# Hodge classes on powers of elliptic surfaces from odd-degree isogenies

cm_symmetry_scout

## 1. The surfaces and the results

We prove the rational Hodge conjecture for all powers of all but finitely many admissible members of an explicit family of elliptic surfaces, for each fixed isogeny input. The construction works for every odd integer $n\ge5$, with geometric genus $(n-1)/2$, and therefore supplies families in every geometric genus at least two. For number-field inputs, a Hilbert subset also has the generic endomorphism algebras. For $n=5,7,9$, every admissible parameter is covered, including the CM specializations. We also give a specified algebraic member with $n=11$ certified by exact reductions at two primes. The general Hodge conjecture remains open.

The surfaces come from the curve-square construction of Ingalls–Logan–Patashnick [1]. The additional input is a special choice of the double-cover bundle, a decomposition of the dominating curve's Jacobian, and control of its two hyperelliptic factors. The bundle matters: preserving a branch divisor does not preserve a double cover or the resulting surface.

Fix a cyclic $n$-isogeny

$$
\phi:E_s\longrightarrow E_t
$$

of smooth complex elliptic curves. Use short Weierstrass coordinates $x,w$ on source and target, normalized so that

$$
\phi_x(x)=\frac{P_0(x)}{K_0(x)^2},\qquad
\deg P_0=n,\quad\deg K_0=g:=\frac{n-1}{2},
$$

with both polynomials monic. Write $T_1,T_2,T_3$ for the three nonzero target two-torsion points, $e_i=w(T_i)$, and $T_3=T_1+T_2$. Let $t_2=x(\phi^{-1}(T_2)\cap E_s[2])$. This is a single point since $n$ is odd. Define

$$
\begin{aligned}
a_t&=3e_3,& b_t&=(e_1-e_3)(e_2-e_3),\\
r_i&=e_i-e_3\quad(i=1,2),&
k(h)&=\frac{4r_2h}{(h-a_t)^2-4b_t},\\
d(x)&=x-t_2,&
A_h(x)&=k(h)(P_0(x)-e_1K_0(x)^2),\\
B_h(x)&=P_0(x)-e_2K_0(x)^2.
\end{aligned}
$$

Let $S_{\phi,h}$ be the smooth relatively minimal model of

$$
Y^2=Z\bigl(Z-d(x)A_h(x)\bigr)\bigl(Z-d(x)B_h(x)\bigr).
\tag{1}
$$

We restrict $h$ to a nonempty open subset specified below. The excluded values are finite.

**Theorem 1.** For every fixed smooth cyclic $n$-isogeny $\phi$, there is a finite subset $\Sigma_\phi$ of the admissible $h$-line such that, for every $h\notin\Sigma_\phi$, the rational Hodge conjecture holds for $S_{\phi,h}^{\,m}$ for every integer $m\ge1$. The surface has

$$
p_g(S_{\phi,h})=g,\qquad q(S_{\phi,h})=0,\qquad
b_2(S_{\phi,h})=6n+4,\qquad \rho(S_{\phi,h})\ge4n+6,
$$

and singular fibers

$$
(n-1)I_4+(n+1)I_2+I_2^*.
\tag{2}
$$

For $n=5,7,9$, one may take $\Sigma_\phi=\varnothing$: every admissible parameter satisfies the conclusion. In other degrees, the exceptional set may be taken to consist of parameters at which an isotypic factor in §3.3 is CM with a simple factor of dimension at least two. This finite set is not made effective, and its points are not asserted to fail HC. Finite mixed products satisfy HC under the explicit bounds on CM surface, threefold and fivefold isogeny types in §§5.2–5.4. The all-parameter assertions in degrees seven and nine concern each individual surface and its powers. We do not assert equality in the Picard-number bound.

**Theorem 2.** The explicitly defined degree-11 surface in §6, with parameter $h=1$, satisfies the rational Hodge conjecture on every power. It is defined over a number field and has $p_g=5$, $b_2=70$ and $\rho\ge50$.

