible component $V'$ of $\varpi^{-1}(V)$; it maps onto $V$, and the components of $\varpi^{-1}(V)$ are its translates. Put \[ \varpi^!V=\frac{|G_{0,\rm tot}|}{|G_{0,\rm tot}\cdot V'|\,\deg(V'\to V)}\sum_{V''\in G_{0,\rm tot}\cdot V'}V'', \] a rational algebraic cycle on $P$, invariant under $G_{0,\rm tot}$, with $\varpi_*\varpi^!V=|G_{0,\rm tot}|\,V$; extend $\varpi^!$ linearly to cycles. \begin{corollary}\label{cor:coneup} Under the hypotheses of Lemma~\ref{lem:cone}, the class $[\varpi^!C(Z)]\in H^{\htot}(P,\Q)$ is algebraic and has a nonzero component on the line $\bigotimes_kU_{\psi_k}$. \end{corollary} \begin{proof} By the projection formula for $\varpi$ (naturality of the evaluation pairing) and Lemma~\ref{lem:cone}, \[ \int_P[\varpi^!C(Z)]\cup\varpi^*u=\langle\varpi_*[\varpi^!C(Z)],u\rangle_Y=|G_{0,\rm tot}|\,\langle[C(Z)],u\rangle_Y\ne0 . \] The class $[\varpi^!C(Z)]$ is $G_{0,\rm tot}$-invariant, so it lies in $H^{\htot}(P)^{G_{0,\rm tot}}=\varpi^*H^{\htot}(Y)$, on which $\Gamma_{\rm tot}$ acts through any lift to $\prod_k\hat G_k$. These lifts are automorphisms of $P$ and preserve its Poincar\'e pairing, so a class in $H^{\htot}(P)^{G_{0,\rm tot}}$ of character $\chi$ pairs to zero with $\varpi^*u$ unless $\chi=\bpsi$. Hence $[\varpi^!C(Z)]$ has a nonzero $\bpsi$-component, and by Lemma~\ref{lem:L1} and K\"unneth the $\bpsi$-part of $H^{\htot}(P)^{G_{0,\rm tot}}$ is the line $\bigotimes_kU_{\psi_k}$. \end{proof} \begin{proposition}\label{prop:Ualg} Assume (H1)--(H4). Let $\cU$ be the rational subspace of $H^{\htot}(\prod_kD_k^{h_k},\Q)$ whose complexification is the sum of the distinct lines $\bigotimes_kU_{\psi_k^t}$. Then $\cU$ is spanned by classes of algebraic cycles. \end{proposition} \begin{proof} By Corollary~\ref{cor:coneup}, the rational algebraic class $x=[\varpi^!C(Z)]$ on $P$ lies in $H^{\htot}(P,\Q)^{G_{0,\rm tot}}$ and has a nonzero component on $\bigotimes_kU_{\psi_k}$. The group $\Gamma_{\rm tot}$ acts on $H^*(P)^{G_{0,\rm tot}}$ through automorphisms of $P$ in $\prod_k\hat G_k$, which preserve algebraic classes. Let $e_{[\bpsi]}\in\Q[\Gamma_{\rm tot}]$ be the central idempotent belonging to the Galois orbit $O$ of $\bpsi$, \[ e_{[\bpsi]}=\frac1{|\Gamma_{\rm tot}|}\sum_{\gamma\in\Gamma_{\rm tot}}c_\gamma\,\gamma,\qquad c_\gamma=\sum_{\chi\in O}\chi(\gamma)^{-1}\in\Q, \] the coefficients being rational because $O$ is Galois stable. Each $\gamma$ acts on $H^*(P)^{G_{0,\rm tot}}$ as the pullback by a lift $\hat\gamma\in\prod_k\hat G_k$, that is, by the algebraic correspondence given by the graph of $\hat\gamma$. So $e_{[\bpsi]}x=\frac1{|\Gamma_{\rm tot}|}\sum_\gamma c_\gamma\hat\gamma^*x$ is a rational combination of algebraic classes, a rational algebraic class, and it is nonzero, because its component on $\bigotimes_kU_{\psi_k}$ is that of $x$. By Lemma~\ref{lem:L1} and K\"unneth, the $\bpsi^t$-part of $H^{\htot}(P)^{G_{0,\rm tot}}$ is the line $\bigotimes_kU_{\psi_k^t}$, so $e_{[\bpsi]}x\in\cU$. It remains to see that $\cU$ is generated by any nonzero vector as a $\Q[\Gamma_{\rm tot}]$-module; this is Lemma~\ref{lem:simple} below, applied to $M=H^{\htot}(P,\Q)^{G_{0,\rm tot}}$ and $\vartheta_0=\bpsi$, whose isotypic parts in the orbit are the one-dimensional lines just named. The $\Q[\Gamma_{\rm tot}]$-span of $e_{[\bpsi]}x$ consists of algebraic classes, so it is all of $\cU$. \end{proof} \begin{lemma}\label{lem:simple} Let $\Gamma$ be a finite abelian group, $\vartheta_0$ a character of $\Gamma$ with Galois orbit $O=\{\vartheta_0^t\}$, and $M$ a finite-dimensional $\Q[\Gamma]$-module with $\dim_\C(M\otimes\C)_\chi=1$ for every $\chi\in O$. Then $\mathcal U_O=e_O M$, whose complexification is $\bigoplus_{\chi\in O}(M\otimes\C)_\chi$, is a simple $\Q[\Gamma]$-module, and any nonzero vector of it generates it. \end{lemma} \begin{proof} By Wedderburn's theorem for the commutative semisimple algebra $\Q[\Gamma]$, $\Q[\Gamma]=\prod_{O'}\Q[\Gamma]e_{O'}$ over the Galois orbits $O'$ of characters, with $\Q[\Gamma]e_{O'}\cong\Q(\chi)$ for $\chi\in O'$, a field of degree $|O'|$. The factor for $O$ acts on $\mathcal U_O$ through the field $\Q(\vartheta_0)$ and the other factors act by $0$. Since $\dim_\Q\mathcal U_O=|O|=[\Q(\vartheta_0):\Q]$, $\mathcal U_O$ is a one-dimensional $\Q(\vartheta_0)$-vector space; so it is simple, and any nonzero vector spans it over $\Q(\vartheta_0)=\Q[\Gamma]e_O$. \end{proof} In our situation $\bpsi$ has order $\operatorname{lcm}_ke_k=M_0$, because its restriction to $\Gamma_k$ is the faithful $\psi_k$; so $t\mapsto\bpsi^t$ is injective on $(\Z/M_0)^\times$ and the lines $\bigotimes_kU_{\psi_k^t}$, $t\in(\Z/M_0)^\times$, are pairwise distinct. \subsection{Abel--Prym}\label{subsec:abelprym} For each slot let $J_k=J(D_k)$, of dimension $g_k$, with theta class $\Theta_k$, and let $\Xi_k\colon D_k^{h_k}\to J_k$ be the sum map after a choice of base point. \begin{lemma}[{After Schoen \cite[proof of Corollary~3.1, pp.~24--25]{Schoen88}}]\label{lem:abelprym} There is an algebraic correspondence $T_k$ with rational coefficients from $D_k^{h_k}$ to $J_k$ such that $T_k\circ\Xi_k^*$ is the identity on $H^{h_k}(J_k,\Q)$. In particular $T_k$ maps $U_{\psi_k^t}$ isomorphically onto $\det H^1(D_k,\C)_{\psi_k^t}$ for every $t$. \end{lemma} \begin{proof} First, $h_k\le g_k$: $\psi_k$ and $\bar\psi_k$ are distinct because $e_k\ge3$, both eigenspaces have dimension $h_k$, and their sum lies in $H^1(D_k)$, of dimension $2g_k$. For $a\in H^{h_k}(J_k)$ the projection formula gives $\Xi_{k*}\Xi_k^*a=a\cup\Xi_{k*}(1)$. The image of $\Xi_k$ is the subvariety $W_{h_k}$ of effective divisor classes of degree $h_k$, onto which $\Xi_k$ has degree $h_k!$, and Poincar\'e's formula gives $[W_{h_k}]=\Theta_k^{g_k-h_k}/(g_k-h_k)!$ (Griffiths--Harris \cite[p.~350]{GH}, as used by Schoen); so $\Xi_{k*}(1)=c_k\Theta_k^{g_k-h_k}$ with $c_k=h_k!/(g_k-h_k)!$. Cup product with $\Theta_k^{g_k-h_k}$ is an isomorphism $L_k\colon H^{h_k}(J_k)\to H^{2g_k-h_k}(J_k)$ by hard Lefschetz. Its inverse is a polynomial in the Lefschetz operator and its adjoint $\Lambda$, and Lieberman's Theorem~3 \cite[p.~372]{Lieberman} states that on an abelian variety $\Lambda$ maps algebraic classes to algebraic classes, through an algebraic correspondence; so $L_k^{-1}$ is induced by an algebraic correspondence with rational coefficients, as in Schoen's proof. Put $T_k=c_k^{-1}L_k^{-1}\circ\Xi_{k*}$. The last statement follows because $U_{\psi^t}=\Xi_k^*\det H^1(D_k)_{\psi^t}$ (Schoen's Remark~1.4). \end{proof} \begin{proof}[Proof of Theorem~\ref{thm:C}] The external product $\boxtimes_kT_k$ is an algebraic correspondence with rational coefficients from $\prod_kD_k^{h_k}$ to $\prod_kJ_k$. By Lemma~\ref{lem:abelprym} it maps each line $\bigotimes_kU_{\psi_k^t}$ isomorphically onto $\ell^{(t)}$, so it maps $\cU$ onto $W$. By Proposition~\ref{prop:Ualg}, $\cU$ is spanned by algebraic classes, hence so is $W$. \end{proof} \begin{remark}\label{rem:ABscope} The conclusion holds at every cover $D\to X$ with the given data. The theorem does not cover slots with $s_k\le1$, for which there is no special fibre to localise to. It says nothing about whether every balanced combination of characters of every group satisfies (H2) and (H3); in the application these are certified (Proposition~\ref{prop:slots}). \end{remark} %% ------------------------------------------------------------------ \section{The Hermitian form and dominance}\label{sec:herm} %% ------------------------------------------------------------------ \subsection{The Hermitian form as a parabolic cup product} To compute the discriminant of the family we need its first cohomology and Hermitian form in terms of the monodromy tuple alone. Let $\pi\colon C\to\PP^1$ be a cover in $\Hur$, $U_C=\PP^1-\{p_j\}$, and $\mathcal R=\pi_*\Q|_{U_C}$, a local system with fibre $\Q[G]$. Put $L'=\Hom_G(V_\sigma,\mathcal R\otimes K)$, a local system of $K$-vector spaces of rank $d$. The fibre pairing of $\mathcal R$ (multiplication followed by the sum over the fibre), made conjugate-linear in the first variable, restricts on $V_\sigma\otimes L'$ to $h_V\otimes h_{L'}$ for a flat positive definite Hermitian form $h_{L'}$ on $L'$, unique up to a positive rational factor. Let \[ \Vp=H^1_{\rm par}(U_C,L'),\qquad \mathcal S(x,y)=\int_{U_C}h_{L'}(x\cup y),\qquad h=\vartheta\mathcal S, \] the parabolic cohomology (the image of compactly supported cohomology), a $K$-space of dimension $m_\sigma$, with its skew-Hermitian cup product $\mathcal S$ and the Hermitian form $h$. \begin{proposition}[{\cite[Proposition~7.1]{SevernQ3}}]\label{prop:herm} There is a $K$-linear or $K$-antilinear isomorphism $H_1(A_\sigma(C),\Q)\cong\Vp$ under which the Hermitian form $H$ of $A_\sigma(C)$ is a nonzero rational multiple of $h$, or of $\bar h$. In particular $H$ and $h$ have the same discriminant. \end{proposition} \begin{proof} \emph{Step 1.} On $H_1(A,\Q)$ the form $\mathcal T=H/2\vartheta$ is the unique $K$-sesquilinear skew-Hermitian form with $\Tr_{K/\Q}\mathcal T=E$. \emph{Step 2.} Let $N'=H_1(J(C),\Q)_\varepsilon$ and $M=\Hom_{K[G]}(V_\sigma,N')$. Evaluation $V_\sigma\otimes_KM\to N'$ is a $G$-equivariant isomorphism, because $\End_{K[G]}(V_\sigma)=K$. A $G$-invariant sesquilinear form on $N'$ pulls back to $h_V\otimes\psi_M$ for a unique sesquilinear form $\psi_M$ on $M$, since the $G$-invariant sesquilinear forms on $V_\sigma$ are the multiples of $h_V$. Applied to $\mathcal T$, which is $G$-invariant by the uniqueness in Step~1, this gives $\mathcal T=h_V\otimes\psi_M$, and $\mathcal T|_{A_\sigma}=h_V(v,v)\,\psi_M$ under $m\mapsto v\otimes m$, because $e$ is Rosati-fixed. \emph{Step 3: cohomology of $C$ as parabolic cohomology.} Let $j:U_C\hookrightarrow\PP^1$. Since $\pi$ is finite and $C$ is smooth, $\pi_*K_C=j_*(\mathcal R\otimes K)$: at a branch point the stalk of $\pi_*K_C$ is the space of functions on the fibre, which is the space of invariants of the local monodromy on $\mathcal R\otimes K$. As $\pi$ is finite, $H^1(C,K)=H^1(\PP^1,j_*(\mathcal R\otimes K))$. For a local system $\mathsf L$ on $U_C$, the sequences $0\to j_!