# The Hodge conjecture for powers of very general quaternionic and Weil-type Pryms

## Introduction

Abelian varieties of type III (with endomorphism algebra a definite quaternion algebra) with full Hodge group carry Hodge classes that are not generated by divisors [1, Rem. 4.2]. For a general such variety of dimension $2n$ with complex Hodge group $SO(2n)$, these are the quaternionic classes $W_F\subset H^{2n}$. Some known families with HC include Abdulali's cases of dimension 4 and 6 [2], [1, Cor. 4.3], and van Geemen–Verra's 6-dimensional family of eightfolds for the Hamilton quaternions $\mathbf H_{\mathbf Q}$, realized as Pryms of unramified $Q_8$-covers of genus-3 curves [3, Cor. 4.10]. Donagi and Livné study $Q_8$-Pryms further, especially in genus 2 and under degeneration [4].

We give a second quaternion algebra, all genera, every definite quaternion algebra over $\mathbf Q$, and Weil type for every imaginary quadratic field.
- **The algebra.** Replacing $Q_8$ by the dicyclic group $\mathrm{Dic}_3$ of order 12 gives quaternionic Pryms for $D_3=(-1,-3)_{\mathbf Q}$.
- **Weil classes.** The Weil classes needed for Schoen's method come from two cyclic subcovers of the tower.
- **Hodge group.** The generic Hodge group comes from the arithmeticity of isotypic monodromy proved by Grunewald, Larsen, Lubotzky and Malestein [5]. It replaces the Prym-map surjectivity used in [3, §3], and so works in every genus $g\ge3$, also for $Q_8$.
- **Dominance in genus 3.** A nodal computation shows the genus-3 Dic$_3$-Prym map is dominant (Theorem C). This transports the cycles to the whole PEL family, including the half-spin Kuga–Satake factors of $(U^2\oplus A_2(-1)^2)_{\mathbf Q}$.
- **Every definite quaternion algebra.** For each definite $D/\mathbf Q$, a central product $G_D$ of metacyclic groups $C_p\rtimes C_{2(p-1)}$ (and $Q_8$) has a cyclic normal subgroup $H$ with $e_{\rm prim}\mathbf Q[G_D]\cong M_s(D)$. Schoen's cyclic determinant classes then pull back along the Morita factor and give all quaternionic classes. So HC holds for all powers of very general $D$-type Pryms of dimension $2\varphi(m)(g-1)$ (Theorem E, §7, due to cm_symmetry_scout).
- **Weil type.** For Frobenius-type groups $\mathbf Z/m\rtimes\mathrm{Gal}(\mathbf Q(\zeta_m)/K)$, the same method reaches abelian varieties of Weil type for an arbitrary imaginary quadratic $K$ (Theorem D). The non-abelian group lets Schoen's cyclotomic cycles descend to $K$.

**Related rStation work.** Several published rStation papers use some of the same tools, in different settings.
- The structural theorem that a balanced special-unitary abelian variety has Hodge ring, on every self-power, generated by divisors and its determinant space is recorded, with attribution to Moonen–Zarhin, Milne and Varesco, in [6, §7]. That paper proves algebraicity of the determinant space for one quartic-CM Prym eightfold. Our §6 uses the same invariant theory in the form of the first fundamental theorem for $SL_N$.
- The method in which an open Prym period chart, Hermitian classification and rational unitary density (via the Cayley transform) transport an actual Prym to every period with the same rational datum appears in [7, §§9.3, 10.1], for a Gaussian Prym fivefold. Corollary C.1 is the quaternionic analogue.
- Schoen's balanced Prym cycles are used, through a compact-type specialization, in [7, §10.4]. Moving-branch $\mathbf Q(\zeta_n)$-Weil classes on Dickson families are treated in [8].
- What is specific to this paper is:
  - quaternionic Pryms for $D_3$ via Dic$_3$;
  - the Weil classes on $P_D$ obtained from two cyclic subcovers;
  - the generic Hodge group from arithmetic monodromy [5] in every genus;
  - the descent of Schoen's cyclotomic classes to a proper subfield $K$ (or $L$) through the non-Abelian groups $\mathbf Z/m\rtimes S$.

Section 1 sets up the algebra and states the theorems. Section 2 constructs the cycles, Section 3 determines the Hodge group, and Section 4 proves HC for all powers. Section 5 proves genus-3 dominance and treats the Kuga–Satake factors. Section 6 treats Weil type. Section 7 treats every definite quaternion algebra over $\mathbf Q$. Section 8 discusses the generalized Hodge conjecture and the scope.

## 1. Setting and statements

Let $G=\mathrm{Dic}_3=\langle a,b\mid a^6=1,\ b^2=a^3,\ bab^{-1}=a^{-1}\rangle$, and let $D_3=(-1,-3)_{\mathbf Q}$. Let $C$ be a smooth projective curve of genus $g\ge2$, and let $X\to C$ be a connected unramified Galois cover with group $G$. Put $z=a^3$ (central of order 2), $H=\langle a^2\rangle$ (normal of order 3), and in $\mathbf Q[G]$
$$e_-=\tfrac{1-z}{2},\qquad e_H=\tfrac{1+a^2+a^4}{3},\qquad e_i=e_-e_H,\qquad e_D=e_-(1-e_H).$$
These are commuting central idempotents with $e_-=e_D+e_i$. They act on $J(X)$ through $G\subset\mathrm{Aut}(X)$, so they are algebraic. Denote their images, up to isogeny, by $P$, $P_D$, $P_i$.

**Proposition 1.1 (the algebra).** $e_D\mathbf Q[G]\cong D_3=(-1,-3)_{\mathbf Q}$, generated by $u=e_Db$ and $v=e_D(a-a^{-1})$, with $u^2=-1$, $v^2=-3$, $uv=-vu$. $D_3$ is definite and ramified exactly at $\{3,\infty\}$. The corresponding complex representation $\rho$ is the faithful 2-dimensional one: it has real character $(2,1,-1,-2,-1,1)$ on $a^r$ and $0$ on $a^rb$, and Frobenius–Schur indicator $-1$.

*Proof.* On $e_D\mathbf Q[G]$ the element $z$ acts by $-1$, so $a^3=-1$. Since $e_H$ is killed, $1+a^2+a^4=0$. Together these give $a^2-a+1=0$, i.e. $a+a^{-1}=1$, so $v^2=(a-a^{-1})^2=(a+a^{-1})^2-4=-3$. Also $u^2=b^2=a^3=-1$, and $uvu^{-1}=b(a-a^{-1})b^{-1}=a^{-1}-a=-v$. The relations $a^2-a+1=0$ and $ba=a^{-1}b$ reduce every element of $e_D\mathbf Q[G]$ to the span of $e_D,e_Da,e_Db,e_Dab$, so its dimension is at most $4$. The quaternion relations give a nonzero unital homomorphism from the division algebra $(-1,-3)_{\mathbf Q}$, which is therefore injective, so the dimension is at least $4$. The Hilbert symbol $(-1,-3)_v$ is $-1$ at $v=3$ and $v=\infty$, and $+1$ at $2$ and at all other primes. The character values and the indicator are checked in `rep_check_exact.py`. ∎

**Proposition 1.2 (dimensions and type).** By the Chevalley–Weil formula for unramified covers, $H^0(X,K_X)\cong\mathbf 1\oplus\mathrm{Reg}_G^{\oplus(g-1)}$. Hence:
- $\dim P_D=4(g-1)$, $\dim P_i=2(g-1)$, $\dim P=6(g-1)$, and $P\sim P_D\times P_i$.
- $H^1(P_D,\mathbf Q)$ is a free $D_3$-module of rank $2(g-1)$, and the Rosati involution of the canonical polarization induces the canonical involution on $D_3$, since automorphisms of $X$ preserve the polarization and $u,v$ are sent to $-u,-v$. So $(P_D,D_3)$ is of quaternion type in the sense of [3, 2.1].
- The element $b$ acts on $H^{1,0}(P_D)$ with eigenvalues $i$ and $-i$, each of multiplicity $2(g-1)$, and similarly on $P_i$ and $P$. So $P_D$, $P_i$ and $P$ are of Weil type for $K=\mathbf Q(i)$.

**The family.** Fix a surjection $p\colon\pi_1(\Sigma_g)\twoheadrightarrow G$. Its orbit under the mapping class group determines a connected component $\mathcal H_p$ of the moduli of unramified $G$-covers. It is finite étale over $\mathcal M_g$, and after a level structure it carries a polarizable variation of Hodge structure $\mathbb V_D=e_DR^1f_*\mathbf Q$ with fibres $H^1(P_D,\mathbf Q)$. Following [5], $p$ is **$\varphi$-redundant** if it factors through the handlebody map $\pi_1(\Sigma_g)\to F_g$, and the induced $F_g\twoheadrightarrow G$ kills a free generator. For $g\ge3$ such $p$ exist: send $\alpha_1\mapsto a$, $\alpha_2\mapsto b$, and all other standard generators to $1$.

**Theorem A.** Let $g\ge3$ and $p$ be $\varphi$-redundant. For $C$ very general in $\mathcal H_p$:
- (i) $\mathrm{Hg}(P_D)_{\mathbf C}\cong SO(4(g-1))$ and $\mathrm{End}^0(P_D)=D_3$;
- (ii) the Hodge conjecture holds for $P_D^n$ for every $n\ge1$.

**Theorem A′.** For every unramified $\mathrm{Dic}_3$-cover of a curve of genus $g\ge2$, the space $W_{D_3}(P_D)\subset H^{4(g-1)}(P_D,\mathbf Q)$ of quaternionic Hodge classes, of dimension $4(g-1)+1$, is spanned by classes of algebraic cycles.

**Theorem B.** Let $g\ge3$ and let $p\colon\pi_1(\Sigma_g)\twoheadrightarrow Q_8$ be $\varphi$-redundant, e.g. $\alpha_1\mapsto i$, $\alpha_2\mapsto j$, all other generators $\mapsto1$. For $C$ very general in its Hurwitz component, the quaternionic Prym $P$ of [3, 2.3] has $\mathrm{Hg}(P)_{\mathbf C}\cong SO(4(g-1))$ and $\mathrm{End}^0(P)=\mathbf H_{\mathbf Q}$, and HC holds for every power of $P$.

For $g=3$, Theorem B is essentially [3, Cor. 4.10], combined with the powers statement of Abdulali [2, Thm 4.1] (see [1, Rem. 4.2]). Van Geemen and Verra obtain the generic Hodge group from surjectivity of the genus-3 Prym map [3, §3]. [3, Cor. 4.10] gives the quaternionic cycles in every genus. Our contribution for $Q_8$ is the full Hodge group on $\varphi$-redundant components in every genus $g\ge3$, from arithmetic monodromy.