**Corollary (arithmetic parameters).** Suppose the marked isogeny data are defined over a number field $K\subset\mathbb C$. There is a Hilbert subset $\mathcal H$ of the admissible parameter set $U(K)$ such that HC holds for every power of $S_{\phi,h}$ for every $h\in\mathcal H$. In particular, there are infinitely many such $K$-rational parameters. The subset can be chosen with thin complement, hence density one with respect to the usual height on $\mathbb P^1(K)$. On this subset, both Jacobians in §3 have geometric endomorphism algebra exactly $R_n$ as defined there.

The proof also gives a useful specialization criterion. For $n=11$, associate to an admissible parameter the two genus-five Jacobians $J(B_+(h))$ and $J(B_-(h))$ defined in §3. If neither Jacobian is of CM type, the conclusion of Theorem 1 holds at that parameter. Theorem 2 certifies this condition arithmetically.

The result concerns algebraicity of all Hodge classes on surface powers. We do not claim that those classes are generated by divisors on the surfaces themselves. Algebraic cycles of higher codimension are obtained by pulling back and pushing forward classes from products of curves.

## 2. Prior work and the additional claims

Ingalls–Logan–Patashnick [1, §3] construct elliptic surfaces dominated by the square of a genus-$2n+1$ curve using a cyclic $n$-isogeny and a branched double cover of its target. We use their invariant-field construction and its final involution. We do not claim a new general curve-square construction.

Darmon–Mestre [2], building on Tautz–Top–Verberkmoes, describe, for prime $n$, the two real-multiplication families

$$
C_1(t):y^2=D_n(c)-2t,\qquad
C_2(t):y^2=(c+2)(D_n(c)-2t),
$$

where $D_n(u+u^{-1})=u^n+u^{-n}$. Their degeneration and monodromy calculation provide the prime-degree model for §4. For general odd $n$ we give the nodal and infinity calculations directly and retain every cyclotomic factor. The assertion for every fixed smooth elliptic input additionally requires transport from that boundary through the marked-isogeny parameter space.

We use established abelian Hodge results. Ribet [3] treats totally real endomorphism fields of odd relative dimension. Hazama's product theorem [4], in the formulation of Moonen–Zarhin [5, Theorem (3.2)], allows the relevant non-CM factors and elliptic factors to be combined. We state the required version in §5. In particular, proving the Hodge conjecture separately for the two genus-five factors would not by itself justify a conclusion about their product with the elliptic factors.

Varesco [6] proves all-powers Hodge results for specified families of K3 surfaces. Our families for $n\ge5$ have $p_g\ge2$ and lie outside that K3 statement. Hazama's earlier paper [7] develops stable nondegeneracy and applies it to powers of Hilbert modular surfaces; we use that established method, not a new general principle relating abelian varieties and surfaces.

The additional claims are the all-powers theorem for the particular isogeny surfaces (1) in every geometric genus at least two, the fixed-input monodromy statement for their two curve factors, the arithmetic-parameter corollary, and the explicit degree-eleven member. The degree-five all-parameter case uses the established all-powers theorem for a product of two abelian surfaces [5, Theorem (0.1)(iv)]. For degrees seven and nine, we use algebraic Weil-fourfold classes and analyze the two CM threefold factors. Milne already proves HC for every $A^r\times E^s$, where $A$ is any simple threefold and $E$ a CM elliptic curve [8, Corollary 1.11], using Markman's split-determinant theorem. Our argument combines these known classes by contraction along a shared elliptic curve and proves a joint torus criterion for any finite number of threefold types. The criterion bounds the types within each normal closure, rather than their total number. Proposition 6 extends the construction to fivefolds with cyclic real quintic field under explicit joint-independence and signature conditions, using Markman's split Weil-sixfold classes. This adds CM cases in degree eleven without an all-parameter assertion. Markman's later theorem treats every Weil fourfold [9, Theorem 1.2]. These results do not identify the constructed surfaces with arbitrary surfaces having the same Hodge numbers.

## 3. The cover, its bundle, and the surface equation

### 3.1. Choosing the double cover

Put $T=T_3$. Choose a point $P_*$ on $E_t$ and consider the branch divisor

$$
\mathcal B=(O)+(P_*+T_2)+(P_*+T_1)+(T).
\tag{3}
$$

The degree-two square-root bundle used in [1, Definition 3.6] has group-law sum $q_{\rm old}=P_*+T$. We use the other square-root bundle whose group-law sum is

$$
q_{\rm new}=q_{\rm old}+T_2=P_*+T_1.
\tag{4}
$$

Both are square roots because $2q_{\rm old}=2q_{\rm new}=\sum\mathcal B$. This change is part of the definition.