\mathsf L\to j_*\mathsf L\to\bigoplus_j(\mathsf L^{T_j})_{p_j}\to0$ and $j_*\mathsf L\to Rj_*\mathsf L\to(R^1j_*\mathsf L)[-1]$, whose third terms are skyscrapers, show that $H^1_c(U_C,\mathsf L)\to H^1(\PP^1,j_*\mathsf L)$ is surjective and $H^1(\PP^1,j_*\mathsf L)\to H^1(U_C,\mathsf L)$ is injective; so $H^1(\PP^1,j_*\mathsf L)$ is the parabolic cohomology $H^1_{\rm par}(U_C,\mathsf L)$. Under these identifications the cup product of $C$ followed by integration is the cup product $H^1_c(U_C,\mathcal R)\times H^1(U_C,\mathcal R)\to H^2_c(U_C,\Q)=\Q$ with the fibre pairing of $\mathcal R$, which is the pushforward of the multiplication on $\Q_C$ followed by the trace $\pi_*\Q_C\to\Q$. The functor $\Hom_{K[G]}(V_\sigma,-)$ is exact and commutes with these operations, so it identifies the $\sigma$-part of the skew-Hermitian lift of the cup product with $h_V\otimes\mathcal S$. \emph{Step 4.} Poincar\'e duality on $C$ and the identification of $H_1$ with $H^1$ by $E$ are $G$-equivariant and carry $\mathcal T$ to $\pm$ the lift of Step~3. Depending on the normalisation of the $K$-action on cohomology (through $\iota$ or through its transpose), they carry the $\sigma$-component to the $\sigma$-component or to the $\bar\sigma$-component; accordingly the induced isomorphism $M\cong\Vp$ is $K$-linear or $K$-antilinear, and $\psi_M=c\,\mathcal S$ or $\psi_M=c\,\bar{\mathcal S}$ with $c\in K^\times$. Both forms are skew-Hermitian, so $c=\bar c$ is rational. In either case the discriminants agree: scaling a Hermitian form of even dimension by a rational number multiplies its determinant by a norm, and $\det\bar h=\det h$. So the choice of normalisation does not matter for the discriminant. \end{proof} The form $\mathcal S$ is computable from the tuple. Let $\mathsf W$ be a local system of $K$-vector spaces on $\PP^1$ minus $r$ points with a flat Hermitian form $h_{\mathsf W}$ and unitary monodromy $T_1,\dots,T_r$ along standard loops with $\ell_1\cdots\ell_r=1$, generating a finite group. Put $Q_i=T_1\cdots T_{i-1}$. A parabolic cocycle is given by vectors $a_i$ with $\sum_iQ_i(1-T_i)a_i=0$; its value on $\ell_i$ is $c_i=(1-T_i)a_i$. Put $F_i=\sum_{j=stealth] \foreach \a/\n in {90/0,0/1,-90/2,180/3} { \draw[->] (0,0) -- (\a:2.9); \node at (\a:3.3) {$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 the plane $\PP^1-\{\infty\}$. 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 $r$, so $e_r=e_0$); 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 $(b_0,p_i,b_\infty)$ in this order: for each $i$ one triangle with edges $f_i,g_i,e_{i-1}$ and one with edges $f_i,g_i,e_i$. (The Euler characteristic is $(r+2)-3r+2r=2$.) 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 $p_i$; then $\ell_1\cdots\ell_r$ is out along $e_0$ and back along $e_r=e_0$, which is trivial. 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 $(f\cup f')[v_0,v_1,v_2]=h_{\mathsf W}\big(f[v_0,v_1],f'[v_1,v_2]\big)$ \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 $h_{\mathsf W}(f[b_0,p_i],f'[p_i,b_\infty])$ of the two triangles combine, by the cocycle condition on each triangle, into \[ \mathcal S_{\rm simp}(f,f')=\sum_ih_{\mathsf W}\big(v^f_{e_i}-v^f_{e_{i-1}},\ v^{f'}_{g_i}\big), \] with $v^f_\epsilon$ the value of $f$ on the chosen lift of $\epsilon$. 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 $\mathcal S_{\rm simp}$ and summing gives \[ \mathcal S_{\rm simp}(f,f')=-\sum_i\bigl[h_{\mathsf W}(c_i,a'_i)+h_{\mathsf W}(Q_ic_i,F'_i)\bigr]+h_{\mathsf W}\Bigl(\sum_iQ_ic_i,\ v^{f'}_{e_0}\Bigr), \] 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 \[ \overline{\mathcal S(b,a)}=\sum_ih_{\mathsf W}\bigl(c_i,\ Q_i^{-1}F'_i+a'_i\bigr)=\sum_i\bigl[h_{\mathsf W}(c_i,a'_i)+h_{\mathsf W}(Q_ic_i,F'_i)\bigr], \] and $\mathcal S_{\rm simp}(f,f')=-\overline{\mathcal S(b,a)}$. It remains to show $\mathcal S(a,b)+\overline{\mathcal S(b,a)}=0$. 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 \[ \begin{aligned} \mathcal S(a,b)+\overline{\mathcal S(b,a)}&=\sum_i\bigl[h_{\mathsf W}(F_{i+1},F'_{i+1})-h_{\mathsf W}(F_i,F'_i)\bigr]\\ &=h_{\mathsf W}(F_{r+1},F'_{r+1})-h_{\mathsf W}(F_1,F'_1)=0, \end{aligned} \] because $F_1=0$ and $F_{r+1}=\sum_iQ_ic_i=0$. So $\mathcal S_{\rm simp}=\mathcal S$, and $\mathcal S$ 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 $\mathcal S(a,b)$ 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} \subsection{The codifferential and dominance}\label{subsec:codiff} Let $C\in\Hur$ with branch set $B_C$, $|B_C|=r$, and put $\cD_\sigma(C)=\Hom_G(\sigma,H^0(C,K_C))$ and $\cD_{\bar\sigma}(C)=\Hom_G(\bar\sigma,H^0(C,K_C))$, each of dimension $n$ by Proposition~\ref{prop:mult}. 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 $r-3$. Let \[ \mu_C\colon\cD_\sigma(C)\otimes\cD_{\bar\sigma}(C)\to H^0\big(\PP^1,K_{\PP^1}^{\otimes2}(B_C)\big) \] be multiplication of differentials followed by contraction of $\sigma\otimes\bar\sigma$ 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 $\cHn$ at the period point of $A_\sigma(C)$ is $\cD_\sigma(C)\otimes\cD_{\bar\sigma}(C)$, of dimension $n^2$, and $\mu_C$ is, up to a nonzero normalization, the codifferential at $C$ of the period map of the family $A_\sigma$. \end{lemma} \begin{proof} The codifferential of the period map of $J(C)$ along the Hurwitz family is multiplication of differentials $H^0(K_C)\otimes H^0(K_C)\to H^0(K_C^{\otimes2})$ 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 $\sigma$- and $\bar\sigma$-components and lands in the invariants is $\mu_C$. The domain $\cHn$ parametrizes the $\sigma$-eigenspace of $H^{1,0}$, which is $\cD_\sigma(C)$ with compatible conventions, and its tangent space is $\Hom(H^{1,0}_\sigma,H^{0,1}_\sigma)\cong\cD_\sigma(C)^\vee\otimes\overline{\cD_{\bar\sigma}(C)}$, whose dual, via the polarization, is $\cD_\sigma(C)\otimes\cD_{\bar\sigma}(C)$. \end{proof} \begin{proposition}[Rank at one point]\label{prop:rank} Let $S$ be a connected component of $\Hur$ and suppose $\rk\mu_{C_1}=n^2$ for one $C_1\in S$. Then the period map of the family $A_\sigma$ over $S$ is a submersion at $C_1$ and has generic rank $n^2$. \end{proposition} \begin{proof} The locus where $\rk\mu_C=n^2$ is Zariski open in the irreducible $S$ and contains $C_1$; $n^2=\dim\cD_\sigma(C)\otimes\cD_{\bar\sigma}(C)$ is the maximum. \end{proof} A dimension count $r-3\ge n^2$ is necessary for this and not sufficient; the rank must be certified at a point. %% ------------------------------------------------------------------ \section{Spreading}\label{sec:spread} %% ------------------------------------------------------------------ Let $\cM=\cM(K,n,\fd)$ be the moduli space of Subsection~\ref{subsec:moduli}. \begin{lemma}\label{lem:locus} Let $\mathcal W=\bigwedge^{2n}_KR^1f_*\Q\subset R^{2n}f_*\Q$ 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}; we give the details. Fix a relatively ample line bundle on the smooth projective family $f\colon\mathcal A\to\cM$. For each Hilbert polynomial, the relative Hilbert scheme of $n$-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 \cite[pp.~90--91]{VoisinII}. A rational algebraic class in $H^{2n}(\mathcal A_t,\Q)$ is a rational combination of classes of $n$-dimensional subschemes of $\mathcal A_t$, so it suffices to consider finite tuples of such subschemes. Fix a finite tuple $\mathcal H_1,\dots,\mathcal H_k$ of components, let $T$ be an irreducible component of $\mathcal H_1\times_\cM\dots\times_\cM\mathcal H_k$, let $\widetilde T\to T$ be a resolution of singularities, and let $g\colon\widetilde T\to\cM$ be the composite. Then $\widetilde T$ is smooth and connected, and $g$ is proper, so $g(\widetilde T)$ is a closed algebraic subset of $\cM$. The universal subschemes pull back to subschemes $\mathcal Z_l\subset\mathcal A\times_\cM\widetilde T$, flat over $\widetilde T$, and the classes $[\mathcal Z_{l,u}]\in H^{2n}(\mathcal A_{g(u)},\Q)$, $u\in\widetilde T$, are global flat sections of the local system $g^*R^{2n}f_*\Q$ on $\widetilde T$: the fibre classes of a flat family of subschemes of a smooth projective family are locally constant, as used in \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 along a path from $u$ to $u'$ is an isomorphism carrying each of these global sections to itself, so it carries $\mathcal F_u$ onto $\mathcal F_{u'}$ and $g^*\mathcal W_u$ onto $g^*\mathcal W_{u'}$; in particular $\dim\mathcal F_u$ does not jump. Hence the set of $u$ with $g^*\mathcal W_u\subseteq\mathcal F_u$ is either all of $\widetilde T$ or empty. 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 some component $T$, which is then good, and $t\in g(\widetilde T)$. So $\cM^{\rm alg}$ is the union of the closed sets $g(\widetilde T)$ over the good $T$, of which there are countably many. \end{proof} \begin{theorem}\label{thm:transfer} Let $S$ be a connected component of $\Hur$ such that $W_K(A_\sigma(C))$ is spanned by algebraic classes for every $C\in S$, and such that $\rk\mu_{C_1}=n^2$ at some $C_1\in S$. Then the Weil classes of every polarized abelian $2n$-fold of Weil type for $K$ with discriminant $\fd(S)$ are algebraic. \end{theorem} \begin{proof} Choose a small polydisc $\Delta\ni C_1$ over which the covers form an analytic family. Over the simply connected $\Delta$ the spaces $H_1(A_\sigma(C),\Q)$ with $\iota$ and $H$ form a trivial flat bundle of Hermitian spaces of discriminant $\fd(S)$, so Landherr's theorem gives a $K$-isometry $m\colon H_1(A_\sigma(C_1),\Q)\to(V,H)$, which we transport over $\Delta$. Sending $C$ to the complex structure of $H_1(A_\sigma(C),\R)$, transported by $m$, gives the period map $\Phi\colon\Delta\to\cHn$, a submersion at $C_1$ by Lemma~\ref{lem:codiff} and Proposition~\ref{prop:rank}. 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 $z$: $H_1(A_z,\Z)=\Lambda$, with complex structure $z$ and polarization $\Tr_{K/\Q}(H/2\vartheta)$ scaled as in Subsection~\ref{subsec:moduli}. The lattice $m(H_1(A_\sigma(C),\Z))$ is commensurable with $\Lambda$; if $N_1m(H_1(A_\sigma(C),\Z))\subseteq\Lambda$ for an integer $N_1\ge1$, then $N_1m$ is a $K$-linear map of lattices compatible with the complex structures, so it is a $K$-linear isogeny $A_\sigma(C)\to A_z$. Since $m$ is an isometry of Hermitian forms, it pulls the polarization of $A_z$ back to a positive rational multiple of that of $A_\sigma(C)$, which changes neither the Weil plane nor the class of the discriminant. 