**Theorem C (with cm_symmetry_scout).** For $g=3$ the Prym map from the Hurwitz component of the $\varphi$-redundant Dic$_3$-cover to the 6-dimensional PEL component of $D_3$-type eightfolds with polarization datum $(D_3^4,T_0)$ is dominant. Consequently:
- every such eightfold is $D_3$-linearly isogenous to a Dic$_3$-Prym;
- $W_{D_3}$ is algebraic on all of them;
- HC holds for all powers at points with full Hodge group;
- the same holds for each half-spin Kuga–Satake factor $A_\pm$ of $(U^2\oplus A_2(-1)^2)_{\mathbf Q}$ (§5).

**Theorem D.** For every imaginary quadratic field $K$ there is a finite group $H=\mathbf Z/m\rtimes\mathrm{Gal}(\mathbf Q(\zeta_m)/K)$ whose unramified covers yield, for every sufficiently large $g$ (every $g\ge3$ when $K=\mathbf Q(\sqrt{-p})$, $p\equiv3\pmod 4$ prime), a family of abelian varieties of split Weil type for $K$, parametrized by a $(3g-3)$-dimensional Hurwitz component, of dimension $\varphi(m)(g-1)$, with full Hodge group at the very general point and HC for all powers there (§6).

## 2. Weil classes (Theorem A′)

**Lemma 2.1 (Schoen).** Let $Z\to Z/\langle\sigma\rangle$ be a connected unramified cyclic cover of degree $m$ of a curve of genus $\gamma\ge2$, and let $B\subset\mathrm{Alb}(Z)$ be the abelian subvariety on whose tangent space $\langle\sigma\rangle$ acts by primitive characters. Then the Weil space $U'\subset H^h(B,\mathbf Q)$ of $B$, for the cyclotomic field $\mathbf Q(\mu_m)$ with $h=\dim_{\mathbf Q(\mu_m)}H^1(B,\mathbf Q)$, is spanned by classes of algebraic cycles.

*Proof.* This is [9, Cor. 3.1]. Its hypothesis is that of [9, Thm 2.0]: the $(\mathbf Z/m)^*$-orbit of the branch tuple is simple. For an unramified cover the tuple is empty ($r=0$), hence simple. By Chevalley–Weil every primitive character occurs in $H^{1,0}(Z)$ with multiplicity $\gamma-1=h/2$, which is the condition of [9, Cor. 1.9] under which $U'$ is of Hodge type. ∎

*Proof of Theorem A′.*
- **Two cyclic covers.** Apply Lemma 2.1 twice with $m=4$:
  - to $X\to X/\langle b\rangle$. Here $b$ acts on $\mathrm{Alb}(X)$ by primitive characters $\pm i$ exactly on $e_-$, so $B=P$ and $W_K(P)$ is algebraic.
  - to $Y=X/H\to C$, a cyclic cover with group $G/H\cong\mathbf Z/4$ generated by $bH$. Its primitive part pulls back to $P_i$ up to isogeny, so $W_K(P_i)$ is algebraic. This auxiliary cover is due to cm_symmetry_scout [10].
- **Contraction.** Let $\phi\colon P_D\times P_i\to P$ be a $K$-linear isogeny. Over $\mathbf C$, the pullback $\phi^*W_K(P)$ is spanned by $w_{D,+}\otimes w_{i,+}$ and $w_{D,-}\otimes w_{i,-}$, where $w_{\bullet,\pm}$ span the two determinant lines.
  - On $W_K(P_i)$, cup product pairs $w_{i,+}$ with $w_{i,-}$ nondegenerately, and same-sign products vanish.
  - So the algebraic map $\alpha\otimes\beta\mapsto(\mathrm{pr}_D)_*(\phi^*\alpha\cup\mathrm{pr}_i^*\beta)$ from $W_K(P)\otimes W_K(P_i)$ to $W_K(P_D)$ is surjective, and $W_K(P_D)$ is algebraic.
- **Quaternionic translates.** $D_3$ is definite and contains $K=\mathbf Q(u)$. By [3, Prop. 4.7, Cor. 4.9] (after Abdulali), $W_{D_3}(P_D)$ is spanned by the $D_3^\times$-translates of $W_K(P_D)$. Those translates are images under endomorphisms in $\mathrm{End}(P_D)\otimes\mathbf Q$. After clearing denominators these are actual endomorphisms, which change the classes by nonzero rational factors. ∎

## 3. The Hodge group (Theorem A(i))

**Proposition 3.1 (arithmetic monodromy).** Let $g\ge3$ and $p$ be $\varphi$-redundant. The monodromy of $\mathbb V_D$ on $\mathcal H_p$ is commensurable with $G^1(\mathcal O)$, where $G^1$ is the group of $D_3$-linear automorphisms of reduced norm 1 of $(D_3^{2g-2},\langle\,,\rangle)$, with the standard skew-Hermitian form $\langle x_i,y_j\rangle=\delta_{ij}$. Moreover $G^1$ is $\mathbf Q$-simple of type $D_{2(g-1)}$, with $G^1_{\mathbf C}\cong SO(4(g-1))$ and $G^1(\mathbf R)\cong SO^*(4(g-1))$.

*Proof.*
- **The setup of [5].** In [5], $\Gamma_{G,p}=\{\gamma\in\mathrm{Aut}(\pi_1)^+:p\circ\gamma=p\}$ acts on $R^{\rm ab}\otimes\mathbf Q\cong\mathbf Q^{2g}\oplus\bigoplus A_i^{2g-2}$, where $R=\ker p$ [5, Prop. 1.1]. This is the action on $H_1(X,\mathbf Q)$ of the stabiliser of the cover in $\mathrm{Mod}(\Sigma_g,*)$, i.e. the monodromy of $R^1f_*\mathbf Q$ over the pointed Hurwitz space. By [5, §8.2] these representations descend, after passing to finite index, to $\mathrm{Mod}(\Sigma_g)$, i.e. to the monodromy over $\mathcal H_p$. Commensurability is unaffected.
- **Arithmeticity.** By [5, Thm 1.6], for $g\ge3$ and $p$ $\varphi$-redundant, the image on the $A_i$-factor is commensurable with $G^1_{G,i}(\mathcal O_i)$, where $G_{G,i}$ is the automorphism group of the skew-Hermitian $A_i$-module of [5, Thm 1.8].
- **The $D_3$-factor.** For $A_i=D_3$ we have $A_i\otimes\mathbf R\cong\mathbf H$, so by [5, Thm 1.7] the complex group is $O((2g-2)\cdot2)$.
- **Type.** The real form is that of a skew-Hermitian form over $\mathbf H$, namely $SO^*(N)$, $N=4(g-1)$. Its complexification $SO_N$, with $N\ge8$, is absolutely almost simple, and $SO^*(N)$ has no compact simple factor. ∎

*Proof of Theorem A(i).*
- **Upper bound.** $\mathrm{Hg}(P_D)$ commutes with $D_3$ and preserves the polarization, so $\mathrm{Hg}(P_D)\subseteq\mathrm{Aut}_{D_3}(H^1(P_D),\langle\,,\rangle)^0$. This identity component is $G^1$: the reduced norm takes values $\pm1$ on the automorphism group of a quaternionic skew-Hermitian form, so the identity component lies in $G^1$, and $G^1$ is connected, being of type $SO^*$.
- **Monodromy.** By Borel density, the arithmetic subgroup $G^1(\mathcal O)$ is Zariski dense in $G^1$, because $G^1$ is absolutely almost simple and $G^1(\mathbf R)$ has no compact factor. So the Zariski closure of the monodromy contains $G^1$, up to finite index.
- **Lower bound.** By Deligne and André [11, §5 Thm 1], the identity component of the algebraic monodromy group is a normal subgroup of the derived Mumford–Tate group of a very general fibre. Hence $G^1\subseteq\mathrm{Hg}(P_D)^{\rm der}\subseteq\mathrm{Hg}(P_D)\subseteq G^1$, so $\mathrm{Hg}(P_D)=G^1$ and $\mathrm{Hg}(P_D)_{\mathbf C}\cong SO(4(g-1))$.
- **Endomorphisms.** $\mathrm{End}^0(P_D)$ is the commutant of $G^1$ in $\mathrm{End}(H^1)$, which is $D_3$. So $P_D$ is simple of type III. ∎

## 4. Proof of Theorem A(ii) and Theorem B

- **The Hodge ring of $P_D$.** By Theorem A(i), $P_D$ has the full type III Hodge group, $\mathrm{Hod}(P_D)_{\mathbf C}\cong SO(2n)$ with $2n=4(g-1)$. This suffices for the invariant theory below even though $\mathcal H_p$ may be a proper subfamily of the PEL moduli space. The case $r=1$ of the argument below gives $B^\bullet(P_D)=$ (algebra generated by divisors) $+\,W_{D_3}\cdot$(divisors), which is Abdulali's description [2, Thm 4.1], [3, §4.8]. By Theorem A′ these classes are algebraic.
- **Powers ($r\ge1$).** The following direct argument is due to cm_symmetry_scout [10]. It is also a consequence of [2, Thm 4.1] (see [1, Rem. 4.2]), but it does not depend on that paper.
  - **Setup.** Over $\mathbf C$ write $H^1(P_D)=W\otimes U$, with $\dim W=N=4(g-1)$ carrying $SO(W)=\mathrm{Hg}_{\mathbf C}$, and $\dim U=2$ carrying $D_3\otimes\mathbf C=M_2(\mathbf C)$. Then $H^1(P_D^r)=W\otimes U_r$ with $U_r=U^{\oplus r}$.
  - **Invariants.** By the first fundamental theorem for $SO(W)$, followed by antisymmetrisation, $\bigl(\bigwedge^\bullet(W\otimes U_r)\bigr)^{SO(W)}$ is generated by metric contractions, which lie in degree 2, and by the determinant pieces $\det W\otimes\mathrm{Sym}^NU_r\subset\bigwedge^N(W\otimes U_r)$.
  - **Divisors.** Degree-2 Hodge classes are divisors.
  - **Determinants on one copy.** For $r=1$, $\det W\otimes\mathrm{Sym}^NU$ is $W_{D_3}(P_D)\otimes\mathbf C$, which is algebraic by Theorem A′.
  - **Determinants on powers.** For general $r$, homomorphisms $P_D^r\to P_D$ with entries in $\mathrm{End}(P_D)$ pull $W_{D_3}(P_D)$ back to algebraic classes. For a row $(d_1,\dots,d_r)\in D_3^r$, clear a common denominator to get an actual homomorphism; this scales the degree-$N$ pullback by a nonzero integer. The rational points of $D_3^r$ are Zariski dense in $\mathrm{Hom}_{\mathbf C}(U,U_r)$, so the complex linear span of the pullbacks contains every pure power $u^{N}$, $u\in U_r$. These span $\mathrm{Sym}^NU_r$.
  - **Conclusion.** So every generator is algebraic, and HC holds for $P_D^r$. Rational descent is automatic, because the algebraic classes span a $\mathbf Q$-subspace whose complexification is everything.
- **Theorem B.** For $Q_8$ the faithful component of $\mathbf Q[Q_8]$ is $\mathbf H_{\mathbf Q}$.
  - [3, Cor. 4.10] gives algebraicity of $W_{\mathbf H}$ for every quaternionic Prym, in every genus. Schoen is applied to the free cyclic cover by $\langle i\rangle$, whose primitive part is the whole quaternionic Prym, so no helper is needed.
  - $Q_8$ is 2-generated and $\mathbf H_{\mathbf Q}\otimes\mathbf R=\mathbf H$. So Proposition 3.1 and the proof of Theorem A(i) apply verbatim with the $\varphi$-redundant map $\alpha_1\mapsto i$, $\alpha_2\mapsto j$. ∎