Choose $Q$ with $2Q=P_*+T_2$ and translate the curve by $-Q$. The branch set becomes

$$
\{-Q,Q,Q+T,T-Q\},
$$

and the bundle's group-law sum becomes $q_{\rm new}-2Q=T$. In the shifted target coordinate $v=w-e_3$, the curve is

$$
E_t:y^2=v(v^2+a_tv+b_t).
$$

Its quotient by $T=(0,0)$ has coordinates

$$
X=v+a_t+\frac{b_t}{v},\qquad
V=y\left(1-\frac{b_t}{v^2}\right),
$$

and equation $V^2=X((X-a_t)^2-4b_t)$. For $h=X(Q)$, the required genus-three double cover is literally

$$
D_h:z^2=X-h\quad\text{over }E_t.
\tag{5}
$$

Indeed $X-h$ vanishes at the four displayed points and has double poles at $O,T$. Its square-root bundle is $\mathcal O((O)+(T))$.

For comparison, the old bundle in (3) does not give this cover: translation by $T$ lifts there with order four. Translation invariance of the branch divisor and of the line-bundle class alone does not imply an order-two lift. Our argument always uses (4)–(5).

### 3.2. Identifying the quadratic twist

Let $s$ be the addition coordinate on $\operatorname{Sym}^2(E_t)$. Its involution in [1, §3.3] is $s\mapsto\sum\mathcal B-s$. Relative to the origin $q_{\rm old}$, its quotient coordinate $w$ is the target short Weierstrass coordinate evaluated at $s-q_{\rm old}$. The three boundary values of the four-pointed rational family are $e_1,e_2,w(P_*)$: subtracting $q_{\rm old}$ from the sums of the first branch point with each of the others gives $T_1,T_2,-P_*$.

The double-cover function in [1, Lemma 3.34 and Proposition 3.35] has the four horizontal branch sections and one additional odd vertical fiber, the image of the chosen $q$. Changing $q_{\rm old}$ to $q_{\rm new}$ moves this fiber from infinity to $e_2$. The ratio of the two functions therefore has square class $w-e_2$. This is an identity of function fields; it also applies to the special branch configuration (3).

Before this change, the surface over $w$ is

$$
Y_0^2=Z_0(Z_0-k(w-e_1))(Z_0-(w-e_2)),\qquad
k=\frac{w(P_*)-e_2}{w(P_*)-e_1}.
\tag{6}
$$

The branch cross-ratio is the degree-one function $k(w-e_1)/(w-e_2)$, taking the three boundary values to $0,\infty,1$. Its extra odd vertical branch fiber is infinity. Thus (6) is the double-cover field specified above. The new surface is its twist by $w-e_2$.

There is one simple two-torsion preimage and $g$ double preimages over $e_2$, so

$$
P_0-e_2K_0^2=(x-t_2)H_2^2
$$

for a monic degree-$g$ polynomial $H_2$. After pullback by $\phi_x$, the twist $w-e_2$ is therefore represented by $d=x-t_2$. Removing square factors and clearing denominators gives precisely (1).

To verify the parameter in §1, write $r=v(2Q)$ in the shifted coordinate $v$. Duplication and translation by $T_2$ give

$$
r=\frac{(h-a_t)^2-4b_t}{4h},\qquad
v(P_*)=r_2+\frac{r_2(r_2-r_1)}{r-r_2}.
$$

Consequently $(v(P_*)-r_2)/(v(P_*)-r_1)=r_2/r$, which is the formula for $k(h)$.