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, and since $\cM$ is irreducible it is all of $\cM$. Every polarized $2n$-fold of Weil type for $K$ with discriminant $\fd(S)$ is $K$-isogenous to a fibre of $\mathcal A$ (Lemma~\ref{lem:isogeny}). \end{proof} %% ------------------------------------------------------------------ \section{The group $W(E_8)\cdot\mu_4$ and the proof of Theorem~A}\label{sec:datum} %% ------------------------------------------------------------------ \subsection{The group and the class datum} Let $K=\Q(i)$ and $W=W(E_8)$, the Weyl group of $E_8$, of order $696\,729\,600$, acting on $\Q^8$; we use the basis of fundamental weights in Bourbaki's numbering (Appendix~\ref{app:tuple}). Let \[ G=W\cdot\mu_4=W\cup iW\subset GL_8(K), \] of order $1\,393\,459\,200$, the central product of $W$ and $\mu_4$ over $\{\pm1\}$, and let $\sigma$ be its natural representation on $K^8$, $d=8$. It is absolutely irreducible, because $W$ is, and its character field is $K$, since $\operatorname{tr}\sigma(iw)=i\operatorname{tr}(w)$. So $\sigma$ satisfies the assumptions of Subsection~\ref{subsec:pt}. The class datum has $r=12$ points: ten reflections of $W$, an element $g_{11}=ix_1$ with $x_1\in W$ of characteristic polynomial $\Phi_4^4$ (so $x_1^2=-1$), and an element $g_{12}=ix_2$ with $x_2\in W$ of characteristic polynomial $\Phi_2^2\Phi_6\Phi_{12}$. Elements $ix$ and $iy$ of $G$ are conjugate exactly when $x$ and $y$ are conjugate in $W$; the classes of $x_1$ and $x_2$ are those of the elements $a^5$ and $-a^{-1}d^5a$ of Appendix~\ref{app:tuple}. Let $\mathbf g=(g_1,\dots,g_{12})$ be the tuple of Appendix~\ref{app:tuple}: it has product $1$, it generates $G$, and its entries lie in these classes. Let $C_0$ be the cover of Appendix~\ref{app:rank}, whose monodromy tuple is $\mathbf g$, and $S$ the connected component of the Hurwitz space containing $C_0$, of dimension $12-3=9$. \emph{Why this datum.} For sixfolds ($n=3$), Proposition~\ref{prop:mult} gives $\sum_jc_\sigma(g_j)=2d+6=22$, and dominance needs at least $n^2+3=12$ branch points with $c_\sigma\ge1$. A reflection has $c_\sigma=1$; the two non-reflection points have $c_\sigma=4$ and $8$, and $10+4+8=22$, with exactly $12$ points. So this datum is at the edge of what the count allows. \begin{lemma}\label{lem:sixfold} For every $C\in\Hur$, $\sigma$ occurs in $H^1(C)$ with multiplicity $6$ and Hodge numbers $(3,3)$, so $A_\sigma(C)$ is a polarized abelian sixfold of Weil type for $K$. \end{lemma} \begin{proof} A reflection has $c_\sigma=1$ and eigenvalue angles $(0^7,\tfrac12)$, so age $\tfrac12$. The element $ix_1$ has eigenvalues $i\cdot(\pm i)=\mp1$, four of each, so $c_\sigma=4$ and age $2$. The eigenvalues of $x_2$ are $-1$ (twice), the primitive sixth roots of unity and the primitive twelfth roots of unity; multiplying by $i$ never gives $1$, so $c_\sigma(ix_2)=8$, and the eight angles of $ix_2$ sum to $4$. Proposition~\ref{prop:mult} gives $m_\sigma=-16+10+4+8=6$. The ages of the inverses are $c_\sigma$ minus the ages, with the same total $22-11=11$, so both Hodge numbers equal $11-8=3$. Conclude with Lemma~\ref{lem:weiltype}. \end{proof} The characters of $G$: the irreducible characters of $W$ are all real and orthogonal, and those that are $-1$ on the central element $-1\in W$ extend to $G$ in two complex conjugate ways, $i\mapsto\pm i$. Exactly $45$ of the resulting pairs $\{\tau,\bar\tau\}$ have $m_\tau>0$, and there are no quaternionic $\tau$, so $X^*(T)=\Z^{45}$, with $e_\sigma$ a coordinate vector (Appendix~\ref{app:witness}). The representation $\sigma$ is the only one with $m_\tau=6$. \subsection{Dominance and the discriminant} \begin{proposition}\label{prop:dominance} $\rk\mu_{C_0}=9$, and $\fd(S)=[-3]$. \end{proposition} \begin{proof} The rank is the certificate of Appendix~\ref{app:rank}: a $9\times9$ minor of $\mu_{C_0}$, in residue coordinates, is enclosed in a complex ball that excludes $0$. For the discriminant, Proposition~\ref{prop:herm} reduces $\fd(S)$ to $\det h$ for $\Vp=H^1_{\rm par}(U_{C_0},L')$, computed from the tuple by Lemma~\ref{lem:cocycle} as exact linear algebra over $K$. \emph{The tuple used.} The computation uses the tuple $T_j=g_{13-j}^T$, $j=1,\dots,12$: the transposes in reverse order (the two scalar points first, then the ten reflections), as stored in the supplementary files. Since every $g\in W$ preserves the Gram matrix $H^{-1}$ of the fundamental weights, with $H$ the Cartan matrix, we have $g^T=H^{-1}g^{-1}H$, so $(T_j)$ is the conjugate by $H$ of $(g_{12}^{-1},\dots,g_1^{-1})$, and each $T_j$ preserves $H$. The tuple $(g_{12}^{-1},\dots,g_1^{-1})$ is the monodromy of the same local system with respect to the opposite orientation of $\PP^1$ (the standard generators read backwards). Reversing the orientation identifies the two parabolic cohomology groups and changes the sign of the intersection pairing; for a $6\times6$ Hermitian matrix this changes neither the determinant nor the signature $(3,3)$. So the computation below gives $\det h$ and the signature of $h$ for $\mathbf g$. \emph{The basis.} Put $Q_j=T_1\cdots T_{j-1}$ and let $E_j$ be the matrix of pivot columns of $1-T_j$ (a basis of $\operatorname{im}(1-T_j)$), of $8,4,1,\dots,1$ columns. A parabolic cocycle is recorded by vectors $a_j$ with $c_j=E_ja_j$, and the cocycle relation is $\sum_jQ_jE_ja_j=0$; the solutions form the right kernel $Z^1$ of the $8\times22$ matrix $(Q_1E_1\mid\dots\mid Q_{12}E_{12})$, of dimension $14$. We take for $Z^1$ the reduced row echelon basis of this kernel (in Sage, \texttt{right\_kernel\_matrix()}). The form of Lemma~\ref{lem:cocycle}, with $h_{\mathsf W}=H$ and times $\vartheta=i$, is Hermitian of rank $6$ on $Z^1$, its radical being the coboundaries, and the first six basis vectors span a complement of the radical. On them its Gram matrix is \[ h_0=\begin{pmatrix} -2&1+i&0&-1-i&2+i&-1+i\\ 1-i&2&2i&0&-1-3i&2\\ 0&-2i&0&i&0&-1-i\\ -1+i&0&-i&0&i&0\\ 2-i&-1+3i&0&-i&-1&\tfrac32+\tfrac12i\\ -1-i&2&-1+i&0&\tfrac32-\tfrac12i&0 \end{pmatrix}, \] of determinant $-3$ and signature $(3,3)$. The class of $-3$ in $\Q^\times/\Nm K^\times$ is $[-3]$. The script is \texttt{independent-det/independent\_det.py} in the supplementary files; it asserts each step above (that each $T_j$ preserves $H$, that the product is $1$, that $\dim Z^1=14$, that the form is Hermitian of rank $6$) and prints this block. The six basis vectors are printed in Appendix~\ref{app:gram}. A second program, written separately, computes $\det h=-6$ for the same tuple in another basis; $(-6)/(-3)=2=\Nm(1+i)$, so the classes agree. \end{proof} The discriminant is computed on the component $S$ of $C_0$, where the rank is certified, so no comparison between components is needed. We do not claim that the class datum has a single Hurwitz component. \subsection{The supplier identity} \begin{proposition}[The integer witness]\label{prop:witness} In $X^*(T)=\Z^{45}$, \[ e_\sigma=\textstyle\sum_{i=1}^{95}n_i\,a(\chi_i)+\sum_{l=1}^{3}a_{H'_l}(\bn_l) \] with integers $n_i$, where the $(H'_i,\chi_i)$ are Schoen-admissible suppliers and the $(H'_l,\bn_l)$ are balanced product suppliers with $H'_l=\langle i\rangle C_W(w_l)$, for $w_1$ of order $3$ with $|C_W(w_1)|=311\,040$, $w_2$ of order $12$ with $|C_W(w_2)|=288$, and $w_3$ of order $6$ with $|C_W(w_3)|=864$. Of the Schoen suppliers, $94$ come from a search pool described in Appendix~\ref{app:witness}, and one comes from $H'=\langle i\rangle N_W(P_3)$, $P_3$ a Sylow $3$-subgroup of $W$, $|H'|=7776$. \end{proposition} \begin{proof} This is the certificate of Appendix~\ref{app:witness}: an integer matrix with $45$ columns whose rows are characters of explicit suppliers, and an integer coefficient vector whose product with it is $e_\sigma$. Schoen-admissibility of each Schoen row is checked from its exponent counts (positive even $h$, simplicity of the exponent tuple in the sense of Subsection~\ref{subsec:schoen}, order $m\ge3$), and balance of each product row at $t=1,5,7,11$. \end{proof} \begin{proposition}[Slot data]\label{prop:slots} The three product suppliers of Proposition~\ref{prop:witness} are product-admissible (Definition~\ref{def:galeadm}): the curves $C/H'_l$ have genus $3259$, $5\,341\,793$ and $1\,730\,347$; every slot character has order $4$ or $12$; every slot has at least $20$ branch points; and $e'_k\in\{4,12\}$. This holds for every integral lift of the given rows. \end{proposition} \begin{proof} The rows of the witness are characters $a_{H'_l}(\bn)$; the integral lift $\bn$ is not unique, and different lifts give different slots. The certificate of Appendix~\ref{app:witness}, Table~\ref{tab:slots}, computes, for every linear character $\chi$ of $H'_l$ of order at least $3$ with $\dim V_\chi>0$ (so for every character that can occur in any integral lift), the branch data of $C/\ker\chi\to C/H'_l$ from the double cosets $H'_l\backslash G/\langle g_j\rangle$, and checks $s\ge2$ and $e'\mid12$. It also checks that the imbalance $I_\chi(t)=p_{\chi^t}-q_{\chi^t}$ satisfies $I_\chi(t)=\varepsilon(t)I_\chi(1)$ with $\varepsilon=(+,+,-,-)$ at $t=1,5,7,11$. Hence $\delta_t(\bn)=\varepsilon(t)\delta_1(\bn)$ for every lift, and $\delta_1(\bn)=\delta_T(a_{H'_l}(\bn))$ depends only on the row; the certificate checks that it vanishes for each row. Since every character that occurs has order dividing $12$, the units $t$ modulo $12$ cover every simultaneous Galois conjugate. \end{proof} \begin{example}[A product supplier]\label{ex:gale3231} The smallest of the three is the row for $w_1$, of order $3$, with $|C_W(w_1)|=311\,040$ and $H'=\langle i\rangle C_W(w_1)$ of order $622\,080$. The quotient $X=C/H'$ has genus $g_X=3259$. One integral lift has two slots, both with negative coefficient, so the slot characters are the inverses $\psi_1=\bar\chi_2$ and $\psi_2=\bar\chi_7$ of two linear characters of $H'$ (numbered as in the certificate): \begin{center} \begin{tabular}{cccccc} slot & order $e$ & branch pts.