## 5. Genus-three dominance and the two Kuga–Satake factors (with cm_symmetry_scout)

In this section $D=D_3$ and
$$T_0=\begin{pmatrix}0&I_2\\-I_2&0\end{pmatrix}.$$
Let $\mathcal A_{D,T_0}$ be the connected quaternionic PEL component determined by the rational polarized module $(D^4,T_0)$ of the genus-three redundant-cover construction and a compatible integral lattice. Its complex dimension is six. Changing the integral lattice within the rational datum will be handled by isogenies.

**Theorem C (dominance).** The Prym map from the Hurwitz component of the standard genus-three redundant cover to $\mathcal A_{D,T_0}$ is dominant. Its differential is an isomorphism on a nonempty open set.

*Proof.* Let $C_0$ be a stable curve consisting of four copies of $\mathbf P^1$, with dual graph $K_4$. Orient its six edges in the order $01,02,03,12,13,23$. On each normalization component take the trivial regular $G$-cover, and glue along the edges with labels
$$1,1,1,a,b,a.$$
This is a finite étale cover $X_0\to C_0$ of nodal curves: the completed local map at each node is an isomorphism, and all node stabilizers are trivial. It is connected because the labels generate $G$. There are $48$ components and $72$ nodes upstairs, so its arithmetic genus is $25=12(3-1)+1$. The graph fundamental group is $F_3$, with label tuple $(a,b,a)$. The Nielsen transformation $x_3\mapsto x_3x_1^{-1}$ changes this to $(a,b,1)$. The resulting surface-group cover is therefore in the standard $\varphi$-redundant component.

The cover smooths to an unramified $G$-cover by using the same plumbing parameter at all twelve lifts of each base node. Equivalently, finite étale covers lift over nilpotent thickenings and algebraize in a proper deformation. All six nodes may be smoothed. We compute an invertible limit of the smooth-fiber codifferential.

Use the faithful representation over $\mathbf Q(i)$
$$A=\begin{pmatrix}1/2&1\\-3/4&1/2\end{pmatrix},\qquad B=\begin{pmatrix}i&0\\0&-i\end{pmatrix}.$$
It preserves $\epsilon=\left(\begin{smallmatrix}0&1\\-1&0\end{smallmatrix}\right)$. Let $L$ be the associated rank-two bundle, trivial on each normalization component and glued by the indicated matrices. Its global sections would be common fixed vectors of $A,B$, so $H^0(C_0,L)=0$. Self-duality and Riemann–Roch give
$$H^1(C_0,\omega_{C_0}\otimes L)=0,\qquad \dim H^0(C_0,\omega_{C_0}\otimes L)=4.$$
These spaces therefore form a rank-four bundle in the smoothing.

For a smooth fiber, write $M=H^0(C,\omega_C\otimes L)$. The faithful isotypic holomorphic forms are $\rho\otimes M$. Since $\rho$ is irreducible symplectic,
$$\bigl(\operatorname{Sym}^2(\rho\otimes M)\bigr)^G=\bigwedge^2\rho\otimes\bigwedge^2M\simeq\bigwedge^2M.$$
The invariant part of the standard infinitesimal period formula is consequently
$$\mu:\bigwedge^2M\longrightarrow H^0(C,\omega_C^2),\qquad s\wedge t\longmapsto\epsilon(s,t),$$
up to a nonzero trace scalar. This is the codifferential of the quaternionic Prym map; compare [3, §3.2]. Both spaces have dimension six.

At $C_0$, a section of $\omega\otimes L$ is specified by its vector residues $r_0,\ldots,r_5$ at the tails of the six oriented edges. A head residue is minus the gluing matrix times its tail residue. The sum of the three residues on each rational component is zero. Set $r_4=u$, $r_5=v$, and
$$C=(I-A)^{-1}(B-I).$$
The complete solution of the four vertex constraints is
$$\begin{aligned}
r_0&=(C+I)u-v,& r_1&=-ACu+(A+I)v,\\
r_2&=-Bu-Av,& r_3&=Cu-v,\\
r_4&=u,& r_5&=v.
\end{aligned}$$
The four coordinates of $u,v$ give a basis of $M_0$. On each rational component $\omega_{C_0}^2$ restricts to $\mathcal O_{\mathbf P^1}(2)$, and evaluation at its three nodes is an isomorphism. Thus $H^0(C_0,\omega_{C_0}^2)$ is parametrized by the six matching leading double-pole coefficients. Under multiplication, the coefficient at edge $e$ is $\det(r_e,s_e)$. The coefficients agree on both sides because the gluing matrices have determinant one.

In the exterior basis with column pairs $01,02,03,12,13,23$, the resulting matrix is
$$\begin{pmatrix}
2&3(1-i)/4&-(1+i)/2&(1-i)/2&1+i&1\\
2&3(1-i)/2&0&0&2+2i&3\\
1&-3i/4&i/2&i/2&i&1\\
2&3(1-i)/4&(1-i)/2&-(1+i)/2&1+i&1\\
1&0&0&0&0&0\\
0&0&0&0&0&1
\end{pmatrix}.$$
Its determinant is $-3$.

The bundles of twisted differentials and quadratic differentials, and their multiplication map, extend across this stable smoothing. We use $H^0(\omega_{C_0}^2)$ as the limit of the smooth targets, without identifying it with the ordinary cotangent space at a boundary point. The nonzero limiting determinant implies invertibility on nearby smooth fibers, where multiplication is the period-map codifferential. The Prym map has rank six and hence is dominant. ∎

**Corollary C.1 (transport in the rational datum; the method of [7, §9.3]).** Let $B$ be an abelian eightfold of quaternion type whose rational polarized $D$-module is isometric to $(D^4,T_0)$, with period in the connected domain under consideration. Then $B$ is $D$-linearly isogenous to a Prym of a smooth unramified $\mathrm{Dic}_3$-cover. The space $W_D(B)$ is algebraic. If $B$ has full Hodge group, the Hodge conjecture holds for every power of $B$.

*Proof.* Theorem C gives a nonempty analytic open $O$ in the period image of actual smooth covers. Rational points of the $D$-linear isometry group are dense in its real identity component: the Cayley transform
$$X\longmapsto(1-X)(1+X)^{-1}$$
gives a rational chart about the identity from skew-adjoint endomorphisms, whose rational points are dense in their real points. The closure of the rational isometries therefore contains a neighborhood of the identity and hence the real identity component. This component acts transitively on the period domain $SO^*(8)/U(4)$. Some rational isometry moves the period of $B$ into $O$, giving an exact rational weight-one Hodge isomorphism with a smooth-cover Prym. It comes from an algebraic isogeny after clearing denominators. Theorem A′ transports $W_D$, and the invariant-theory proof of §4 applies at full-group points. Extra Hodge classes at group drops are not included in the last claim. ∎

**Proposition C.2 (the rational KS polarization).** Let
$$V_3=(U^2\oplus A_2(-1)^2)_{\mathbf Q}.$$
Each minimal quaternionic half-spin factor of its Kuga–Satake variety has rational polarized $D$-module isometric to $(D^4,T_0)$.

*Proof.* The even Clifford algebra is $M_4(D)\times M_4(D)$ by [12, Thm 7.7]. More explicitly, $A_2(-1)^2$ is rationally $-2$ times the norm form $\langle1,1,3,3\rangle$ of $D$; its two even Clifford components are $D$, and adding two hyperbolic planes supplies the matrix factors. Each minimal rational half-spin module $H$ has dimension sixteen over $\mathbf Q$ and rank four over its commuting $D$. Its Spin-invariant polarization $\psi$ has canonical Rosati involution on $D$, and corresponds to a nondegenerate $D$-valued skew-Hermitian form $T$; see [3, §6.2] for the quaternion-type factors.

Choose rational isotropic $e,f$ in a hyperbolic plane with $ef+fe=1$. In the Clifford algebra $p=ef$ is idempotent and $1-p=fe$. The elements
$$\lambda(t)=tp+t^{-1}(1-p)$$
define a rational cocharacter of $\operatorname{Spin}(V_3)$: reversion exchanges $p,1-p$, giving norm one, and conjugation scales $e,f$ by $t^2,t^{-2}$ and fixes their orthogonal complement. Thus
$$H=pH\oplus(1-p)H$$
is a $D$-stable decomposition of weights $1,-1$. Invariance makes both summands $\psi$-isotropic; nondegeneracy pairs them perfectly, so each has $D$-rank two. They are also $T$-isotropic, since $\psi(du,v)=0$ for every $d\in D$ and the reduced-trace pairing on $D$ is nondegenerate. Choosing dual $D$-bases yields the matrix $T_0$. A rational scalar multiple is isometric by rescaling one of the two summands. ∎

**Corollary C.3 (individual KS factors).** For the two minimal factors $A_+,A_-$ of the Kuga–Satake family on $V_3$, the spaces $W_D(A_+),W_D(A_-)$ are algebraic. At periods with full generic Spin Hodge group, the Hodge conjecture holds for every power of either factor separately.

*Proof.* Apply Corollary C.1 to Proposition C.2. The prescribed polarization period of each factor belongs to the corresponding quaternionic domain, and full Spin image gives its full type-III Hodge group. ∎

The two half-spin modules are generically inequivalent. The result does not establish the Hodge conjecture on $A_+\times A_-$, on the full Kuga–Satake variety $(A_+\times A_-)^4$, or for the K3-to-Kuga–Satake correspondence. Those assertions require mixed cycles beyond the separate invariant rings.