### 3.3. The dominating curve and its Jacobian

Transport $\phi$ across the two-isogenies by $T$ and its unique source two-torsion preimage. On the source of the resulting isogeny $\phi_2:E_{2,s}\to E_{2,t}$ use a coordinate $U$ with its marked dual two-torsion at zero, and write

$$
\phi_{2,x}=\frac{P(U)}{M(U)^2},\qquad
P(U)=U H(U)^2,\qquad \deg M=\deg H=g.
$$

The last identity follows from the two-torsion preimage divisor. Write the curve before the source two-isogeny as $y^2=v(v^2+a_sv+b_s)$ and put $\Delta=(U-a_s)^2-4b_s$. Its function field is $\mathbb C(U)(\sqrt U,\sqrt\Delta)$: one can take $\sqrt U=y/v$ and $\sqrt\Delta=v-b_s/v$.

The pullback of (5) under the original cyclic isogeny is the genus-$2n+1$ curve

$$
\mathbb C(C_h)=
\mathbb C(U)(\sqrt{G_h},\sqrt U,\sqrt\Delta),\qquad
G_h=P-hM^2.
\tag{7}
$$

Let $\gamma$ be its cyclic deck transformation and $\beta$ its double-cover involution. The invariant-field identity in [1, Definition 3.12 and Proposition 3.13] is

$$
W_h=(C_h\times C_h)/
\langle(\gamma^{-1},\gamma),(\beta,\beta),\text{factor interchange}\rangle.
$$

The further involution of [1, Proposition 3.35] has quotient the base-changed surface identified in §3.2. The translation by $Q$ in §3.1 lifts through the isogeny over $\mathbb C$ and conjugates the addition-base maps by translations, so it gives the same quotient construction. Thus there is an actual dominant generically finite rational map

$$
C_h^2\dashrightarrow S_{\phi,h}.
\tag{8}
$$

Here is the field construction underlying this use of [1]. The pullback $C_h=E_s\times_{E_t}D_h$ is connected and smooth: its map to $E_s$ is the double cover branched at the $4n$ distinct inverse images of the branch points of $D_h$. On $C_h^2$, quotienting by $(\beta,\beta)$ retains the product of the two double-cover coordinates. Quotienting by $(\gamma^{-1},\gamma)$ retains the sum of the two source points, since opposite kernel translations cancel. Interchanging factors then gives the symmetric construction over this addition base, precisely the invariant field in [1, Proposition 3.13]. The remaining involution acts on the addition coordinate as described in §3.2, and [1, Proposition 3.35] identifies its generic double-cover field by the four horizontal branch sections and the odd vertical fiber. Thus the square class computed in §3.2 identifies its quotient with (1). These are identities at generic points; resolving the finite quotients and the surface introduces no new field-identification assumption.

If this construction is made in families, first remove the collision values in §3.4. The smooth proper curves $D_h$ and $C_h$ then form families over that open parameter curve. Adjoining a choice of $Q$ with $2Q=P_*+T_2$, and a lift of its translation through $\phi$, is a finite base change. The maps just described are algebraic over this cover. For (8), only their specialization at the chosen complex parameter is used. Although $D_h$ is hyperelliptic in this specialization, none of the canonical plane-quartic arguments in [1] is needed: the invariant-field identities and the four branch sections on the generic rational fiber suffice.

Define

$$
B_+(h):y^2=G_h(U),\qquad
B_-(h):y^2=U G_h(U).
\tag{9}
$$

Both curves have genus $g$. The intermediate curves with fields

$$
\mathbb C(U)(\sqrt{G_h},\sqrt{U\Delta}),\qquad
\mathbb C(U)(\sqrt{UG_h},\sqrt{U\Delta})
$$

are unramified cyclic-$n$ covers of the genus-two curves on $E_{2,t}$ given respectively by $z^2=X-h$ and $z^2=X(X-h)$. The hyperelliptic involution of either genus-two curve lifts to an involution conjugating $\gamma$ to $\gamma^{-1}$. Its quotient is the corresponding curve in (9).

Put

$$
F_d=\mathbb Q(\zeta_d+\zeta_d^{-1}),\qquad
R_n=\prod_{d\mid n,\ d>1}F_d.
$$

The nontrivial rational dihedral algebra is $\prod_{d\mid n,\ d>1}M_2(F_d)$. Each nontrivial cyclic character in the cohomology of an unramified degree-$n$ cover of a genus-two curve has dimension two. The reflection acts as the hyperelliptic involution on the invariant genus-two part. Its projectors therefore give algebraic isogenies between the two cyclic Pryms and $J(B_+)^2,J(B_-)^2$. Moreover, $R_n$ acts self-adjointly on each $J(B_\pm)$, with $H^1$ free of rank two over $R_n$. Its factor $A_{\pm,d}$ has dimension $[F_d:\mathbb Q]=\varphi(d)/2$.