\ $s$ & $h=2g_X-2+s$ & $d$ & $2p_t-h$ at $t=1,5,7,11$\\\hline $\psi_1$ & $4$ & $6732$ & $13\,248$ & $1$ & $(-4,-4,4,4)$\\ $\psi_2$ & $12$ & $20$ & $6536$ & $1$ & $(4,4,-4,-4)$ \end{tabular} \end{center} The imbalances are given in the orientation convention of the certificate, $p_t=g_X-1+\sum_b\{t\alpha_b/e\}$ with $\{\cdot\}$ the fractional part; this is the opposite of the convention of Section~\ref{sec:AB}, which negates all of them. Neither slot is of Hodge type, and the exponent tuples of the cyclic covers are not simple (Subsection~\ref{subsec:schoen}); their product is of Hodge type at every $t$. In the orientation convention of the certificate the line $\det H^1(D)_{\psi_1}$ has Hodge type $(6622,6626)$ and $\det H^1(D)_{\psi_2}$ type $(3270,3266)$; reversing the convention swaps each pair. The line $\ell_\bn$ has degree $19\,784$ and type $(9892,9892)$. The proof of Theorem~\ref{thm:C} turns this into a Fermat statement. The first slot gives the Fermat hypersurface of degree $4$ and dimension $s-2=6730$, the second that of degree $12$ and dimension $18$. By Lemma~\ref{lem:special}(c) the two Fermat lines have types $(3363,3367)$ and $(11,7)$, and their product, on the product of dimension $6748=2\nu$, has type $(3374,3374)$: a Hodge class. Pulled back along $[Y_j]\mapsto[Y_j^3]$ from $X^{6730}_{12}$ on the first factor, it is a Hodge class on $X^{6730}_{12}\times X^{18}_{12}$, algebraic by Shioda's theorem. The cone over a representing cycle in the product of the two rank-$3259$ vector bundles has dimension $3374+2\cdot3259=9892$, half of $\htot=19\,784$, and Lemma~\ref{lem:cone} and the Abel--Prym correspondence give the algebraicity of $W_\bn$ on $J(D)^2$. No integral lift of this row uses characters from a single cyclic group, so the statement is about a non-cyclic abelian cover. \end{example} \begin{example}[A Schoen supplier that needs Lemma~\ref{lem:schoenR}]\label{ex:schoen3215} The Schoen supplier from $H'=\langle i\rangle N_W(P_3)$, $|H'|=7776$, has a character $\chi$ of order $m=4$. The quotient $C/H'$ has genus $q=384\,479$, the cyclic cover $C/\ker\chi\to C/H'$ has $s=88$ branch points, all with exponent $2=m/2$, and $h=769\,044$. The exponent tuple is simple ($2\equiv-2$), and $h$ is positive and even, so the supplier is Schoen-admissible. Since $4\mid m$ and every exponent is $m/2$, Schoen's Sublemma~2.9 finds no prime $p\mid m$ ramified at the branch points, and Lemma~\ref{lem:schoenR} is used. This supplier removes an index-$3$ obstruction in the lattice generated by the search pool. \end{example} \subsection{Proof of Theorem~A} By Lemma~\ref{lem:sixfold}, $A_\sigma(C)$ is a polarized abelian sixfold of Weil type for $K=\Q(i)$ for every $C\in\Hur$. By Propositions~\ref{prop:witness} and \ref{prop:slots}, condition (L) of Definition~\ref{def:lattice} holds, so by Corollary~\ref{cor:everymember} the Weil classes of $A_\sigma(C)$ are algebraic for every $C\in\Hur$; here Corollary~\ref{cor:galesupply}, that is Theorem~C, supplies the three product suppliers at every $C$, and Lemma~\ref{lem:schoen}, with Lemma~\ref{lem:schoenR}, the $95$ Schoen suppliers. By Proposition~\ref{prop:dominance}, $\rk\mu_{C_0}=9=n^2$ and $\fd(S)=[-3]$. Theorem~\ref{thm:transfer} gives the Weil classes of every polarized abelian sixfold of Weil type for $\Q(i)$ with discriminant $[-3]$. \qed %% ------------------------------------------------------------------ \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 $\Lambda=\bigwedge(\mathsf W\otimes\mathsf U_1)\otimes\bigwedge(\mathsf W^*\otimes\mathsf U_2)$, 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}[{\cite[Lemma~11.1]{SevernQ3}}]\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 of invariant theory 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 (see also \cite[Ch.~II]{Weyl}). We use it in multilinear form. An $SL(\mathsf W)$-invariant element of $\mathsf W^{\otimes p}\otimes\mathsf W^{*\otimes q}$ is an invariant multilinear function on $\mathsf W^{*p}\times\mathsf W^q$, that 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 homogeneous of degree one in each variable is a linear combination of products of scalar products and determinants 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 three kinds of elementary invariants, each occupying some of the tensor factors: the contraction $\sum_iw_i\otimes w_i^*$ in one factor $\mathsf W$ and one factor $\mathsf W^*$; the determinant tensor $\sum_{\pi\in S_N}\operatorname{sgn}(\pi)\,w_{\pi(1)}\otimes\dots\otimes w_{\pi(N)}$ in $N$ factors $\mathsf W$; and the dual determinant tensor in $N$ factors $\mathsf W^*$. The homogeneous piece $\bigwedge^p(\mathsf W\otimes\mathsf U_1)\otimes\bigwedge^q(\mathsf W^*\otimes\mathsf U_2)$ of $\Lambda$ 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 $SL(\mathsf W)$-equivariant and surjective. Since $SL(\mathsf W)$ is reductive, the invariants of the image are the image of the invariants, so they are spanned by the images of $I\otimes u_1\otimes\dots\otimes u_p\otimes u'_1\otimes\dots\otimes u'_q$ with $I$ a product of elementary 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 $\sum_\pi\operatorname{sgn}(\pi)(w_{\pi(1)}\otimes u_{k_1})\wedge\dots\wedge(w_{\pi(N)}\otimes u_{k_N})$, which is the full polarization of the homogeneous polynomial map $x\mapsto N!\det(\mathsf W\otimes x)$ of degree $N$, evaluated at $(u_{k_1},\dots,u_{k_N})$; by the polarization identity it is a linear combination of values $\det(\mathsf W\otimes x)$. The same holds for the dual determinant tensors. So every invariant is a combination of products of the stated generators. \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 $\theta(u,u')$ are Hodge classes of degree two, hence combinations of divisor classes. Let $\mathcal O=\{y\in K:\iota(y)\in\End(A)\}$, an order of $K$. For $\mathbf y=(y_1,\dots,y_m)\in\mathcal O^m$ the homomorphism $f_{\mathbf y}\colon A^m\to A$, $(a_k)\mapsto\sum_k\iota(y_k)a_k$, pulls the Weil classes of $A$ back to $\det(\mathsf W\otimes\tilde{\mathbf y})$ and $\det(\mathsf W^*\otimes\overline{\tilde{\mathbf y}})$, with $\tilde{\mathbf y}\in\C^m=\mathsf U_1$ the image of $\mathbf y$ under the given embedding of $K$; these are algebraic. The set of all $\tilde{\mathbf y}$, $\mathbf y\in\mathcal O^m$, contains $\Z^m$, which is Zariski dense in $\C^m$: a polynomial vanishing on $\Z^m$ is zero, by induction on $m$. The map $x\mapsto\det(\mathsf W\otimes x)$ is polynomial, so a linear form vanishing on its values at a Zariski dense set vanishes on all its values. Hence these classes span all $\det(\mathsf W\otimes x)$, $x\in\mathsf U_1$, and likewise for $\mathsf W^*$. By Lemma~\ref{lem:sl}, products of algebraic classes span all Hodge classes over $\C$, and rational descent as in Corollary~\ref{cor:everymember} finishes the proof. \end{proof} \begin{proof}[Proof of Theorem~B] Theorem~A and Theorem~\ref{thm:powers} with $n=3$. \end{proof} %% ------------------------------------------------------------------ \appendix \section{The monodromy tuple}\label{app:tuple} %% ------------------------------------------------------------------ Matrices act on column vectors of coordinates with respect to the fundamental weights $\omega_1,\dots,\omega_8$ of $E_8$ in Bourbaki's numbering (the simple roots $\alpha_1,\alpha_3,\alpha_4,\dots,\alpha_8$ form a chain and $\alpha_2$ is joined to $\alpha_4$); column $j$ of a matrix is the image of $\omega_j$. Since the $E_8$ lattice is unimodular, this is a $\Z$-basis of it, and $W$ preserves the Gram matrix of the fundamental weights, the inverse of the Cartan matrix. For example the simple reflection $s_{\alpha_1}$ maps $\omega_1$ to $\omega_1-\alpha_1=-\omega_1+\omega_3$ and fixes the other $\omega_j$; it is the matrix $c$ below. Let \[ a=\left(\begin{smallmatrix}0&0&1&0&0&0&0&0\\1&1&1&1&1&1&0&0\\1&1&1&2&2&2&2&1\\-1&-1&-1&-1&-1&-1&-1&0\\-1&-1&-2&-3&-3&-2&-1&-1\\0&0&0&0&1&0&0&0\\1&1&2&3&2&2&1&0\\1&2&2&3&2&1&1&1\end{smallmatrix}\right),\qquad b=\left(\begin{smallmatrix}-1&-2&-3&-5&-4&-3&-2&-1\\2&3&3&5&4&3&2&1\\2&2&4&5&4&3&2&1\\-2&-2&-3&-4&-4&-3&-2&-1\\0&0&0&0&1&0&0&0\\0&0&0&0&0&1&0&0\\0&0&0&0&0&0&1&0\\0&0&0&0&0&0&0&1\end{smallmatrix}\right), \] \[ c=\left(\begin{smallmatrix}-1&0&0&0&0&0&0&0\\0&1&0&0&0&0&0&0\\1&0&1&0&0&0&0&0\\0&0&0&1&0&0&0&0\\0&0&0&0&1&0&0&0\\0&0&0&0&0&1&0&0\\0&0&0&0&0&0&1&0\\0&0&0&0&0&0&0&1\end{smallmatrix}\right), \] and $d=(abc)^{-1}$. Then $a$ has characteristic polynomial $\Phi_{20}$ and centralizer of order $20$ in $W$, and $a^5$ has characteristic polynomial $\Phi_4^4$ and centralizer of order $46\,080$; $b$ and $c$ are reflections; $d$ has characteristic polynomial $\Phi_1^2\Phi_3\Phi_{12}$; and $a,b,c$ generate $W$. The tuple is \[ \begin{aligned} \mathbf g=\big(&b,\ c,\ aba^{-1},\ aca^{-1},\ a^2ba^{-2},\ a^2ca^{-2},\\ &a^3ba^{-3},\ a^3ca^{-3},\ a^4ba^{-4},\ a^4ca^{-4},\ ia^5,\ -ia^{-1}d^5a\big). \end{aligned} \] Its product is $1$: the first ten entries multiply to $(bca)^5a^{-5}$, and $bca=a^{-1}d^{-1}a$ because $abcd=1$, so the product is $a^{-1}d^{-5}a\cdot a^{-5}\cdot a^5\cdot a^{-1}d^5a=1$. So $g_{11}=ix_1$ with $x_1=a^5$, and $g_{12}=ix_2$ with $x_2=-a^{-1}d^5a$, of characteristic polynomial $\Phi_2^2\Phi_6\Phi_{12}$. The twelve matrices generate a group of order $1\,393\,459\,200$, that is all of $G$; this was checked in GAP \cite{GAP} from their permutation action on the $480$ vectors formed by the roots of $E_8$ and their multiples by $i$. The ten reflections alone generate a subgroup of order $51\,840$, so the two other entries are needed for generation. The matrices $a,b,c$ were found by a search in $W$ for $a$ with characteristic polynomial $\Phi_{20}$ and pairs of reflections $(b,c)$ with $d=(abc)^{-1}$ of characteristic polynomial $\Phi_1^2\Phi_3\Phi_{12}$ and $\langle a,b,c\rangle=W$. Replacing $d$ by a solution with characteristic polynomial $\Phi_2^2\Phi_6\Phi_{12}$ gives the conjugate class datum, exchanged with this one by the automorphism of $G$ that fixes $W$ and maps $i$ to $-i$. %% ------------------------------------------------------------------ \section{The rank certificate}\label{app:rank} %% ------------------------------------------------------------------ \subsection*{The cover} Let $M$ be a Fuchsian system $dY=\big(\frac{B_0}{z}+\frac{B_1}{z-1}+\frac{B_c}{z-c_0}\big)Y\,dz$ of rank $8$ on $\PP^1_z$, with $c_0=2+i$ and residues $B_0,B_1,B_c\in M_8(\C)$, whose monodromy around $0,1,c_0,\infty$ is exactly $(a,b,c,d)$ up to simultaneous conjugation. Its pullback along $s\mapsto s^5=z$, tensored with the rank-one system $d\phi=\frac14\big(\frac{ds}{s}\big)\phi$ (monodromy $i$ at $s=0$ and $-i$ at $s=\infty$), has twelve singular