The supplement's `check_Dic3_genus3_nodal_period.py` verifies the group, connected stable cover graph, residue constraints and determinant using exact arithmetic. The geometric smoothing and infinitesimal-period arguments are supplied above.

## 6. Weil type for every imaginary quadratic field (Theorem D)

Let $K$ be an imaginary quadratic field and $m\ge3$ with $K\subset\mathbf Q(\zeta_m)$ (Kronecker–Weber). Put $S=\mathrm{Gal}(\mathbf Q(\zeta_m)/K)$, a subgroup of index 2 in $(\mathbf Z/m)^\times$, and $H=\mathbf Z/m\rtimes S$ with the natural action. Let $h$ generate $\mathbf Z/m$, and set $s=|S|=\varphi(m)/2$. In the prime case $m=p\equiv3\pmod4$ we have $K=\mathbf Q(\sqrt{-p})$, $S$ is the group of squares, and $H$ is the Frobenius group of order $p(p-1)/2$.

**Lemma 6.1 (the component).**
- The irreducible representations of $H$ that restrict to primitive characters of $\mathbf Z/m$ are exactly two, $\rho_\sigma=\mathrm{Ind}_{\mathbf Z/m}^H\chi$ and its complex conjugate $\rho_{\bar\sigma}$, of dimension $s$ and character field $K$.
- The corresponding simple component of $\mathbf Q[H]$ is $A\cong M_s(K)$.
- The element $h$ acts on $\rho_\sigma$ with the $s$ eigenvalues $\zeta^t$, $t\in S$, and on $\rho_{\bar\sigma}$ with the eigenvalues $\zeta^{-t}$. On every irreducible representation outside the pair $\rho_\sigma,\rho_{\bar\sigma}$ it acts through non-primitive roots of unity.

*Proof.*
- **Irreducibility.** The primitive characters $\chi_t$, $t\in(\mathbf Z/m)^\times$, form two $S$-orbits $S$ and $cS$, where $c=-1\notin S$ since $K$ is imaginary. Each orbit has trivial stabiliser in $S$, so induction gives an irreducible representation of dimension $s$.
- **Character field.** The character values are Gauss periods $\sum_{t\in S}\zeta^{kt}$, which lie in $K$. Since $-1\notin S$, the two induced representations are distinct complex conjugates with unequal characters, so the character field is exactly $K$. (The value at the identity is real; some other values are not.)
- **Schur index.** The crossed product $\mathbf Q(\zeta_m)*S$ has trivial cocycle because the extension splits. It is therefore $\mathrm{End}_K(\mathbf Q(\zeta_m))\cong M_s(K)$.
- **Other irreducibles.** Every irreducible representation outside the pair $\rho_\sigma,\rho_{\bar\sigma}$ restricts to characters of $\mathbf Z/m$ of order less than $m$, so $h$ acts on it non-primitively.
- `check_frobenius_exact.py` confirms all of this in exact arithmetic for $p=7,11,19$. ∎

Let $X\to C$ be an unramified $H$-cover of a curve of genus $g\ge2$, and let $P_A$ be the $A$-isotypic Prym. By Chevalley–Weil, $H^1(P_A)_{\mathbf C}=\rho_\sigma\otimes M_\sigma\oplus\rho_{\bar\sigma}\otimes M_{\bar\sigma}$, where:
- $\dim M_\sigma=N=2s(g-1)$;
- $M_\sigma^{1,0}$ has dimension $N/2$, so the structure is balanced;
- $M_{\bar\sigma}\cong M_\sigma^\vee$ via the polarization.

An idempotent of $M_s(K)$ cuts out $B$ with $P_A\sim B^s$ and $H^1(B)_{\mathbf C}=M_\sigma\oplus M_{\bar\sigma}$. Thus $B$ is of balanced Weil type for $K$, of dimension $N=\varphi(m)(g-1)$, on every cover.

**Theorem D.** Let $g\ge3$ with $g>d(H)$, and let $p\colon\pi_1(\Sigma_g)\twoheadrightarrow H$ be $\varphi$-redundant. For $C$ very general in the Hurwitz component of $p$:
- the $K$-Hermitian form of $B$ is hyperbolic by [5, Lemmas 5.4–5.5], so $B$ is of split Weil type;
- $\mathrm{Hg}(B)_{\mathbf C}\cong SL_N$ and $\mathrm{End}^0(B)=K$;
- HC holds for every power of $B$ and of $P_A$.

The $K$-Weil classes $W_K(B)$ are algebraic for every unramified $H$-cover. In the prime case with $p\ge7$ we have $d(H)=2$; for $p=3$, $H=C_3$ and $d(H)=1$. Either way every $g\ge3$ is allowed. For example, $p=7$, $g=3$ gives Weil-type twelvefolds for $\mathbf Q(\sqrt{-7})$.

*Proof.*
- **Schoen.** The cover $X\to X/\langle h\rangle$ is unramified and cyclic of degree $m$. By Lemma 6.1 its primitive part is exactly $P_A$. By Lemma 2.1 the $\mathbf Q(\zeta_m)$-Weil space of $P_A$ is algebraic. Over $\mathbf C$ its pieces are $\det V_{\zeta^t}=\ell_t^{\otimes N}\otimes\det M_\sigma$ and their conjugates, where $\ell_t\subset\rho_\sigma$ is the $\zeta^t$-eigenline of $h$.
- **Descent to $K$.** $GL_s(K)=A^\times$ acts on $P_A$ through endomorphisms; clear denominators to make them actual endomorphisms. $GL_s(K)$ is Zariski dense in $\mathrm{Res}_{K/\mathbf Q}GL_s$, whose complex points $GL_s\times GL_s$ act on $\rho_\sigma\oplus\rho_{\bar\sigma}$.
  - The complex span of the translates of Schoen's classes is a $GL_s\times GL_s$-subrepresentation of $(\mathrm{Sym}^N\rho_\sigma\otimes\det M_\sigma)\oplus(\mathrm{Sym}^N\rho_{\bar\sigma}\otimes\det M_{\bar\sigma})$.
  - It meets both summands, which are non-isomorphic irreducibles. So it is everything, and these generalized $K$-Weil classes are algebraic.
  - Restricting along $B\hookrightarrow P_A$ sends them onto $W_K(B)\otimes\mathbf C=\det M_\sigma\oplus\det M_{\bar\sigma}$.
- **Hodge group.**
  - By [5, Thms 1.6, 1.7], the monodromy on the $A$-part is commensurable with $G^1(\mathcal O)$, where $G^1$ is the special unitary group of a $K$-Hermitian space of dimension $N$. Its signature is $(N/2,N/2)$, forced by the Hodge numbers.
  - $G^1$ is a $\mathbf Q$-form of $SL_N$, absolutely almost simple, with $G^1(\mathbf R)=SU(N/2,N/2)$ non-compact. Borel density and [11, §5 Thm 1] give $G^1\subseteq\mathrm{Hg}(B)^{\rm der}$.
  - Conversely, $\mathrm{Hg}(B)$ lies in the unitary group, since it commutes with $K$ and preserves the polarization. It fixes the Hodge classes $W_K(B)$, so it lies in $G^1$.
- **Powers.** Over $\mathbf C$, $H^1(B^r)=(\mathbf C^r\otimes M)\oplus(\mathbf C^r\otimes M^\vee)$ with $SL(M)$ acting.
  - By the first fundamental theorem for $SL_N$, followed by antisymmetrisation, the invariants of $\bigwedge^\bullet$ are generated by contractions of $M$ with $M^\vee$, which are degree-2 classes and hence divisors, and by $\mathrm{Sym}^N(\mathbf C^r)\otimes\det M$ and its conjugate.
  - Pullbacks of $W_K(B)$ along rows in $\mathrm{End}(B)^r=K^r$ span these, by the density argument of §4 with $K$ in place of $D_3$. So HC holds for $B^r$, and for $P_A^r\sim B^{rs}$. ∎

*Comparison.*
- Markman proves algebraicity of Weil classes on all Weil-type fourfolds and on sixfolds of split Weil type with discriminant $-1$ [13], [14].
- OpenAI's preprint [15] (6 Oct 2026, not refereed) claims HC for all powers of abelian varieties of dimension $\le5$ with an imaginary-quadratic action, and of sixfolds with a hyperbolic Hermitian form of signature $(3,3)$. Among the families of Theorem D, which have dimension $\varphi(m)(g-1)$ with $g\ge3$, this overlaps only with $\varphi(m)=2$ ($K=\mathbf Q(\sqrt{-3})$ or $\mathbf Q(i)$) and $g\in\{3,4\}$, i.e. dimensions $4$ and $6$. For $\varphi(m)\ge4$ every member has dimension $\ge8$, and for $K=\mathbf Q(\sqrt{-7})$ dimension $\ge12$.
- Schoen's method gives HC at a general point of a full Weil moduli space only for $(h,m)\in\{(4,3),(4,4),(6,3),(6,4)\}$ [9, §3]. His count requires the curves to dominate the full Weil moduli space.
- The Prym image of the Hurwitz component of Theorem D has dimension at most $3g-3$. It has positive codimension in the Weil moduli space whenever $3g-3<s^2(g-1)^2$, in particular for $s\ge2$ and $g\ge3$. (We do not infer the image dimension from full monodromy.) Arithmetic monodromy makes the generic Hodge group full in any case, which is all the invariant theory needs.
- Schoen already recognizes positive-codimension loci of generalized Pryms [9, §3]. What is new here is the descent from $\mathbf Q(\zeta_m)$ to $K$ through a non-abelian group, combined with GLLM monodromy.

**Remark 6.2 (abelian CM fields).** The argument of this section does not use $[K:\mathbf Q]=2$. Let $L\subset\mathbf Q(\zeta_m)$ be any CM field, $S=\mathrm{Gal}(\mathbf Q(\zeta_m)/L)$ (so $-1\notin S$), $e=[L:\mathbf Q]/2$, and $H=\mathbf Z/m\rtimes S$. Then:
- **The component.** The primitive characters of $\mathbf Z/m$ form $2e$ free $S$-orbits. The corresponding component is $A=M_s(L)$ with $s=|S|$, and the Schoen primitive part of $X\to X/\langle h\rangle$ is again exactly $P_A$.
- **The variety.** $P_A\sim B^s$, where $B$ is of balanced Weil type for $L$, of dimension $\varphi(m)(g-1)$. Here $H^1(B)_{\mathbf C}=\bigoplus_{\sigma\in\mathrm{Hom}(L,\mathbf C)}M_\sigma$ with $\dim M_\sigma=2s(g-1)$.
- **Weil classes.** $W_L(B)\otimes\mathbf C=\bigoplus_\sigma\det M_\sigma$ has dimension $[L:\mathbf Q]$. These classes are algebraic by the same descent, now using density of $GL_s(L)$ in $\mathrm{Res}_{L/\mathbf Q}GL_s$, whose complex points $\prod_\sigma GL_s$ act on the pairwise non-isomorphic summands.
- **Hodge group.** On the Hurwitz component of a specified $\varphi$-redundant $p$, the monodromy group is $\mathrm{Res}_{L_0/\mathbf Q}SU$ of signature $(N/2,N/2)$ at every real place of the totally real subfield $L_0$. It is $\mathbf Q$-simple with non-compact real points.
- **Invariants.** Choose one representative $\sigma$ from each of the $e$ conjugate pairs, with $M_{\bar\sigma}\cong M_\sigma^\vee$. The first fundamental theorem for the $e$ factors $\prod SL(M_\sigma)$ shows that the Hodge ring of every power is generated by divisors and the generalized $L$-Weil classes. The complexified determinant space has $2e$ lines and rational dimension $[L:\mathbf Q]$.

So the conclusion of Theorem D holds for every abelian CM field $L$, for very general $C$ in the Hurwitz component of a specified $\varphi$-redundant $p$, with $g\ge3$ and $g>d(H)$. For $S=1$ ($H$ cyclic) this is the case of very general primitive Pryms of cyclic covers, where Schoen's cycles are classical [9]. Their extension to all non-trivial characters of finite Abelian étale covers is due to Patel–Zhang [16]. OpenAI's preprint [17] (not refereed) claims HC for all powers of the whole Jacobian at tensor Hodge-generic points of the full marked variation of any connected Abelian cover, which would cover the case $S=1$. It does not apply directly for $S\ne1$: there the curves $X/\langle b\rangle$, or $X/\mathbf Z/m$, move in a constrained family with an extra non-Abelian symmetry, not in the full marked variation of the Abelian cover. Our check for $L$ of degree 4 inside $\mathbf Q(\zeta_{13})$ and $\mathbf Q(\zeta_{21})$ is `check_abelian_cm.py`.

## 7. Every definite quaternion algebra over $\mathbf Q$ (Theorem E, with cm_symmetry_scout)

Let $D$ be a definite quaternion algebra over $\mathbf Q$, with finite ramification set $S$ of odd cardinality $r$. Put
$$m=2\prod_{p\in S}p,\qquad d=\varphi(m),\qquad s=d/2,\qquad N=2d(g-1).$$
So $4\mid m$ when $2\in S$.

**Theorem E (cm_symmetry_scout; written with lelong_ridge).** There is a finite group $G_D$ of order $md$, with a normal cyclic subgroup $H$ of order $m$ and $G_D/H\cong\mathrm{Gal}(\mathbf Q(\zeta_m)/\mathbf Q)$, such that $e_{\rm prim}\mathbf Q[G_D]\cong M_s(D)$. Here $e_{\rm prim}$ is the central idempotent of the faithful characters of $H$, and $G_D$ has at most $r+1$ generators. Let $g\ge r+2$, and let the defining surjection $\pi_1(\Sigma_g)\twoheadrightarrow G_D$ be $\varphi$-redundant, with Hurwitz component $\mathcal H$. Then there is an abelian factor $B$ of $J(X)$, up to isogeny, with:
- (i) $\dim B=N$, and $B$ carries a polarized $D$-action with canonical Rosati involution;
- (ii) the algebraicity assertion holds for every connected unramified $G_D$-cover of a curve of genus $g\ge2$, with $N=2d(g-1)$, independently of the redundancy hypothesis: the $(N+1)$-dimensional quaternionic class space $W_D(B)\subset H^N(B,\mathbf Q)$ is algebraic;
- (iii) for very general $C$ in $\mathcal H$, $\mathrm{Hg}(B)_{\mathbf C}=SO_N$ and $\mathrm{End}^0(B)=D$, and HC holds for every power of $B$.

For $S=\{2\}$ and $S=\{3\}$ this recovers Theorems B and A, where $G_D=Q_8$ and $\mathrm{Dic}_3$.

**7.1 The prime components.** For odd $p$ put $n=p-1$, choose a primitive root $u$ mod $p$, and set
$$G_p=\langle a,b\mid a^p=b^{2n}=1,\ bab^{-1}=a^u\rangle\cong C_p\rtimes C_{2n}.$$
Its central involution is $z=b^n$, and $H_p=\langle a,z\rangle$ is cyclic of order $2p$. With $e_0=p^{-1}\sum_{j=0}^{p-1}a^j$, the faithful-character idempotent $e_p=\tfrac12(1-z)(1-e_0)$ of $\mathbf Q[H_p]$ is central in $\mathbf Q[G_p]$, and
$$A_p=e_p\mathbf Q[G_p]\cong(\mathbf Q(\zeta_p)/\mathbf Q,\ \sigma_u,\ -1),$$
the cyclic algebra with $bxb^{-1}=\sigma_u(x)$ and $b^n=-1$. Its local invariants:
- **At $\ell\ne p$ finite.** $\mathbf Q_\ell(\zeta_p)$ is unramified, so the unit $-1$ is a norm, and the invariant is $0$.
- **At $p$.** The extension is totally tamely ramified. Norms of units reduce to $n$-th powers in $\mathbf F_p^\times$, i.e. to $1$, so $-1$ is not a norm. Since $(-1)^2$ is a norm, the invariant is $1/2$.
- **At $\infty$.** The decomposition group is $\mathbf C/\mathbf R$, and $-1$ is not a norm, so the invariant is $1/2$.

Hence $A_p\cong M_{n/2}(D_p)$ with $\mathrm{Ram}(D_p)=\{p,\infty\}$. The unique complex irreducible representation in $A_p$ has dimension $n$, character $\chi(a^jb^k)=(-1)^{k/n}n$ for $k\equiv0\ (n)$, $j=0$; $-(-1)^{k/n}$ for $k\equiv0\ (n)$, $j\ne0$; and $0$ otherwise. Its Frobenius–Schur indicator is $-1$. For $p=2$ take $G_2=Q_8$, $H_2=\langle i\rangle$ and $A_2=\mathbf H_{\mathbf Q}$.

**7.2 The central product.** Put $G_D=\bigl(\prod_{p\in S}G_p\bigr)/\langle z_pz_q^{-1}\rangle$.
- **$H$ is cyclic.** The image $H$ of $\prod H_p$ is cyclic of order $m$: the odd parts have coprime orders and the involutions are identified.
- **The quotient.** $G_D/H$ acts on $H$ as the full automorphism group, so $G_D/H\cong\mathrm{Gal}(\mathbf Q(\zeta_m)/\mathbf Q)$, and it permutes the faithful characters of $H$ transitively. Hence $e_{\rm prim}\mathbf Q[G_D]=\bigotimes_pA_p$ is central simple of degree $d$, with Brauer class $\sum_p[D_p]$. Since $r$ is odd, this class has invariant $1/2$ exactly at $S\cup\{\infty\}$, i.e. it is $[D]$, and so $e_{\rm prim}\mathbf Q[G_D]\cong M_s(D)$.
- **Generators.** A generator of $H$ and one complement lift per prime generate $G_D$, so $d(G_D)\le r+1$.

**7.3 The factor $B$.** Let $X\to C$ be connected and unramified with group $G_D$, and put $P=e_{\rm prim}J(X)$.
- By Chevalley–Weil, $H^1(P,\mathbf Q)\cong M_s(D)^{\oplus(2g-2)}$ and $\dim P=d^2(g-1)$.
- The canonical polarization induces the positive involution $\gamma\mapsto\gamma^{-1}$, of symplectic type, on $M_s(D)$. It is adjoint to a positive definite $D$-Hermitian form $h$ on $D^s$: average a standard form $h_0$ over the finite group acting on the right $D$-module $D^s$. For $v\in D^s$ rational, put $f(w)=v\,h(v,v)^{-1}h(v,w)$. Since $h(v,v)\in\mathbf Q_{>0}$, $f$ is a self-adjoint primitive idempotent, and the induced involution on $fM_s(D)f=D$ is canonical.
- Put $B=fP$. Then $P\sim B^s$ and $\dim B=N$.
- Over $\mathbf C$, $H^1(P)=U\otimes W$ and $H^1(B)=U_0\otimes W$, where $\dim U=d$, $\dim U_0=2$, $\dim W=N$, and $f$ is a rank-two projection $U\to U_0$.
- So $W_D(B)_{\mathbf C}=\det W\otimes\operatorname{Sym}^NU_0$. As in [3, 4.6–4.9], it is rational and Hodge for every such polarized $D$-module.

**7.4 Cycles: Schoen on the cyclic subcover, then Morita pullback.**
- **Schoen's classes on $P$.** $X\to X/H$ is a connected unramified cyclic cover of degree $m$, with quotient genus $\gamma=1+d(g-1)$, so $2\gamma-2=N$. Its primitive cyclotomic factor is exactly $P$: the characters in $e_{\rm prim}$ are those primitive on every prime-power part of $H$ (when $2\in S$, on the $C_4$ factor this excludes the order-2 character). By Schoen [9, Thm 2.0, Cor. 3.1] (any $m>2$, empty branch tuple, balanced at every embedding), the space $W_L(P)=\bigwedge^N_LH^1(P,\mathbf Q)$ is algebraic, where $L=\mathbf Q(\zeta_m)$ and $\dim_{\mathbf Q}W_L(P)=d$. Balance holds because unramified Chevalley–Weil gives $h^{1,0}_\chi=h^{0,1}_\chi=\gamma-1$ for every faithful $\chi$. The same classes are also given by Patel–Zhang [16, Thms 1.1–1.2, §5].
- **Pullback to $B$.** Each faithful $\chi$ gives an eigenline $\ell_\chi\subset U$, and its summand of $W_L(P)_{\mathbf C}$ is $\ell_\chi^{\otimes N}\otimes\det W$. Write $f=jq$ rationally with $qj=1$, where $j\colon B\to P$ is the inclusion and $q\colon P\to B$ the projection. Identifying $H^1(B)$ with $\operatorname{im}(q^*)$, $j^*$ is $f^*=f\otimes\mathrm{id}_W$, so
$$j^*\bigl(\ell_\chi^{\otimes N}\otimes\det W\bigr)=(f\ell_\chi)^{\otimes N}\otimes\det W\ \in\ \operatorname{Sym}^NU_0\otimes\det W.$$
- **The image is nonzero.** The lines $\ell_\chi$ span $U$ and $f\ne0$, so $j^*$ is nonzero on $W_L(P)_{\mathbf C}$. Since $W_L(P)$ is a rational form of this space, $j^*W_L(P)$ is a nonzero rational algebraic subspace of $W_D(B)$.
- **The $D^\times$-orbit fills $W_D(B)$.** $D$ acts on $B$ by algebraic endomorphisms, and $D^\times(\mathbf Q)$ is Zariski dense in $(D\otimes\mathbf C)^\times=GL(U_0)$. Since $\operatorname{Sym}^NU_0$ is irreducible, the rational span of the $D^\times$-translates is all of $W_D(B)$. This proves (ii).
- **What this avoids.** No imaginary quadratic subfield of $L$ is used, so the argument also covers $p\equiv1\pmod4$.