points: $s=0$, the fifth roots of $1$, the fifth roots of $c_0$, and $s=\infty$, with monodromy tuple $\mathbf g$. Let $C_0$ be the $G$-cover of $\PP^1_s$ branched at these twelve points with monodromy tuple $\mathbf g$. We know of no rigid local system that pulls back to this datum, so a non-rigid system on four points, whose character variety has dimension $12$, is used instead. \subsection*{Existence of $M$} The system $M$ is found numerically, by continuation from a system with abelian monodromy, and then shown to exist exactly near the numerical solution. The residues at $1$ and $c_0$ have rank one; after conjugating so that some entries are fixed (this removes the action of simultaneous conjugation), they and an invertible matrix $Q$ that should conjugate the monodromy of $M$ to $(a,b,c)$ form $87$ unknowns, and $87$ polynomial-in-the-monodromy equations express this conjugacy: two trace conditions making the residues at $1$ and $c_0$ of trace $\tfrac12$, so that their monodromies are reflections; seven rows of the eigenvector equation for each of the eight eigenvectors of $a$ ($56$ equations); $14$ equations for each of $b$ and $c$ matching the reflecting vectors; and $\det Q=1$. Here $B_0=\operatorname{diag}(\beta_1,\dots,\beta_8)$ with $\beta_k=m_k/20$, $m_k\in\{1,3,7,9,11,13,17,19\}$, and the residues at $1$ and $c_0$ are $u_1v_1^T$ and $u_2v_2^T$, with $u_1$ and one entry of $u_2$ fixed. Each eigenvector equation drops one row, and each reflection equation is imposed on all but one row and on one pivot row of $M-1$. A zero of this system gives the exact conjugacy $M_kQ=Qt_k$ at all three finite poles provided the following side conditions hold: for each eigenvalue $\lambda_j$ of $a$, the seven kept rows of $M_0-\lambda_j$ have rank seven; for the poles $1$ and $c_0$, the seven kept rows of $M_k+1$ have rank seven and the chosen pivot row of $M_k-1$ is nonzero. The certificate checks these minors and entries in balls over the whole ball of radius $\rho$ below, and none of the enclosures contains $0$. The monodromy is computed by Taylor expansion of solutions in ball arithmetic \cite{Arb} at $256$ bits, with a rigorous bound on the tail of each series, along $115$ steps of a path whose endpoints match exactly. Let $F$ be the map of the $87$ equations, $\tilde x$ a refined numerical zero with $|F(\tilde x)|$ of order $10^{-37}$, and $Y_0$ an approximate inverse of the Jacobian at $\tilde x$. The map $w\mapsto w-Y_0F(w)$ sends the ball of radius $\rho$ about $\tilde x$ into itself and is a contraction there, by the inequalities $\eta+\kappa\rho\le\rho$ and $\kappa<1$, where $\eta$ bounds $\|Y_0F(\tilde x)\|$ and $\kappa$ bounds the deviation of $Y_0$ times the Jacobian from the identity on the ball (with finite-difference and Cauchy-estimate errors included, on a polydisc of radius $2^{-40}$). With every bound rounded outward: \[ \eta\approx1.05\cdot10^{-35},\qquad\kappa\approx0.0056,\qquad\rho\approx1.05\cdot10^{-34}. \] So $F$ has a unique zero in this ball, and $M$ exists exactly, with residues within $\rho$ of $\tilde x$. Gershgorin discs then pin the exponents at $\infty$ to exact rational values. The residues of $M$, to $256$ bits, are in the supplementary files. \subsection*{Differentials from the system} Let $L$ be the pulled-back system on $\PP^1_s$, after a change of the coordinate $s$ by a M\"obius transformation that makes $\infty$ a nonsingular point and a scalar gauge transformation by a power of the coordinate (neither changes the cover), and let $Y$ be a fundamental solution. At a pole $p$ with residue $A_p=P_p\Lambda_pP_p^{-1}$, $\Lambda_p=\operatorname{diag}(\lambda_{p,1},\dots,\lambda_{p,8})$, the local monodromy has finite order, hence is semisimple, and by Levelt's normal form \cite{Levelt} there is a local solution $Y=P_pH_p(s-p)\,(s-p)^{\Lambda_p}$ with $H_p$ holomorphic and $H_p(0)=1$, also where eigenvalues of $A_p$ differ by integers (as at $s=0$, where the exponents are $(m_k+1)/4$). Where no two eigenvalues differ by a positive integer, $H_p$ is the unique solution of the Frobenius recursion. \begin{lemma}[Differentials from the system]\label{lem:dictionary} $\cD_\sigma(C_0)$ is identified with the space of $r\in\C(s)^8$ such that the vector of forms $Y^{-1}r\,ds$ is holomorphic on $C_0$, and $\cD_{\bar\sigma}(C_0)$ with the space of $q\in\C(s)^8$ such that $Y^Tq\,ds$ is holomorphic on $C_0$ (up to exchanging $\sigma$ and $\bar\sigma$, which does not affect $\mu_{C_0}$). Concretely the conditions are \[ \operatorname{ord}_p\big[(P_pH_p)^{-1}r\big]_j\ge\lfloor\lambda_{p,j}\rfloor,\qquad \operatorname{ord}_p\big[(P_pH_p)^{T}q\big]_j\ge\lfloor-\lambda_{p,j}\rfloor \] at each pole and each $j$, holomorphy elsewhere, and vanishing to order $2$ at $\infty$. Under these identifications $\mu_{C_0}(r\otimes q)=q^Tr\,ds^2$. \end{lemma} \begin{proof} Pulled back to $C_0$ minus the ramification points, $Y$ is single-valued, because its monodromy factors through $\sigma$ and the loops that lift to closed loops on $C_0$ map to $1$ in $G$; and $g^*Y=Y\sigma'(g)$ for $g\in G$, with $\sigma'\in\{\sigma,\bar\sigma\}$ depending on conventions. For a rational vector $r$, the vector $\omega=Y^{-1}r\,ds$ of meromorphic forms then transforms by $g^*\omega=\sigma'(g)^{-1}\omega$, so its components span the image of a $G$-map from $\sigma'^\vee$ to the meromorphic forms on $C_0$. Conversely a $G$-map from $\sigma'^\vee$ to $H^0(K_{C_0})$ is a vector $\omega$ of holomorphic forms with this transformation law, and $r=Y\omega/ds$ is $G$-invariant, hence descends; it is meromorphic at the poles because $Y$ and $Y^{-1}$ have moderate growth at a Fuchsian singularity, and at the regular point $\infty$, where $ds$ has a double pole, it vanishes to order at least two. So $r$ is rational. Since $\sigma$ is unitary, $\sigma'^\vee\cong\bar\sigma'$; the same argument with $Y^T$ gives the second space, and $q^TY\cdot Y^{-1}r=q^Tr$ is the invariant contraction. For the order condition: the local monodromy at $p$ has finite order $e$, so $e\lambda_{p,j}\in\Z$; in a local coordinate $y$ on $C_0$ with $s-p=y^e$, the $j$-th component of $Y^{-1}r\,ds$ 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 $y$, which is $\ge0$ exactly when $\operatorname{ord}_p[(P_pH_p)^{-1}r]_j\ge\lfloor\lambda_{p,j}\rfloor$. \end{proof} The following two lemmas turn these conditions into finite linear algebra. \begin{lemma}\label{lem:kernel} Put $M_p=\max(0,-\min_j\lfloor\lambda_{p,j}\rfloor)$ for $\cD_\sigma$, and the same with $-\lambda_{p,j}$ for $\cD_{\bar\sigma}$. \begin{enumerate} \item Every $r\in\cD_\sigma(C_0)$ is uniquely of the form $r=\sum_p\sum_{m=1}^{M_p}c_{p,m}(s-p)^{-m}$ with $c_{p,m}\in\C^8$, and likewise for $\cD_{\bar\sigma}(C_0)$. \item $\cD_\sigma(C_0)$ is the kernel of the linear map $\Phi_A$ from the space of these coefficients to the Laurent coefficients that the conditions require to vanish, and similarly for $\Phi_B$; both kernels have dimension $3$. \item If a square submatrix $\Phi_{R,F'}$ of size $n_A-3$, 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 $\ker\Phi_A$; if $\Phi_{R,F'}$ is certified invertible in ball arithmetic, the same formula computed in balls encloses this basis. \end{enumerate} \end{lemma} \begin{proof} (1) Away from the poles $Y$ is holomorphic and invertible, so $r$ is holomorphic there. At a pole, $P_pH_p$ is holomorphic and invertible, so $\operatorname{ord}_pr\ge\min_j\lfloor\lambda_{p,j}\rfloor\ge-M_p$. Since $r$ vanishes at $\infty$, it is the sum of its principal parts, and principal parts at distinct points are independent. (2) Each condition is the vanishing of finitely many Laurent coefficients, which are linear in the unknowns, and involve the Taylor coefficients of $H_p$ only to bounded order. By (1) the kernel maps isomorphically onto $\cD_\sigma(C_0)$, which has dimension $3$ by Proposition~\ref{prop:mult}. (3) The $x_f$ lie in the kernel $\ker\Phi_R$ of the rows $R$ and are independent. Since $\Phi_{R,F'}$ is invertible, the rows $R$ have rank $n_A-3$, so $\ker\Phi_R$ has dimension $3$ and is spanned by the $x_f$; it contains $\ker\Phi_A$, of dimension $3$, so they are equal. Ball inversion encloses the inverse of every matrix in the ball. \end{proof} \begin{lemma}\label{lem:residues} Let $B\subset\C$ be a set of $12$ points. The map sending $g(s)\,ds^2\in H^0(\PP^1,K_{\PP^1}^{\otimes2}(B))$ to its residues $(\operatorname{Res}_pg)_{p\in B}\in\C^{12}$ is injective, and its image is the $9$-dimensional subspace cut out by $\sum_p\rho_pp^k=0$ for $k=0,1,2$. \end{lemma} \begin{proof} The function $g$ has at most simple poles on $B$ and is otherwise holomorphic on $\C$; at $\infty$, with $s=1/t$, $ds^2=t^{-4}dt^2$, so $g=O(s^{-4})$. Hence $g=\sum_p\rho_p/(s-p)$, which proves injectivity, and expanding, $g=\sum_{k\ge0}(\sum_p\rho_pp^k)s^{-k-1}$, so $g=O(s^{-4})$ is exactly the three relations. \end{proof} So $\mu_{C_0}$, in \emph{residue coordinates}, is the $9\times12$ matrix of residues of $q_b^Tr_a\,ds^2$ at the twelve poles, for bases $r_a$ of $\cD_\sigma(C_0)$ and $q_b$ of $\cD_{\bar\sigma}(C_0)$, and its rank is that of $\mu_{C_0}$. \subsection*{The rank} The Frobenius expansions are computed to order $12$, with exact rational exponents. Here $\Phi_A$ is $45\times48$ and $\Phi_B$ is $117\times120$, and square pivot blocks of sizes $45$ and $117$ are certified invertible by bounds $\|I-RS\|_\infty<1$ for an approximate inverse $R$. The residue relations of Lemma~\ref{lem:residues} hold within the balls, which is a consistency check. The $9\times9$ minor of the residue matrix on columns $4$ to $12$ is \[ 0.105574412595065626-0.044099913684798359\,i \] with coordinate radii below $7\cdot10^{-18}$. It is not zero, so $\rk\mu_{C_0}=9$. For the conjugate class datum the same procedure gives a minor of modulus greater than $5.4\cdot10^{-12}$, with radii below $6\cdot10^{-27}$. %% ------------------------------------------------------------------ \section{The supplier certificates}\label{app:witness} %% ------------------------------------------------------------------ \subsection*{The character group} $G$ is built in GAP 4.14 \cite{GAP} with its character table library, from the simple reflections of $W$ in the basis of Appendix~\ref{app:tuple} and the scalar $i$. Its $45$ pairs of non-real irreducible characters with $m_\tau>0$ give $X^*(T)=\Z^{45}$, and $e_\sigma$ is the second coordinate vector. A supplier $(H',\chi)$ contributes the row $a(\chi)$, computed by Frobenius reciprocity from the restrictions $\tau|_{H'}$; the program checks each row's $h$ against the Chevalley--Weil count $h=\sum_\tau m_\tau\langle\tau|_{H'},\chi\rangle$ and against $2q-2+s$. \subsection*{The witness} The witness consists of an integer matrix with $3296$ rows and $45$ columns, an integer coefficient vector with $98$ nonzero entries, and the target $e_\sigma$. The rows are: $3214$ Schoen-admissible suppliers from a search pool; $16$ Schoen-admissible suppliers of order $4$ on $\langle i\rangle N_W(P_3)$; and $66$ balanced product rows on the three groups $\langle i\rangle C_W(w_l)$. The $98$ nonzero coefficients sit on $94$ pool rows, on one row from $\langle i\rangle N_W(P_3)$ (row $3215$, Example~\ref{ex:schoen3215}), and on one product row for each group (rows $3231$, $3269$ and $3244$ for $w_1$, $w_2$ and $w_3$; Example~\ref{ex:gale3231} is row $3231$). Table~\ref{tab:coef} lists them with their signs and their lengths in decimal digits; the integers themselves fill about $72\,000$ digits and are in the supplementary files. Checking the identity takes seconds in GAP. \begin{table}[ht] \caption{The nonzero coefficients of the witness: row index, sign, number of decimal digits. Rows $1$ to $1125$ are pool rows; the last four are the supplier on $\langle i\rangle N_W(P_3)$ and the product rows for $w_1$, $w_3$, $w_2$.}\label{tab:coef} \small \begin{tabular}{rcr|rcr|rcr|rcr|rcr} row&&dig.&row&&dig.&row&&dig.&row&&dig.&row&&dig.\\\hline 1&$+$&753&21&$+$&752&41&$+$&753&226&$+$&753&459&$+$&753\\ 2&$-$&753&22&$+$&753&42&$-$&752&236&$+$&751&463&$-$&753\\ 3&$+$&753&23&$+$&753&43&$+$&752&243&$-$&753&467&$+$&752\\ 4&$+$&753&24&$+$&753&44&$-$&753&246&$-$&1&469&$+$&753\\ 5&$-$&752&25&$+$&753&45&$+$&753&290&$+$&752&471&$-$&753\\ 6&$-$&753&26&$-$&753&46&$-$&752&313&$+$&753&473&$+$&751\\ 7&$-$&753&27&$-$&753&76&$-$&752&314&$+$&753&477&$-$&753\\ 8&$+$&753&28&$+$&753&78&$+$&753&330&$-$&753&478&$-$&752\\ 9&$-$&753&29&$-$&753&79&$+$&753&332&$-$&753&492&$+$&753\\ 10&$+$&752&30&$-$&753&88&$+$&753&338&$+$&753&493&$-$&753\\ 11&$+$&753&31&$+$&753&105&$+$&753&345&$+$&752&536&$-$&752\\ 12&$-$&753&32&$-$&753&116&$-$&753&364&$-$&753&574&$+$&752\\ 13&$-$&753&33&$-$&753&126&$+$&753&366&$+$&752&857&$+$&753\\ 14&$+$&752&34&$-$&753&129&$-$&753&403&$-$&753&1125&$+$&752\\ 15&$-$&753&35&$+$&752&156&$+$&752&404&$+$&753&3215&$-$&752\\ 16&$-$&753&36&$+$&753&195&$-$&753&425&$-$&753&3231&$-$&753\\ 17&$+$&752&37&$+$&752&208&$+$&752&426&$-$&753&3244&$+$&752\\ 18&$+$&752&38&$-$&1&217&$+$&752&434&$-$&752&3269&$-$&753\\ 19&$+$&753&39&$-$&753&221&$-$&753&437&$-$&753&&&\\ 20&$-$&753&40&$+$&753&224&$+$&752&457&$-$&753&&& \end{tabular} \end{table} The search pool consists of the Schoen-admissible suppliers on: cyclic subgroups $\langle iw\rangle$ and subgroups $\langle i,w\rangle$ for all $112$ classes of $W$; subgroups of the normalizers of the cyclic groups generated by $g_{12}$ and its square; the groups $\langle i\rangle C_W(w)$ and $\langle i\rangle N_W(\langle w\rangle)$ for all classes; subgroups of $C_G(g_{12}^3)$ meeting the classes of order $12$; overgroups of a fixed $3$-subgroup inside $3$-local subgroups; subgroups of the normalizers of $\langle y\rangle$ and $\langle y^2\rangle$ for $y=iw$ with $w$ in two classes of elements of $W$; and the groups $\langle i\rangle W_S\langle-1\rangle$ and $\langle i\rangle N_W(W_S)$ for reflection subgroups $W_S$ of rank $7$ and $8$. The pool spans a sublattice of rank $42$ with an elementary divisor $3$, and the functional of Appendix~\ref{app:schoenfails} vanishes on it. The $16$ suppliers on $\langle i\rangle N_W(P_3)$ remove the elementary divisor, and the three product rows raise the rank to $44$; the target lies in the integer span. \subsection*{The suppliers} Table~\ref{tab:suppliers} identifies the supplier $(H',\chi)$ of each of the $94$ pool rows with a nonzero coefficient and of row $3215$, with its branch data $m$, $q$, $s$ and $h=2q-2+s$. Except in the three rows discussed below, each identification is an exact match of the row of the witness with the vector $a(\chi)$ computed from $(H',\chi)$. \begin{itemize} \item \emph{Notation.} $\mathcal C_c$ is the $c$-th conjugacy class of $W$ in the ordering of the character table of $W(E_8)$ in the GAP character table library \cite{GAP}, and $\zeta_n=e^{2\pi i/n}$. Recall that $g_{12}$ is the last entry of the tuple $\mathbf g$ (Appendix~\ref{app:tuple}). In two rows, $y=iw$ with $w\in\mathcal C_{53}$. \item \emph{Abelian rows} ($18$ rows). Here $H'=\langle i,w\rangle$ or $H'=\langle iw\rangle$, and $\chi$ is given by its values on the generators. \item \emph{Rows $\le N_G(\langle g_{12}\rangle)$, $N_G(\langle g_{12}^2\rangle)$, $C_G(g_{12}^3)$ or $N_G(\langle y^2\rangle)$.} $H'$ is a subgroup of the named group, given by its isomorphism type. The $\chi$ column gives the order of $\chi$, then $\chi(i)$ (or $i\notin H'$), then $|\ker\chi|$. \item \emph{Rows $\langle i\rangle\cdot H_0$.} Here $H'=\langle i\rangle H_0$ with $H_0\le W$ as named and $|H'|=2|H_0|$, and $\chi$ is given by $\chi(i)=-i$ and $\chi_0=\chi|_{H_0}$, with $\chi_0(-1)=-1$, its order and $|\ker\chi_0|$. In the rows $W_S\langle-1\rangle$, $S$ is a set of nodes of the extended Dynkin diagram of $E_8$, node $9$ being the negative of the highest root. \end{itemize} In the three rows $492$, $493$ and $857$, whose groups $H_0$ have orders $161\,280$ and $5\,806\,080$, the character $\chi_0$ was not singled out among the linear characters of $H_0$. For these rows the vector is the one recorded with the pool, together with its $q$, $s$ and $h$. Row $3215$, which is not a pool row, is the single Schoen supplier $(\langle i\rangle N_W(P_3),\chi)$ of Proposition~\ref{prop:witness} and Example~\ref{ex:schoen3215}. Here $\chi=\chi_0\cdot(i\mapsto i)$, with $\chi_0$ the linear character of order $2$ of $N_W(P_3)$ with $\chi_0(-1)=-1$; it is listed in the last line of Table~\ref{tab:suppliers}. The three remaining nonzero rows ($3231$, $3244$, $3269$) are not single suppliers. Each is a $\delta$-balanced integer combination of the linear characters of $\langle i\rangle C_W(w_l)$, the product supplier of Proposition~\ref{prop:slots}. {\scriptsize \setlength{\tabcolsep}{1.5pt} \begin{longtable}{rcp{0.26\textwidth}rp{0.14\textwidth}rrrr} \caption{The suppliers of the $94$ nonzero pool rows and of row $3215$: row, sign of the coefficient, $H'$, $|H'|$, $\chi$, $m$, $q$, $s$, $h$.}\label{tab:suppliers}\\ row & & $H'$ & $|H'|$ & $\chi$ & $m$ & $q$ & $s$ & $h$\\\hline \endfirsthead row & & $H'$ & $|H'|$ & $\chi$ & $m$ & $q$ & $s$ & $h$\\\hline \endhead 1 & $+$ & $\le N_G(\langle g_{12}\rangle)$, $C_{4}$ & 4 & $4$; $-i$; $1$ & 4 & 769\,305\,601 & 0 & 1\,538\,611\,200\\ 2 & $-$ & $\le N_G(\langle g_{12}\rangle)$, $C_{4}\times C_{2}$ & 8 & $4$; $-i$; $2$ & 4 & 381\,024\,001 & 14\,515\,200 & 776\,563\,200\\ 3 & $+$ & $\langle i,w\rangle$, $w\in\mathcal C_{4}$ & 8 & $\chi(i)=-i$, $\chi(w)=1$ & 4 & 384\,652\,801 & 0 & 769\,305\,600\\ 4 & $+$ & $\le N_G(\langle g_{12}\rangle)$, $C_{12}$ & 12 & $4$; $-i$; $3$ & 4 & 256\,435\,201 & 0 & 512\,870\,400\\ 5 & $-$ & $\le N_G(\langle g_{12}\rangle)$, $C_{4}\times S_{3}$ & 24 & $4$; $-i$; $6$ & 4 & 124\,588\,801 & 14\,515\,200 & 263\,692\,800\\ 6 & $-$ & $\le N_G(\langle g_{12}\rangle)$, $C_{4}$ & 4 & $4$; $i\notin H'$; $1$ & 4 & 769\,305\,601 & 0 & 1\,538\,611\,200\\ 7 & $-$ & $\le N_G(\langle g_{12}\rangle)$, $C_{4}\times C_{2}$ & 8 & $4$; $-i$; $2$ & 4 & 384\,652\,801 & 0 & 769\,305\,600\\ 8 & $+$ & $\langle i,w\rangle$, $w\in\mathcal C_{70}$ & 16 & $\chi(i)=-i$, $\chi(w)=1$ & 4 & 192\,326\,401 & 0 & 384\,652\,800\\ 9 & $-$ & $\le N_G(\langle g_{12}^2\rangle)$, $C_{12}\times C_{2}$ & 24 & $4$; $-i$; $6$ & 4 & 127\,008\,001 & 4\,838\,400 & 258\,854\,400\\ 10 & $+$ & $\le N_G(\langle g_{12}\rangle)$, $C_{4}\times C_{2}$ & 8 & $4$; $-i$; $2$ & 4 & 384\,648\,193 & 9\,216 & 769\,305\,600\\ 11 & $+$ & $\le N_G(\langle g_{12}\rangle)$, $C_{12}$ & 12 & $12$; $-i$; $1$ & 12 & 256\,435\,201 & 0 & 512\,870\,400\\ 12 & $-$ & $\le N_G(\langle g_{12}^2\rangle)$, $C_{12}$ & 12 & $4$; $-i$; $3$ & 4 & 256\,435\,201 & 0 & 512\,870\,400\\ 13 & $-$ & $\langle i,w\rangle$, $w\in\mathcal C_{19}$ & 16 & $\chi(i)=-i$, $\chi(w)=1$ & 4 & 192\,326\,401 & 0 & 384\,652\,800\\ 14 & $+$ & $\langle i,w\rangle$, $w\in\mathcal C_{30}$ & 24 & $\chi(i)=-i$, $\chi(w)=1$ & 4 & 128\,217\,601 & 0 & 256\,435\,200\\ 15 & $-$ & $\le N_G(\langle g_{12}\rangle)$, $C_{12}\times C_{2}$ & 24 & $4$; $-i$; $6$ & 4 & 128\,216\,065 & 3\,072 & 256\,435\,200\\ 16 & $-$ & $\langle iw\rangle$, $w\in\mathcal C_{23}$ & 20 & $\chi(iw)=\zeta_{20}^{15}$ & 4 & 153\,861\,121 & 0 & 307\,722\,240\\ 17 & $+$ & $\le N_G(\langle g_{12}^2\rangle)$, $C_{12}\times C_{3}$ & 36 & $4$; $-i$; $9$ & 4 & 85\,478\,401 & 0 & 170\,956\,800\\ 18 & $+$ & $\langle iw\rangle$, $w\in\mathcal C_{4}$ & 4 & $\chi(iw)=\zeta_{4}^{3}$ & 4 & 769\,305\,601 & 0 & 1\,538\,611\,200\\ 19 & $+$ & $\le N_G(\langle g_{12}\rangle)$, $C_{4}\times S_{3}$ & 24 & $4$; $-i$; $6$ & 4 & 128\,217\,601 & 0 & 256\,435\,200\\ 20 & $-$ & $\le N_G(\langle g_{12}\rangle)$, $C_{4}\times C_{2}\times C_{2}$ & 16 & $4$; $-i$; $4$ & 4 & 190\,509\,697 & 7\,262\,208 & 388\,281\,600\\ 21 & $+$ & $\le N_G(\langle g_{12}\rangle)$, $C_{2}\times C_{4}\times S_{3}$ & 48 & $4$; $-i$; $12$ & 4 & 62\,293\,633 & 7\,259\,136 & 131\,846\,400\\ 22 & $+$ & $\le N_G(\langle g_{12}^2\rangle)$, $C_{12}\times S_{3}$ & 72 & $4$; $-i$; $18$ & 4 & 41\,529\,601 & 4\,838\,400 & 87\,897\,600\\ 23 & $+$ & $\le N_G(\langle g_{12}\rangle)$, $C_{4}\times S_{3}$ & 24 & $4$; $i\notin H'$; $6$ & 4 & 128\,217\,601 & 0 & 256\,435\,200\\ 24 & $+$ & $\le N_G(\langle g_{12}\rangle)$, $C_{4}\times C_{2}$ & 8 & $4$; $i\notin H'$; $2$ & 4 & 384\,648\,193 & 9\,216 & 769\,305\,600\\ 25 & $+$ & $\le N_G(\langle g_{12}^2\rangle)$, $C_{12}$ & 12 & $4$; $i\notin H'$; $3$ & 4 & 256\,435\,201 & 0 & 512\,870\,400\\ 26 & $-$ & $\langle iw\rangle$, $w\in\mathcal C_{70}$ & 4 & $\chi(iw)=\zeta_{4}^{3}$ & 4 & 769\,305\,601 & 0 & 1\,538\,611\,200\\ 27 & $-$ & $\le N_G(\langle g_{12}\rangle)$, $C_{4}\times C_{2}$ & 8 & $4$; $-i$; $2$ & 4 & 384\,652\,801 & 0 & 769\,305\,600\\ 28 & $+$ & $\le N_G(\langle g_{12}\rangle)$, $C_{12}$ & 12 & $12$; $-i$; $1$ & 12 & 256\,434\,985 & 648 & 512\,870\,616\\ 29 & $-$ & $\langle i,w\rangle$, $w\in\mathcal C_{72}$ & 16 & $\chi(i)=-i$, $\chi(w)=1$ & 4 & 192\,326\,401 & 0 & 384\,652\,800\\ 30 & $-$ & $\le N_G(\langle g_{12}^2\rangle)$, $C_{12}\times C_{2}$ & 24 & $4$; $-i$; $6$ & 4 & 128\,217\,601 & 0 & 256\,435\,200\\ 31 & $+$ & $\le N_G(\langle g_{12}^2\rangle)$, $C_{4}\times S_{3}$ & 24 & $4$; $-i$; $6$ & 4 & 128\,217\,601 & 0 & 256\,435\,200\\ 32 & $-$ & $\le N_G(\langle g_{12}^2\rangle)$, $C_{12}\times C_{2}$ & 24 & $4$; $-i$; $6$ & 4 & 128\,217\,601 & 0 & 256\,435\,200\\ 33 & $-$ & $\le N_G(\langle g_{12}\rangle)$, $C_{12}\times C_{3}$ & 36 & $12$; $-i$; $3$ & 12 & 85\,478\,329 & 216 & 170\,956\,872\\ 34 & $-$ & $\le N_G(\langle g_{12}\rangle)$, $C_{4}\times C_{2}\times C_{2}$ & 16 & $4$; $-i$; $4$ & 4 & 190\,509\,697 & 7\,262\,208 & 388\,281\,600\\ 35 & $+$ & $\le N_G(\langle g_{12}\rangle)$, $C_{4}\times C_{4}$ & 16 & $4$; $-i$; $4$ & 4 & 192\,324\,097 & 4\,608 & 384\,652\,800\\ 36 & $+$ & $\le N_G(\langle g_{12}^2\rangle)$, $C_{12}\times C_{2}$ & 24 & $4$; $-i$; $6$ & 4 & 127\,008\,001 & 4\,838\,400 & 258\,854\,400\\ 37 & $+$ & $\le N_G(\langle g_{12}^2\rangle)$, $C_{4}\times C_{2}\times C_{2}$ & 16 & $4$; $-i$; $4$ & 4 & 192\,326\,401 & 0 & 384\,652\,800\\ 38 & $-$ & $\le N_G(\langle g_{12}^2\rangle)$, $C_{12}\times C_{2}$ & 24 & $4$; $-i$; $6$ & 4 & 128\,217\,601 & 0 & 256\,435\,200\\ 39 & $-$ & $\le N_G(\langle g_{12}^2\rangle)$, $C_{4}\times S_{3}$ & 24 & $4$; $-i$; $6$ & 4 & 128\,217\,601 & 0 & 256\,435\,200\\ 40 & $+$ & $\le N_G(\langle g_{12}^2\rangle)$, $C_{12}\times C_{2}\times C_{2}$ & 48 & $4$; $-i$; $12$ & 4 & 63\,503\,233 & 2\,420\,736 & 129\,427\,200\\ 41 & $+$ & $\langle i,w\rangle$, $w\in\mathcal C_{97}$ & 48 & $\chi(i)=-i$, $\chi(w)=1$ & 4 & 64\,108\,801 & 0 & 128\,217\,600\\ 42 & $-$ & $\langle i,w\rangle$, $w\in\mathcal C_{93}$ & 40 & $\chi(i)=-i$, $\chi(w)=1$ & 4 & 76\,204\,801 & 2\,903\,040 & 155\,312\,640\\ 43 & $+$ & $\langle i,w\rangle$, $w\in\mathcal C_{95}$ & 48 & $\chi(i)=-i$, $\chi(w)=\zeta_{12}^{6}$ & 4 & 64\,108\,801 & 0 & 128\,217\,600\\ 44 & $-$ & $\le N_G(\langle g_{12}^2\rangle)$, $C_{2}\times C_{4}\times S_{3}$ & 48 & $4$; $-i$; $12$ & 4 & 64\,108\,033 & 1\,536 & 128\,217\,600\\ 45 & $+$ & $\le N_G(\langle g_{12}\rangle)$, $C_{12}\times C_{2}$ & 24 & $12$; $-i$; $2$ & 12 & 127\,007\,893 & 4\,838\,724 & 258\,854\,508\\ 46 & $-$ & $\le N_G(\langle g_{12}^2\rangle)$, $C_{2}\times C_{4}\times S_{3}$ & 48 & $4$; $-i$; $12$ & 4 & 62\,294\,401 & 7\,257\,600 & 131\,846\,400\\ 76 & $-$ & $\langle i,w\rangle$, $w\in\mathcal C_{44}$ & 32 & $\chi(i)=-i$, $\chi(w)=1$ & 4 & 96\,162\,049 & 2\,304 & 192\,326\,400\\ 78 & $+$ & $\langle i,w\rangle$, $w\in\mathcal C_{59}$ & 48 & $\chi(i)=-i$, $\chi(w)=1$ & 4 & 64\,108\,801 & 0 & 128\,217\,600\\ 79 & $+$ & $\langle i,w\rangle$, $w\in\mathcal C_{50}$ & 40 & $\chi(i)=-i$, $\chi(w)=1$ & 4 & 76\,930\,561 & 0 & 153\,861\,120\\ 88 & $+$ & $\le N_G(\langle g_{12}^2\rangle)$, $C_{12}\times C_{6}$ & 72 & $4$; $-i$; $18$ & 4 & 42\,738\,609 & 1\,056 & 85\,478\,272\\ 105 & $+$ & $\le N_G(\langle g_{12}^2\rangle)$, $C_{12}\times C_{2}$ & 24 & $12$; $-i$; $2$ & 12 & 127\,008\,001 & 4\,838\,400 & 258\,854\,400\\ 116 & $-$ & $\le N_G(\langle g_{12}^2\rangle)$, $C_{12}\times C_{6}$ & 72 & $12$; $-i$; $6$ & 12 & 42\,335\,929 & 1\,613\,016 & 86\,284\,872\\ 126 & $+$ & $\le N_G(\langle g_{12}^2\rangle)$, $C_{2}\times C_{4}\times S_{3}$ & 48 & $4$; $-i$; $12$ & 4 & 64\,108\,801 & 0 & 128\,217\,600\\ 129 & $-$ & $\le N_G(\langle g_{12}^2\rangle)$, $C_{2}\times C_{2}\times C_{4}\times S_{3}$ & 96 & $4$; $-i$; $24$ & 4 & 31\,751\,617 & 1\,210\,368 & 64\,713\,600\\ 156 & $+$ & $\le N_G(\langle g_{12}\rangle)$, $C_{8}\times C_{4}$ & 32 & $4$; $i\notin H'$; $8$ & 4 & 96\,162\,049 & 2\,304 & 192\,326\,400\\ 195 & $-$ & $\langle iw\rangle$, $w\in\mathcal C_{23}$ & 20 & $\chi(iw)=\zeta_{20}^{3}$ & 20 & 153\,861\,121 & 0 & 307\,722\,240\\ 208 & $+$ & $\langle iw\rangle$, $w\in\mathcal C_{41}$ & 28 & $\chi(iw)=\zeta_{28}$ & 28 & 109\,900\,801 & 0 & 219\,801\,600\\ 217 & $+$ & $\langle iw\rangle$, $w\in\mathcal C_{48}$ & 36 & $\chi(iw)=\zeta_{36}^{3}$ & 12 & 85\,478\,329 & 0 & 170\,956\,656\\ 221 & $-$ & $\le N_G(\langle g_{12}\rangle)$, $C_{12}\times C_{4}$ & 48 & $4$; $-i$; $12$ & 4 & 64\,107\,973 & 1\,560 & 128\,217\,504\\ 224 & $+$ & $\langle i\rangle\cdot C_W(w),\ w\in\mathcal C_{52}$ & 40 & $\chi(i)=-i$; $\chi_0$: $20$; $1$ & 20 & 76\,929\,409 & 2\,304 & 153\,861\,120\\ 226 & $+$ & $\langle i\rangle\cdot C_W(w),\ w\in\mathcal C_{61}$ & 48 & $\chi(i)=-i$; $\chi_0$: $24$; $1$ & 24 & 64\,107\,841 & 1\,920 & 128\,217\,600\\ 236 & $+$ & $\le N_G(\langle g_{12}^2\rangle)$, $C_{12}\times C_{6}$ & 72 & $4$; $-i$; $18$ & 4 & 42\,739\,129 & 0 & 85\,478\,256\\ 243 & $-$ & $\le C_G(g_{12}^3)$, $C_{12}\times C_{4}$ & 48 & $12$; $-i$; $4$ & 12 & 64\,107\,841 & 1\,980 & 128\,217\,660\\ 246 & $-$ & $\langle i\rangle\cdot C_W(w),\ w\in\mathcal C_{64}$ & 60 & $\chi(i)=-i$; $\chi_0$: $30$; $1$ & 60 & 51\,287\,041 & 0 & 102\,574\,080\\ 290 & $+$ & $\langle i\rangle\cdot C_W(w),\ w\in\mathcal C_{49}$ & 108 & $\chi(i)=-i$; $\chi_0$: $6$; $9$ & 12 & 28\,492\,777 & 0 & 56\,985\,552\\ 313 & $+$ & $\le N_G(\langle y^2\rangle)$, $C_{4}\times (C_{3}\rtimes (C_{4}\times D_{8}))$ & 384 & $4$; $-i$; $96$ & 4 & 8\,013\,281 & 640 & 16\,027\,200\\ 314 & $+$ & $\le C_G(g_{12}^3)$, $C_{4}\times ((C_{2}\times C_{2}\times Q_8)\rtimes C_{3})$ & 384 & $4$; $i\notin H'$; $96$ & 4 & 8\,013\,361 & 288 & 16\,027\,008\\ 330 & $-$ & $\langle i,w\rangle$, $w\in\mathcal C_{93}$ & 40 & $\chi(i)=-i$, $\chi(w)=\zeta_{10}^{2}$ & 20 & 76\,204\,801 & 2\,903\,040 & 155\,312\,640\\ 332 & $-$ & $\le N_G(\langle g_{12}^2\rangle)$, $C_{12}\times C_{2}\times C_{2}$ & 48 & $12$; $-i$; $4$ & 12 & 63\,503\,233 & 2\,420\,736 & 129\,427\,200\\ 338 & $+$ & $\langle i\rangle\cdot C_W(w),\ w\in\mathcal C_{41}$ & 56 & $\chi(i)=-i$; $\chi_0$: $14$; $2$ & 28 & 54\,432\,001 & 2\,073\,600 & 110\,937\,600\\ 345 & $+$ & $\le N_G(\langle g_{12}\rangle)$, $C_{8}\times C_{4}\times C_{2}$ & 64 & $8$; $-i$; $8$ & 8 & 47\,627\,329 & 1\,815\,744 & 97\,070\,400\\ 364 & $-$ & $\langle i\rangle\cdot C_W(w),\ w\in\mathcal C_{102}$ & 96 & $\chi(i)=-i$; $\chi_0$: $12$; $4$ & 12 & 31\,751\,521 & 1\,210\,590 & 64\,713\,630\\ 366 & $+$ & $\langle i\rangle\cdot C_W(w),\ w\in\mathcal C_{40}$ & 96 & $\chi(i)=-i$; $\chi_0$: $6$; $8$ & 12 & 31\,449\,217 & 2\,419\,968 & 65\,318\,400\\ 403 & $-$ & $\langle i\rangle\cdot C_W(w),\ w\in\mathcal C_{51}$ & 160 & $\chi(i)=-i$; $\chi_0$: $10$; $8$ & 20 & 18\,869\,761 & 1\,451\,520 & 39\,191\,040\\ 404 & $+$ & $\langle i\rangle\cdot C_W(w),\ w\in\mathcal C_{94}$ & 240 & $\chi(i)=-i$; $\chi_0$: $10$; $12$ & 20 & 12\,337\,921 & 1\,935\,360 & 26\,611\,200\\ 425 & $-$ & $\le N_G(\langle y^2\rangle)$, $C_{4}\times (((C_{4}\times C_{2})\rtimes C_{2})\times S_{3})$ & 384 & $4$; $-i$; $96$ & 4 & 7\,635\,473 & 1\,512\,256 & 16\,783\,200\\ 426 & $-$ & $\langle i\rangle\cdot C_W(w),\ w\in\mathcal C_{59}$ & 192 & $\chi(i)=-i$; $\chi_0$: $6$; $16$ & 12 & 15\,724\,609 & 1\,209\,984 & 32\,659\,200\\ 434 & $-$ & $\langle i\rangle\cdot C_W(w),\ w\in\mathcal C_{23}$ & 2\,400 & $\chi(i)=-i$; $\chi_0$: $10$; $120$ & 20 & 1\,161\,217 & 483\,840 & 2\,806\,272\\ 437 & $-$ & $\le C_G(g_{12}^3)$, $C_{4}\times ((C_{2}\times C_{2}\times ((C_{4}\times C_{2})\rtimes C_{2}))\rtimes C_{3})$ & 768 & $4$; $i\notin H'$; $192$ & 4 & 4\,006\,681 & 144 & 8\,013\,504\\ 457 & $-$ & $\langle i\rangle\cdot N_W(\langle w\rangle),\ w\in\mathcal C_{106}$ & 432 & $\chi(i)=-i$; $\chi_0$: $2$; $108$ & 4 & 7\,055\,995 & 268\,800 & 14\,380\,788\\ 459 & $+$ & $\langle i\rangle\cdot N_W(\langle w\rangle),\ w\in\mathcal C_{106}$ & 432 & $\chi(i)=-i$; $\chi_0$: $2$; $108$ & 4 & 7\,055\,995 & 0 & 14\,111\,988\\ 463 & $-$ & $\langle i\rangle\cdot N_W(\langle w\rangle),\ w\in\mathcal C_{59}$ & 768 & $\chi(i)=-i$; $\chi_0$: $2$; $192$ & 4 & 3\,742\,057 & 1\,058\,688 & 8\,542\,800\\ 467 & $+$ & $\langle i\rangle\cdot N_W(\langle w\rangle),\ w\in\mathcal C_{59}$ & 768 & $\chi(i)=-i$; $\chi_0$: $2$; $192$ & 4 & 3\,742\,057 & 907\,488 & 8\,391\,600\\ 469 & $+$ & $\langle i\rangle\cdot C_W(w),\ w\in\mathcal C_{19}$ & 1\,536 & $\chi(i)=-i$; $\chi_0$: $4$; $192$ & 4 & 1\,852\,009 & 605\,184 & 4\,309\,200\\ 471 & $-$ & $\langle i\rangle\cdot W_S\langle-1\rangle,\ S=\{1,2,4,5,7,8,9\}$ & 4\,608 & $\chi(i)=-i$; $\chi_0$: $2$; $1152$ & 4 & 585\,829 & 327\,744 & 1\,499\,400\\ 473 & $+$ & $\langle i\rangle\cdot N_W(\langle w\rangle),\ w\in\mathcal C_{51}$ & 640 & $\chi(i)=-i$; $\chi_0$: $2$; $160$ & 4 & 4\,717\,441 & 362\,880 & 9\,797\,760\\ 477 & $-$ & $\langle i\rangle\cdot W_S\langle-1\rangle,\ S=\{1,2,3,4,6,7,8\}$ & 11\,520 & $\chi(i)=-i$; $\chi_0$: $2$; $2880$ & 4 & 226\,801 & 161\,280 & 614\,880\\ 478 & $-$ & $\langle i\rangle\cdot W_S\langle-1\rangle,\ S=\{1,3,4,5,6,8,9\}$ & 17\,280 & $\chi(i)=-i$; $\chi_0$: $2$; $4320$ & 4 & 147\,835 & 120\,960 & 416\,628\\ 492 & $+$ & $\langle i\rangle\cdot W_S\langle-1\rangle,\ S=\{1,2,4,5,6,7,8,9\}$ & 322\,560 & $\chi(i)=-i$; $\chi_0$ not singled out & 4 & 6\,919 & 10\,104 & 23\,940\\ 493 & $-$ & $\langle i\rangle\cdot C_W(w),\ w\in\mathcal C_{67}$ & 11\,612\,160 & $\chi(i)=-i$; $\chi_0$ not singled out & 4 & 103 & 636 & 840\\ 536 & $-$ & $\le N_G(\langle g_{12}\rangle)$, $C_{8}\times (C_{3}\rtimes C_{4})$ & 96 & $4$; $i\notin H'$; $24$ & 4 & 32\,054\,017 & 768 & 64\,108\,800\\ 574 & $+$ & $\le N_G(\langle g_{12}^2\rangle)$, $C_{12}\times Q_8$ & 96 & $12$; $i\notin H'$; $8$ & 12 & 32\,053\,987 & 852 & 64\,108\,824\\ 857 & $+$ & $\langle i\rangle\cdot C_W(w),\ w\in\mathcal C_{67}$ & 11\,612\,160 & $\chi(i)=-i$; $\chi_0$ not singled out & 4 & 103 & 16 & 220\\ 1125 & $+$ & $\le C_G(g_{12}^3)$, $C_{12}\times ((C_{4}\times C_{4})\rtimes C_{2})$ & 384 & $12$; $-i$; $32$ & 12 & 8\,013\,441 & 320 & 16\,027\,200\\\hline 3215 & $-$ & $\langle i\rangle\cdot N_W(P_3)$ & 7\,776 & $\chi(i)=i$; $\chi_0$: $2$; $1944$ & 4 & 384\,479 & 88 & 769\,044\\ \end{longtable} } \subsection*{Independent computations} The product rows, the suppliers on $\langle i\rangle N_W(P_3)$ and the combination step were rebuilt by a second program, written separately, which found its own integer witness with $98$ terms (with different support), confirmed that every row of the original matrix belongs to the corresponding family, and recomputed the product. That program used the same realization of $G$, the same class data and the same definition and admissibility test of the search pool, so it confirms the product rows and the combination step but not every prerequisite. A third computation reconstructed the three product rows and the row from $\langle i\rangle N_W(P_3)$ and verified balance at $t=1,5,7,11$. Table~\ref{tab:claims} lists every computer-checked claim of the paper. \begin{table}[ht] \caption{The computer-checked claims.