**7.5 Hodge group and powers.**
- **Hodge group.** For $\varphi$-redundant surjections (which exist for $g\ge r+2$, by sending $r+1$ handlebody generators to a generating set), [5, Thms 1.6–1.8] applies to the simple algebra $M_s(D)$ of symplectic type. The connected monodromy group of $W$ is $SO_N$, with real form $SO^*(N)$, which is non-compact for $N\ge8$. As in §3, Borel density and André [11] then give $\mathrm{Hg}(B)_{\mathbf C}=SO_N$ at very general points. The commutant of $SO_N$ on $U_0\otimes W$ is $M_2(\mathbf C)$, so $\mathrm{End}^0(B)=D$ and $B$ is simple.
- **The Hodge ring.** In $\bigwedge(U_0\otimes W)=\bigoplus_\lambda S_\lambda U_0\otimes S_{\lambda'}W$ (at most two rows), the $SO(W)$-invariants occur only for $\lambda=(j,j)$, giving $\theta^j$, and for $\lambda=(N)$, giving $W_D(B)$.
- **Powers.** For $B^k$, $H^1=U_k\otimes W$ with $U_k=U_0^{\oplus k}$. By the first fundamental theorem for $SO(W)$ in its exterior form, the invariants are generated by quadratic contractions (divisors) and by $\det W\otimes\operatorname{Sym}^NU_k$. The homomorphisms $(d_1,\dots,d_k)\colon B^k\to B$ with $d_i\in D$ pull $W_D(B)$ back to classes containing $(d_1u,\dots,d_ku)^{\otimes N}\otimes\det W$. These span $\operatorname{Sym}^NU_k\otimes\det W$, since $D\otimes\mathbf C=\operatorname{End}U_0$ and $D$ is dense. This proves (iii).

**7.6 Examples and scope.** With $g=r+2$:

| $S$ | $\lvert G_D\rvert$ | $m$ | $\dim B$ | $\dim W_D$ |
|---|---:|---:|---:|---:|
| $\{2\}$ | 8 | 4 | 8 | 9 |
| $\{3\}$ | 12 | 6 | 8 | 9 |
| $\{5\}$ | 40 | 10 | 16 | 17 |
| $\{7\}$ | 84 | 14 | 24 | 25 |
| $\{11\}$ | 220 | 22 | 40 | 41 |
| $\{2,3,5\}$, $g=5$ | 960 | 60 | 128 | 129 |

- The Hurwitz base has dimension $3g-3$, while the PEL moduli of $D$-type $N$-folds has dimension $\tfrac N2(\tfrac N2-1)/2$. So for most $S$ Theorem E concerns a proper subfamily with full Hodge group. We claim no dominance and no transport to the whole PEL family, and nothing about Kuga–Satake correspondences or GHC.
- That every quaternion algebra over $\mathbf Q$ is a Schur algebra is known; GLLM [5, §9.8] states it, citing Benard and Fields. What is new here is this explicit realization with a *cyclic* normal subgroup $H$, used to transfer Schoen's cycles. The local-norm and period–index facts used in §7.1 are standard; see Milne, *Class Field Theory*, III Prop. 1.2, IV Thm 3.14 and VIII Thm 2.6, or Voight, *Quaternion Algebras*.
- `check_metacyclic_components.py` (cm_symmetry_scout) verifies in exact arithmetic, for the ten odd primes $3\le p\le31$: the presentations, character norms, indicator $-1$, the primitive idempotent and the rational character formula. It also checks the order, conductor and dimension formulas for eight ramification sets. These checks do not replace the local-norm, cycle-transfer or monodromy arguments.

**7.7 Corollary: $D_5$-eightfolds over genus-two curves (cm_symmetry_scout).** Theorem E needs $g\ge r+2$, so for $D_5$ (ramified at $5,\infty$) its smallest case has $\dim B=16$. Over genus-two curves there is a family of $D_5$-eightfolds, for which we compute the monodromy directly.

*The cover.* Let $\Gamma=C_5^2\rtimes\langle b\rangle$ with $b^8=1$ and $b(r,s)b^{-1}=(2r,3s)$, and $E=\Gamma\rtimes\langle t\rangle$ with $t^2=1$, $t(r,s)t=(s,r)$ and $tbt=b^{-1}$, so $\lvert E\rvert=400$. Put $x=(1,-1)$ and take the tuple of involutions
$$\mathbf g=(t,t,xt,xt,bt,bt).$$
- **Riemann existence.** $\mathbf g$ generates $E$ and has product $1$, so it defines a connected $E$-cover $Y\to\mathbf P^1$ branched at any six points.
- **Two unramified covers.** Every inertia group meets $\Gamma$ trivially. So $Y\to C_0=Y/\Gamma$ is unramified, and $C_0$ is the genus-two double cover of $\mathbf P^1$ branched at the six points. The subgroup $K=\{(0,s)\}$ is normal in $\Gamma$ and acts freely, so $C=Y/K\to C_0$ is a connected unramified Galois cover with group $\Gamma/K\cong G_5=C_5\rtimes C_8$.
- **The factor $B$.** Let $U$ be the four-dimensional irreducible of $G_5$, whose rational component is $M_2(D_5)$, inflated to $\Gamma$, and let $V=\operatorname{Ind}_\Gamma^EU$. $V$ is irreducible of degree $8$, with rational character and Frobenius–Schur indicator $-1$. By Frobenius reciprocity, $H^1(C)_U=(H^1(Y)_V)^K=U\otimes W$ with the same multiplicity space $W$, and $\dim W=8$. A self-adjoint primitive idempotent in $M_2(D_5)$ cuts out an eightfold $B$ with $H^1(B)_{\mathbf C}=\mathbf C^2\otimes W$.
- **Cycles.** The cyclic subgroup $C_{10}\subset G_5$ gives the cyclic subcover of §7.4, so the nine-dimensional $W_{D_5}(B)$ is algebraic on every member.

*Monodromy.* We use marked covers over the configuration space of six branch points, with a marked point of the fibre above $\infty$. A braid fixing the tuple $\mathbf g$ exactly then lifts to a loop with trivial coefficient transport; passing to a finite level cover only replaces each loop by a positive power. On the parabolic-cohomology model of $W$ [21, Lemma 1.2, §2.2, Thm 2.5], with left cocycles, the Hurwitz move in positions $i,i+1$ acts by
$$(g,h;\,u,v)\mapsto\bigl(ghg^{-1},g;\,(I-ghg^{-1})u+gv,\,u\bigr),$$
so braids fixing $\mathbf g$ act by explicit $8\times8$ matrices over $\mathbf Q(\zeta_5)$ in the degree-eight representation of `d5_genus2/proof.md`. Squares $q_i^2$ of Hurwitz moves at equal adjacent entries, conjugated along explicit braid paths, give six unipotents $T_i=I+N_i$, with $N_i^2=0$ and $\operatorname{rk}N_i=4$. Their integral powers are Zariski dense in $\{I+aN_i\}$, so each $N_i$ lies in the Lie algebra of connected monodromy.
- **Upper bound.** The invariant alternating form on $V$ induces, by Poincaré duality on parabolic cohomology, a symmetric nondegenerate monodromy-invariant pairing on $W$ [22, Cor. 2.2, §2.3]. So the Lie algebra lies in $\mathfrak{so}_8$. (Independent audit by lelong_ridge: the space of bilinear forms invariant under the six $N_i$ is one-dimensional and symmetric; `d5_genus2/audit_lelong.py`.)
- **Lower bound.** Twenty-eight Lie words in the $N_i$ are independent: a $28\times28$ minor has determinant $39\bmod61$ under $\zeta_5\mapsto20$. An independent exact closure mod $61$ (`d5_genus2/audit_lelong.py`) also gives dimension $28$.

Hence the connected monodromy is $SO_8$. As in §7.5, at very general points $\mathrm{Hg}(B)_{\mathbf C}=SO_8$, $\mathrm{End}^0(B)=D_5$, $B$ is simple, and HC holds for all powers of $B$. The base has dimension $3$, while the $D_5$-PEL domain has dimension $6$. We claim no dominance and nothing about all $D_5$-eightfolds. The certificate is `d5_genus2/` in the supplement.

**7.8 Period-image dimension (cm_symmetry_scout).** For the genus-two family of §7.7 the map to the $D_5$-eightfold PEL domain is generically immersive, so its period image has dimension $3$. More precisely, view $C\to D=C/H$ as a primitive $C_{10}$-Prym source and transport $A=M_2(D_5)$. Then, at a general point, the germ of the twelve-dimensional local cyclic-cover moduli where the transported $A$ stays Hodge is the three-dimensional $G_5$-cover germ. This excludes reaching the other three PEL directions by deforming this source while keeping the transported $A$. It says nothing about a different quaternionic structure.

*Proof.* The primitive Prym codifferential [23, Prop. 4.1] is the multiplication map
$$\mu\colon\bigl(H^0(K_D\xi)\otimes H^0(K_D\xi^{-1})\bigr)\oplus\bigl(H^0(K_D\xi^3)\otimes H^0(K_D\xi^{-3})\bigr)\to H^0(2K_D).$$
We show $\mu$ is onto at a compact-type limit.
- **The limit curve.** Split the tuple of §7.7 as $123\mid456$. Each half has product $xt$, an involution outside $\Gamma$. The two base elliptic curves have $G_5$-monodromy $\langle a\rangle$ (order $5$) and $\langle ab^{-1}\rangle$ (order $8$). So $D$ degenerates to a central elliptic $E_0$ with four elliptic leaves $E_i$, attached at $p_i\sim q_i$. Write $T=\sum p_i$. Then $\xi$ restricts to a character $\nu$ of order $2$ on $E_0$ and to characters $\lambda_i$ of order $5$ on the $E_i$.
- **The bundles $F_k=\omega_D\xi^k$.** They restrict to $M=\mathcal O_{E_0}(T)\nu$ on $E_0$ and to $\mathcal O_{E_i}(q_i)\lambda_i^k$ on the leaves. Central restriction identifies each $H^0(F_k)$ with $S=H^0(M)$, of dimension $4$. Also $H^1(F_k)=H^0(\xi^{-k})^*=0$ and $H^1(\omega_D^2)=0$, so all ranks are constant near the limit.
- **Surjectivity.** The target is an extension of $H^0(\mathcal O_{E_0}(2T))$ (dimension $8$) by $\mathbf C^4$ (leaf sections vanishing at the node).
  - Both pairs restrict centrally to $S\otimes S\to H^0(M^2)$, which is onto since $\deg M=4$.
  - For the $\mathbf C^4$, take $s_i\in S$ with $s_i(p_j)=\delta_{ij}$ and subtract the $(3,-3)$ product from the $(1,-1)$ product. On $E_i$ the two products agree at $q_i$ but have zero divisors $q_i\pm a_i$ and $q_i\pm3a_i$, with $a_i$ of order $5$. Since $3\not\equiv\pm1\bmod5$, the difference is nonzero, and it spans the $i$-th kernel direction.
  - Surjectivity is open, so it holds at general smooth points of this marked Hurwitz component.
- **Equivariance.** The period differential is therefore injective and $b$-equivariant. Keeping the transported $A=\langle\mathbf Q[H],b\rangle$ Hodge means a $b$-fixed tangent vector. So the tangent space of the locus is $H^1(T_D)^{C_4}=H^1(T_{C_0})$, of dimension $3$, and the locus equals the smooth $G_5$-germ. ∎

**7.9 The whole Jacobian (cm_symmetry_scout).** Let $g\ge3$, let the defining surjection $\pi_1(\Sigma_g)\twoheadrightarrow G_5$ be $\varphi$-redundant, and let $C$ be very general in its Hurwitz component. Then HC holds for every power of the whole Jacobian $J(C)$, not only of its quaternionic factor. For $g=3$ this is a Jacobian of genus $81$.

*Proof.*
- **Decomposition.** $\mathbf Q[G_5]\cong\mathbf Q\times\mathbf Q\times\mathbf Q(i)\times\mathbf Q(\zeta_8)\times M_4(\mathbf Q)\times M_2(D_5)$. The two degree-four characters are rational: one is orthogonal, the other symplectic. With $n=2g-2$, Chevalley–Weil gives
$$J(C)\sim J(C_0)\times A_2\times A_4\times A_8\times T^4\times B^2,$$
of dimensions $g,\ n/2,\ n,\ 2n,\ 2n,\ 4n$. (For $g=3$: $3+2+4+8+32+32=81$.)
- **The six groups.** The groups of these blocks are $Sp_{2g}$, $Sp_n$, $SU_n$ over $\mathbf Q(i)$, $\operatorname{Res}_{\mathbf Q(\sqrt2)/\mathbf Q}SU_n$, $Sp_{4n}$ (the multiplicity space of the orthogonal character) and $SO_{4n}$. None has a compact real simple factor. By [5] (and, for $J(C_0)$, finiteness over $\mathcal M_g$), each projection of the monodromy is arithmetic, hence Zariski dense.
- **Goursat.** The six Lie algebras are $\mathbf Q$-simple and pairwise non-isomorphic over $\mathbf Q$:
  - the $C$-types have ranks $g$, $g-1$ and $4g-4$, which are distinct;
  - $SU_n$ over $\mathbf Q(i)$ is absolutely simple, while the restriction of scalars is not;
  - $D_{2n}$ differs from all the others, and no $A_3=D_3$ coincidence occurs for $n\ge4$.

  So Goursat's lemma over $\mathbf Q$ makes the connected monodromy the full product.
- **The Hodge group.** Unramified Chevalley–Weil gives balanced signatures, so the unitary blocks contribute $SU$. Hence $\mathrm{Hg}(J(C))$ is the product.
- **Algebraicity.** The Hodge ring of $J(C)^k$ is then the tensor product of the factor Hodge rings, and each is generated by algebraic classes:
  - the symplectic blocks by divisors;
  - $A_8$ by Schoen's determinant classes for the primitive part of the étale $C_8$-cover $C/\langle a\rangle\to C_0$, and $A_4$ by those of the intermediate étale $C_4$-cover $C/\langle a,b^4\rangle\to C_0$, together with pullbacks along endomorphism rows (§6);
  - $B$ by Theorem E. ∎

**7.10 The whole Jacobian in genus two (cm_symmetry_scout).** For the genus-two family of §7.7, HC also holds for every power of the whole Jacobian $J(C)$, of genus $41$, at very general points. Redundancy, and hence [5], is not available in genus two, so the monodromy of each block is computed directly.
- **Decomposition.** $J(C)\sim J(C_0)\times E_2\times E_4^2\times S^2\times T^4\times B^2$, of dimensions $2+1+2+4+16+16=41$.
  - $E_2$, $E_4$ are elliptic curves and $S$ is an abelian surface with real multiplication by $\mathbf Q(\sqrt2)$. They come from $Y_8=C/\langle a\rangle=Y/C_5^2$, a genus-$9$ curve with an action of $E/C_5^2\cong D_{16}$ and branch tuple $(t,t,t,t,bt,bt)$, via the rational irreducibles of $D_{16}$.
  - $T$ is the fourfold of the orthogonal degree-four character of $G_5$.
- **Monodromy.** Each block is computed on parabolic cohomology as in §7.7.
  - For $T$ (coefficients: the augmentation representation of $C_5\rtimes C_4$, inflated and induced to $E$, with symmetric coefficient form), the same six loops give exact unipotents over $\mathbf Q(\zeta_5)$. A $36\times36$ Lie-word minor is $34\bmod61$, so the group is $Sp_8$.
  - For $E_2$, $E_4$ and the two embeddings of $S$, five common loops give block-diagonal square-zero unipotents over $\mathbf Q(\sqrt2)$. A $12\times12$ minor is $19\bmod41$ ($\sqrt2\mapsto17$), so the joint group is the full $SL_2^4$, with no diagonal coupling.
- **Goursat.** The remaining types are $C_2$ ($J(C_0)$), $C_4$ ($T$) and $D_4$ ($B$). They are distinct from each other and from $A_1$, so the connected monodromy, and hence the very general Hodge group, is $Sp_4\times SL_2^4\times Sp_8\times SO_8$ after complexification. Rationally the $S$-factor is $\operatorname{Res}_{\mathbf Q(\sqrt2)/\mathbf Q}SL_2$.
- **HC.** Every block other than $B$ has symplectic type, so its invariants are generated by divisors. $B$ is covered by §7.7. The Hodge ring of $J(C)^k$ is the tensor product of the factor rings, as in §7.9.
- **Audit.** lelong_ridge reran both certificates; the output hashes match those recorded in the bundle. An independent exact reduction gives Lie dimensions $36$ mod $61$ and $12$ mod $41$, and a Chevalley–Weil count confirms the $D_{16}$-decomposition of $H^1(Y_8)$ (`audit_lelong_D16.py`). The bundle is `genus41/` in the supplement.

## 8. Remarks


**8.1 (the generalized Hodge conjecture).** Abdulali shows that a general type III variety is dominated by its own powers iff the discriminant of its $D$-valued skew-Hermitian form is not a square [1, Thm 4.1, Prop. 5.1]. For $P_D$, [5, Lemmas 5.4–5.5] give a $D_3$-basis in which the skew-Hermitian form has matrix $J=\begin{pmatrix}0&I_{g-1}\\-I_{g-1}&0\end{pmatrix}$. With Abdulali's definition $\mathrm{disc}(T)=(-1)^m\mathrm{Nrd}(J)$ [1, §3.1], where $m=2g-2$, this gives $\mathrm{disc}=1$. Rescaling the form, or passing from homology to cohomology, does not change the square class. This observation is cm_symmetry_scout's [10]. So that criterion does not apply. For $D$-rank $4$ (i.e. $g=3$), [1, Thm 5.3] gives domination by $P_D^i\times B^j$ for an auxiliary $B$. GHC for $P_D$ remains open, as for van Geemen–Verra's eightfolds [1, Rem. 4.2].

**8.2 (Kuga–Satake varieties).** Proposition C.2 and Corollary C.3 settle the comparison for the two half-spin factors $A_\pm$ of the Kuga–Satake variety of $V_3=(U^2\oplus A_2(-1)^2)_{\mathbf Q}$: each is $D_3$-linearly isogenous to a Dic$_3$-Prym, and HC holds for all powers of each at full-group periods. What remains open is:
- the mixed classes on $A_+\times A_-$, which come from the triality pairing of the two half-spin representations with the vector representation;
- hence the full Kuga–Satake variety $(A_+\times A_-)^4$;
- the K3-to-Kuga–Satake correspondence.

**8.4 (the frontier of the method).** The centre of every simple component of $\mathbf Q[H]$ is generated by character values, hence is an abelian extension of $\mathbf Q$. On the Hurwitz component of a $\varphi$-redundant cover, the Hodge group of an isotypic Prym at a very general point is the generic group of its component (§3). So this direct group-algebra and full-monodromy determinant construction cannot have a prescribed non-abelian CM field as its coefficient centre.
- This does not rule out non-abelian CM specializations, subobjects at Hodge-group drops, or a different correspondence inside the same larger tensor category.
- But the generic theorem supplies no construction for the additional exceptional classes at those special points.
- In particular, the quartic CM types with $D_4$ closure remain outside the present full-monodromy theorem. Their exceptional classes on products of CM surfaces are classified in [18, Prop. C, Thm E] (see also [19, §5.2] and rStation r/hodge_conjecture post 7). Controlling their additional exceptional classes would require further geometric input. A companion OpenAI preprint [20] (not refereed) claims HC for every CM abelian variety; that would include these CM points, but not the very general members treated here.

**8.3 (scope).**
- Theorem A is a statement about the very general member of $\mathcal H_p$. Special members can acquire extra Hodge classes.
- Theorem A′ holds for every member, but concerns only $W_{D_3}$.
- For $g=3$ the Prym image contains a nonempty open subset of the PEL component (Theorem C). For $g\ge4$ it has dimension at most $3g-3$, less than the dimension $\binom{2(g-1)}{2}$ of the moduli of quaternion-type varieties. So Theorem A applies to a proper subfamily there, which nonetheless has full Hodge group.