}\label{tab:claims} \small \begin{tabular}{p{0.38\textwidth}p{0.2\textwidth}p{0.32\textwidth}} claim & software & independent check\\\hline $\mathbf g$ generates $G$; the ten reflections generate a group of order $51\,840$ (App.~\ref{app:tuple}) & GAP & none\\ $X^*(T)=\Z^{45}$ and the rows $a(\chi)$ (App.~\ref{app:witness}) & GAP & rows checked against $h=2q-2+s$\\ $W_{\rm coef}\cdot W_{\rm gen}=e_\sigma$ (Prop.~\ref{prop:witness}) & GAP & second witness, same group data\\ admissibility and balance of the product rows; slot data for every lift (Prop.~\ref{prop:slots}) & GAP & third computation of balance\\ $\det h=-3$, signature $(3,3)$ (Prop.~\ref{prop:dominance}) & Sage, exact over $K$ & second program, $\det h=-6$\\ existence of $M$ and $\rk\mu_{C_0}=9$ (App.~\ref{app:rank}) & Python, Arb ball arithmetic & conjugate datum; rerun of the verification\\ Shioda's condition $(P_{12})$ (Subsec.~\ref{subsec:shioda}) & Python, Normaliz & Shioda's verification \cite[p.~182]{Shioda}\\ Schoen suppliers alone fail (App.~\ref{app:schoenfails}; not used) & GAP & separate recomputation \end{tabular} \end{table} \subsection*{Slot data} For each product row, Table~\ref{tab:slots} lists one integral lift (the characters are numbered as in the certificate's enumeration of the linear characters of $H'_l$, and a negative coefficient means that the slot uses the inverse character), the genus of $C/H'_l$, and the slot data. The certificate covers every integral lift (Proposition~\ref{prop:slots}), so the choice of lift does not matter. \begin{table}[ht] \caption{Slot data of the three product suppliers.}\label{tab:slots} \small \begin{tabular}{p{0.13\textwidth}p{0.12\textwidth}p{0.3\textwidth}p{0.34\textwidth}} $w_l$ & genus $g_X$ & lift & slots $(e,s,d,e')$\\\hline $w_1$ (order $3$) & $3259$ & $-\chi_2-\chi_7$ & $(4,6732,1,4)$; $(12,20,1,12)$\\ $w_2$ (order $12$) & $5\,341\,793$ & $\chi_1+\chi_3-\chi_5-\chi_9$ & $(4,1176,1,4)$; $(12,1236,1,12)$ twice; $(12,1176,3,4)$\\ $w_3$ (order $6$) & $1\,730\,347$ & $-\chi_5+9\chi_8+12\chi_9+11\chi_{10}-\chi_{13}+11\chi_{19}$ & $45$ slots, all with $e=12$ and $d=1$; $s\in\{112,124,201\,688,201\,712\}$ \end{tabular} \end{table} For $w_3$ the six characters have $(s,h)$ equal to $(201\,712,3\,662\,404)$ for $\chi_5,\chi_9,\chi_{10}$, $(112,3\,460\,804)$ for $\chi_8$, $(201\,688,3\,662\,380)$ for $\chi_{13}$ and $(124,3\,460\,816)$ for $\chi_{19}$, with imbalances at $t=1$ of absolute value $22$, except $20$ for $\chi_{19}$; the combination has imbalance $0$ at $t=1,5,7,11$. For $w_2$ every slot has imbalance $\pm64$ at $t=1$, and the signs cancel. In the slot with $d=3$ the cyclic cover has reduced order $e'=4$, and Lemma~\ref{lem:special}(a) applies with three components. The slot program was also run on three cases with known answers: a slot with all exponents $0$ fails (H2), the supplier of Example~\ref{ex:schoen3215} is a single slot of Hodge type with $d=2$ and $e'=2$, and a cyclic cover of a genus-one curve with exponents $(2,2,4,4)$ modulo $6$ has $d=2$ and $e'=3$. \subsection*{Supplementary files} The supplementary files are the certificate archive \cite{QiCert}, \texttt{doi:10.5281/zenodo.23549514}. Its README lists every file named below with its path in the archive. The integer witness and a GAP script that checks the identity are in \texttt{witness/}; the witness file \texttt{witness/ncb/ncb\_witness.g} has SHA-256 hash \begin{center} {\footnotesize\texttt{93a68c0760a10eb84f2d0760a258f66075c25aaa230c82ae33c395ded6fcf3ba}}. \end{center} The rank certificate (residues, enclosures and programs) is in \texttt{dominance/}, the discriminant computation in \texttt{independent-det/}, the slot data for every integral lift in \texttt{gale3-slots/}, the files of pool vectors read by the witness in \texttt{pool-scripts/} (byte-identical, by SHA-256, to the copies the witness read when it was built), and the recomputation of Shioda's condition for $m=12$ (Subsection~\ref{subsec:shioda}) in \texttt{shioda-p12/}. The archive also contains the scripts that rebuild the witness from the group (\texttt{qi-gale/}). The full rebuild of the witness takes several hours and about $10$\,GB of memory; the other checks take seconds to minutes. %% ------------------------------------------------------------------ \section{The cocycle basis of Proposition~\ref{prop:dominance}}\label{app:gram} %% ------------------------------------------------------------------ This appendix prints the basis vectors used in the discriminant computation of Proposition~\ref{prop:dominance}. A vector $z$ of the reduced row echelon basis of $Z^1$ is the tuple $(a_1,\dots,a_{12})$ of Proposition~\ref{prop:dominance}, of lengths $8,4,1,\dots,1$, in the tuple $(T_j)$ used there. We write it as $(a_1)\mid(a_2)\mid(a_3,\dots,a_{12})$, with entries in $K=\Q(i)$. For the tuple $\mathbf g$ the first six vectors $z^g_0,\dots,z^g_5$ span a complement of the radical; their Gram matrix is $h_0$. For the conjugate class datum (Appendix~\ref{app:tuple}) the vectors used are $z^h_0,z^h_1,z^h_3,z^h_4,z^h_5,z^h_6$; their Gram matrix also has determinant $-3$ and signature $(3,3)$. The vector $z^h_2$ is printed because it precedes $z^h_6$ in the echelon basis. {\small \begin{align*} z^{g}_{0}&=(1,0,0,0,0,0,0,0)\mid(0,0,0,0)\mid{}\\&\qquad(0,0,0,0,0,0,0,0,\tfrac{1}{2}+\tfrac{1}{2}i,1+i)\\ z^{g}_{1}&=(0,1,0,0,0,0,0,0)\mid(0,0,i,-1+i)\mid{}\\&\qquad(0,1-i,0,0,0,-1+i,0,2,\tfrac{1}{2},0)\\ z^{g}_{2}&=(0,0,1,0,0,0,0,0)\mid(0,0,1,0)\mid{}\\&\qquad(0,0,0,0,0,1,0,0,-\tfrac{1}{2},-1)\\ z^{g}_{3}&=(0,0,0,1,0,0,0,0)\mid(0,0,0,1)\mid{}\\&\qquad(0,0,0,0,0,0,0,-1,-\tfrac{1}{2},0)\\ z^{g}_{4}&=(0,0,0,0,1,0,0,0)\mid(0,0,-1-i,-\tfrac{1}{2}-\tfrac{1}{2}i)\mid{}\\&\qquad(0,-1+i,0,\tfrac{1}{2}+\tfrac{1}{2}i,0,-i,0,0,\tfrac{3}{4}-\tfrac{1}{4}i,1)\\ z^{g}_{5}&=(0,0,0,0,0,1,0,0)\mid(0,0,1,0)\mid{}\\&\qquad(0,-2i,0,-1,0,i,0,1-i,\tfrac{1}{2}i,0)\\ z^{h}_{0}&=(1,0,0,0,0,0,0,0)\mid(0,0,-1,i)\mid{}\\&\qquad(0,2i,0,1,0,i,0,2+i,0,1-i)\\ z^{h}_{1}&=(0,1,0,0,0,0,0,0)\mid(0,0,0,-1)\mid{}\\&\qquad(0,-1-i,0,i,0,-1-i,0,-1,1,i)\\ z^{h}_{2}&=(0,0,1,0,0,0,0,0)\mid(0,0,1,0)\mid{}\\&\qquad(0,0,0,0,0,0,0,-1,1,-1)\\ z^{h}_{3}&=(0,0,0,1,0,0,0,0)\mid(0,0,0,1)\mid{}\\&\qquad(0,1,0,0,0,0,0,1,-1,1)\\ z^{h}_{4}&=(0,0,0,0,1,0,0,0)\mid(0,0,-\tfrac{1}{2}-\tfrac{1}{2}i,-1)\mid{}\\&\qquad(0,-2,0,-\tfrac{1}{2}-\tfrac{1}{2}i,0,0,0,-1,-\tfrac{1}{2}+\tfrac{1}{2}i,-\tfrac{1}{2}-\tfrac{1}{2}i)\\ z^{h}_{5}&=(0,0,0,0,0,1,0,0)\mid(0,0,0,-i)\mid{}\\&\qquad(0,1,0,i,0,1,0,1,-1,i)\\ z^{h}_{6}&=(0,0,0,0,0,0,1,0)\mid(0,0,\tfrac{1}{2}+\tfrac{1}{2}i,1)\mid{}\\&\qquad(0,1-i,0,-\tfrac{1}{2}-\tfrac{3}{2}i,0,0,0,-1-i,\tfrac{3}{2}-\tfrac{3}{2}i,-\tfrac{1}{2}-\tfrac{1}{2}i) \end{align*} } To check $h_0$ from these vectors, set \[ F_1=0,\qquad F_{j+1}=F_j+Q_jE_ja_j, \] with $T_j$, $Q_j$, $E_j$ as in Proposition~\ref{prop:dominance}. Then evaluate the form of Lemma~\ref{lem:cocycle}, \[ h(z,z')=i\sum_{j=1}^{12}\overline{\big(Q_j^{-1}F_j+U_ja_j\big)}^{\,T}H\,E_ja'_j, \] where $H$ is the Cartan matrix of $E_8$ and $U_j$ is the matrix of the same pivot columns of the identity, so that $E_ja_j=(1-T_j)U_ja_j$. Recomputing $h$ on $z^g_0,\dots,z^g_5$ gives the matrix $h_0$ of Proposition~\ref{prop:dominance}. Recomputing it on $z^h_0,z^h_1,z^h_3,\dots,z^h_6$ gives the block for the conjugate datum. The script \texttt{independent-det/independent\_det.py} in the supplementary files does this; these vectors are the first rows of the matrix $Z$ that it computes. %% ------------------------------------------------------------------ \section{Schoen suppliers alone do not suffice}\label{app:schoenfails} %% ------------------------------------------------------------------ This appendix is not used in the proof. It records why a second source of cycles is needed: for this class datum, condition (L) with Schoen-admissible suppliers alone fails. Let $y=iw$ for a suitable element $w\in W$ of order $3$. For a subgroup $H'$ and a linear character $\chi$, Frobenius reciprocity gives a linear functional $f$ on $X^*(T)$ with $f(e_\sigma)=1$ and \[ f(a(\chi))=\frac{|C_G(y)|}{|H'|}\sum_{z\in H'\cap y^G}\operatorname{Im}\chi(z). \] A local analysis in the centralizers $C_{H'}(z)$, $z\in H'\cap y^G$, shows that a Schoen-admissible supplier with $f(a(\chi))\ne0$ must contain one of finitely many configurations of three elements; up to conjugation there are $1248$ of them. A necessary condition that is inherited by overgroups rules out all but $72$, and the subgroups containing these $72$ were enumerated exhaustively: they carry $4496$ characters of the relevant kind, of which $626$ give Schoen-admissible suppliers, and $f$ vanishes on all of them. The configurations, the necessary conditions and all $4496$ characters were recomputed by a separate program; the enumeration itself was checked from its saved state. So $e_\sigma$ is not in the subgroup generated by the characters of Schoen-admissible suppliers. Brauer's induction theorem, which supplies (L) in \cite[\S5]{SevernQ3}, does not apply: the class of $ix_2$ is not conjugate to its inverse. The programs and their outputs are in the certificate archive \cite{QiCert}, directory \texttt{qi-schoen-negative} (subdirectories \texttt{LS1}, \texttt{LS2}, \texttt{LS2R}). %% ------------------------------------------------------------------ \begin{thebibliography}{99} \bibitem{Arb} F. Johansson, \emph{Arb: efficient arbitrary-precision midpoint-radius interval arithmetic}, IEEE Trans. Comput. 66 (2017), 1281--1292. \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. 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