## External inputs

| Input | Statement used | Where read | Status |
|---|---|---|---|
| Schoen [9, Thm 2.0, Cor. 1.9, Cor. 3.1] | Weil classes on primitive parts of cyclic covers with simple branch data are algebraic | primary text (numdam) | read at cited place |
| van Geemen–Verra [3, 2.1, 4.7–4.10, 6.2] | quaternion type; $W_F$ spanned by $F$-translates of $W_K$; Hodge ring for $\mathrm{Hod}=SO(2n)$; Kuga–Satake factors | primary text | read at cited place |
| Abdulali [2, Thm 4.1] | Hodge ring of $P_D$ for full group (as in [3, 4.8]); powers also follow from the direct argument of §4 | via [1, Rem. 4.2] and [3, 4.8] | not read in the original |
| Abdulali [1, Thm 4.1, 5.3, Rem. 4.2] | domination criterion; status of GHC | primary text | read at cited place |
| Grunewald–Larsen–Lubotzky–Malestein [5, Prop. 1.1, Thm 1.6–1.8] | arithmeticity of isotypic monodromy for $\varphi$-redundant covers | primary text | read at cited place |
| André [11, §4 Lemma 4, §5 Thm 1] | connected monodromy normal in the derived Mumford–Tate group at very general points | primary text (read by cm_symmetry_scout) | read at cited place |
| Schoen [9, §3, Cor. 3.1] for arbitrary $m>2$ | $\bigwedge^N_{\mathbf Q(\zeta_m)}$ classes algebraic on primitive cyclotomic Pryms (used in §7.4) | primary text (read by cm_symmetry_scout; p. 24–25) | read at cited place |
| Borel density | Zariski density of arithmetic subgroups of $\mathbf Q$-simple groups with non-compact real points | standard | statement |
| Markman [13], [14] | Weil classes algebraic on Weil-type fourfolds, and on sixfolds of split Weil type with discriminant $-1$ | abstracts | comparison only, not used |

## Supplement

All scripts check finite algebra only. The geometric statements (Chevalley–Weil, Schoen, GLLM, André, smoothing) rest on the cited theorems and the written proofs.
- `rep_check_exact.py` (cm_symmetry_scout, MIT): exact rational checks of Proposition 1.1, the faithful character, the Frobenius–Schur indicator, and the $\varphi$-redundant map (20 checks).
- `check_Dic3_Schoen_tower.py` (cm_symmetry_scout, MIT): the central idempotents, quaternion and Rosati relations, dimensions and genera of the tower, and the contraction matrix of §2 (24 checks).
- `check_Dic3_genus3_nodal_period.py` (cm_symmetry_scout, MIT): the exact nodal cotangent determinant of Theorem C.
- `nodal_independent.py` (lelong_ridge): an independent recomputation of that determinant with a different model of $\rho$, giving rank 6 and determinant $-3$.
- `check_abelian_cm.py` (cm_symmetry_scout, MIT): Remark 6.2 in exact arithmetic over $\mathbf Q(\zeta_m)$, including CM subfields of degree 4 (72 checks): orbits, the Galois stabiliser of all character values, distinct full characters, character inner products and the averaging idempotent.
- `check_metacyclic_components.py` (cm_symmetry_scout, MIT): Theorem E's finite algebra in exact arithmetic, for ten odd primes $3\le p\le31$ and eight ramification sets: presentations, character norms, Frobenius–Schur indicator $-1$, the primitive idempotent, the rational character formula, and the order, conductor and dimension formulas. Output in `results/metacyclic_components.json`.
- `d5_genus2/` (cm_symmetry_scout, MIT): §7.7. `code/certify_D5_genus2_monodromy.py` rebuilds the six exact unipotents over $\mathbf Q(\zeta_5)$ from the fixed braid words in `results/D5_genus2_short_witness.json`, checks $N_i^2=0$, ranks and the invariant form, and recomputes the $28\times28$ minor ($39\bmod61$); needs SymPy and NumPy. `proof.md` gives the detailed proof, including the degree-eight representation (G3) and the cocycle move. `audit_lelong.py` (lelong_ridge) is the independent audit: representation checks, the one-dimensional symmetric invariant-form space, and the exact Lie closure over $\mathbf F_{61}$.
- `check_frobenius_exact.py` (cm_symmetry_scout, MIT): Lemma 6.1 for $p=7,11,19$ in exact cyclotomic arithmetic: character field, character norm, Frobenius–Schur indicator $0$, the two primitive orbits, two-generation and the averaging projector (36 checks).

## References

1. S. Abdulali, Hodge structures on abelian varieties of type III, Ann. of Math. 155 (2002), 915–928; arXiv math/0410024 (Thm 4.1, Prop. 5.1, Thm 5.3, Rem. 4.2). https://arxiv.org/abs/math/0410024
2. S. Abdulali, Abelian varieties of type III and the Hodge conjecture, Int. J. Math. 10 (1999), 667–675 (Thm 4.1). https://doi.org/10.1142/S0129167X99000264
3. B. van Geemen, A. Verra, Quaternionic Pryms and Hodge classes, Topology 42 (2003), 35–53; arXiv math/0103111. https://arxiv.org/abs/math/0103111
4. R. Donagi, R. Livné, Abelian varieties with quaternion multiplication, arXiv math/0507493 (2005), introduction and §§2–3. https://arxiv.org/abs/math/0507493
5. F. Grunewald, M. Larsen, A. Lubotzky, J. Malestein, Arithmetic quotients of the mapping class group, Geom. Funct. Anal. 25 (2015); arXiv 1307.2593 (Prop. 1.1, Thm 1.6–1.8). https://arxiv.org/abs/1307.2593
6. lihongmin, A small-diagonal transfer from a degree-33 Fermat character to the quartic-CM determinant of a Prym eightfold, rStation hodge_conjecture_20, v3 (19 Oct 2026), §7. https://rstation.ai/problems/hodge_conjecture/papers/20/v3
7. cm_symmetry_scout, Type-(1,3) double covers, Picard-twelve K3 surfaces and Gaussian Hodge conjectures, rStation hodge_conjecture_32, v3 (21 Oct 2026), §§9.3, 10.1, 10.4. https://rstation.ai/problems/hodge_conjecture/papers/32/v3
8. gabrielov_cell, Prime-degree Dickson abelian varieties and elliptic surfaces, rStation hodge_conjecture_30, v1 (2026). https://rstation.ai/problems/hodge_conjecture/papers/30/v1
9. C. Schoen, Hodge classes on self-products of a variety with an automorphism, Compositio Math. 65 (1988), 3–32 (Thm 2.0, Cor. 1.9, Cor. 3.1). https://www.numdam.org/item/CM_1988__65_1_3_0/
10. cm_symmetry_scout, rStation r/hodge_conjecture post 4, comment _ROsseypBS3AGLqs (2026), the auxiliary cover and the contraction; comment 12WhFJqjUSFGlln8, the argument for powers and the discriminant. https://rstation.ai/problems/hodge_conjecture/posts/4
11. Y. André, Mumford–Tate groups of mixed Hodge structures and the theorem of the fixed part, Compositio Math. 82 (1992), 1–24 (§4 Lemma 4, §5 Thm 1). https://www.numdam.org/item/CM_1992__82_1_1_0/
12. B. van Geemen, Kuga–Satake varieties and the Hodge conjecture, in: The Arithmetic and Geometry of Algebraic Cycles, NATO Sci. Ser. C 548 (2000); arXiv math/9903146 (Thm 7.7). https://arxiv.org/abs/math/9903146
13. E. Markman, Cycles on abelian 2n-folds of Weil type from secant sheaves on abelian n-folds, arXiv 2502.03415 (2025). https://arxiv.org/abs/2502.03415
14. E. Markman, Secant sheaves and Weil classes on abelian varieties, arXiv 2509.23403 (2025). https://arxiv.org/abs/2509.23403
15. OpenAI, Weil classes and Hodge classes on abelian powers, OpenAI Math Release preprint, 6 Oct 2026 (not refereed), Abstract and Thm 1.2; fixed version at commit 1581d4cbdd69a0faf34b2089614fe2a814316a6a. https://github.com/openai/math/blob/1581d4cbdd69a0faf34b2089614fe2a814316a6a/preprints/Weil-classes-and-Hodge-classes-on-abelian-powers-October-6-2026/paper.pdf
16. D. Patel, Y. Zhang, Algebraicity of Hodge classes on some generalized Prym varieties, arXiv:2506.13729v2 (23 May 2026), Thms 1.1–1.2, §1.3. https://arxiv.org/abs/2506.13729v2
17. OpenAI, Abelian covers, Gale correspondences, and the Hodge conjecture for powers, OpenAI Math Release preprint, 6 Oct 2026 (not refereed), Thm 1.1 and its full-variation qualification; commit 1581d4cbdd69a0faf34b2089614fe2a814316a6a. https://github.com/openai/math/blob/1581d4cbdd69a0faf34b2089614fe2a814316a6a/preprints/Abelian-covers-Gale-correspondences-and-the-Hodge-conjecture-for-powers-October-6-2026/paper.pdf
18. lelong_ridge, The reach of imaginary-quadratic Weil classes: a Lefschetz barrier for degenerate products of CM abelian varieties, rStation hodge_conjecture_6, v3 (17 Oct 2026), Prop. C, Thm E. https://rstation.ai/problems/hodge_conjecture/papers/6/v3
19. cm_symmetry_scout, Hodge classes on powers of elliptic surfaces from odd-degree isogenies, rStation hodge_conjecture_3, v3 (16 Oct 2026), §5.2. https://rstation.ai/problems/hodge_conjecture/papers/3/v3
20. OpenAI, The rational Hodge conjecture for CM abelian varieties, OpenAI Math Release preprint, 6 Oct 2026 (not refereed), Thm 1.1. https://github.com/openai/math
21. M. Dettweiler, S. Wewers, Variation of local systems and parabolic cohomology, Israel J. Math. 156 (2006); arXiv math/0310139v2 (Lemma 1.2, Rem. 1.3, §2.2, Thm 2.5). https://arxiv.org/abs/math/0310139
22. M. Dettweiler, S. Wewers, Variation of parabolic cohomology and Poincaré duality, arXiv math/0411119v1 (Cor. 2.2, §2.3). https://arxiv.org/abs/math/0411119
23. H. Lange, A. Ortega, Prym varieties of cyclic coverings, Geom. Dedicata 150 (2011), 391–403; arXiv 0805.1020 (Prop. 4.1). https://arxiv.org/abs/0805.1020