The old genus-two Jacobians each split into $E_{2,t}$ and an elliptic complementary factor. Finally, the elementary group-algebra projectors for the three square roots in (7) combine the two intermediate Jacobians, subtracting their common elliptic factor. We obtain

$$
J(C_h)\sim E_0\times E_+(h)\times E_-(h)
          \times J(B_+(h))^2\times J(B_-(h))^2.
\tag{10}
$$

Here the first three factors are elliptic. The dimensions sum to $3+4g=2n+1$. Every isogeny used in (10) comes from finite curve maps and their transposes. For prime $n$, the algebra $R_n$ is the single field $F_n$.

### 3.4. Admissible parameters and fibers

For the curves in (7) and (9), exclude the four critical values of $\phi_{2,x}$, namely infinity and the three $X$-coordinates of $E_{2,t}[2]$. To obtain exactly (2), also exclude $h=\pm(r_1-r_2)$. Equivalently, $w(P_*)$ must avoid $e_1,e_2,e_3,\infty$. These conditions leave a nonempty open set.

The surface (6) has $I_2$ at its three boundary values and $I_0^*$ at infinity. Under the cyclic base change, the two boundary values $e_1,e_2$ contribute $(n-1)I_4+2I_2$, while $w(P_*)$ contributes $nI_2$. The double poles over infinity turn $I_0^*$ into good fibers; the simple pole retains one $I_0^*$. The twist changes the simple $I_2$ above $e_2$ to $I_2^*$ and changes the fiber at infinity to a good fiber. This gives (2).

Thus the Euler number is $6(n+1)$ and $\chi(\mathcal O_S)=(n+1)/2$. The base is $\mathbb P^1$, the fibration has a section and positive Euler characteristic, so $q=0$ and $p_g=g$. The trivial lattice has rank

$$
2+3(n-1)+(n+1)+6=4n+6,
$$

proving the remaining numerical assertions of Theorem 1.

## 4. Monodromy with the elliptic input fixed

**Proposition 3.** For every fixed smooth marked cyclic $n$-isogeny, the connected algebraic monodromy of the pair $H^1(B_+)\oplus H^1(B_-)$ as $h$ varies is

$$
G_n\times G_n,\qquad
G_n=\prod_{d\mid n,\ d>1}\operatorname{Res}_{F_d/\mathbb Q}SL_2.
\tag{11}
$$

The individual factors alone suffice for Theorem 1, by the product theorem used in §5. We prove the stronger joint statement to make the independence transparent.

### 4.1. A smooth curve limit at a nodal isogeny

Choose a degeneration of the transported isogeny $\phi_2$ in §3.3 using Tate curves $E_q=\mathbb C^*/q^{\mathbb Z}$ and the isogeny $E_q\to E_{q^n}$ given by $u\mapsto u^n$. Put $q=s^2$ and choose its marked dual two-torsion point to be represented by $s$, whose image is represented by $s^n$. Thus the marked dual point tends to the node. Subtract the Tate $x$-coordinate of each marked point to put it at zero. As $s\to0$, the rational coordinates tend to

$$
U=\frac{u}{(1-u)^2}=\frac1{c-2},\qquad c=u+u^{-1}.
$$

The rational isogeny tends to

$$
\phi_0(U)=\frac1{D_n(2+1/U)-2}
          =\frac{U^n}{M_g(U)^2},
\tag{12}
$$

where

$$
D_n(c)-2=(c-2)R_g(c)^2,\qquad
M_g(U)=U^gR_g(2+1/U).
$$

The coordinates here need not be the normalized Vélu coordinates: their leading coefficient at infinity differs by the nonzero factor $n^{-2}$. A target rescaling only rescales $h$ and does not change monodromy.

The rational maps converge coefficientwise after a local analytic choice of numerator and denominator. Their limiting numerator and denominator in (12) are coprime and retain degree $n$. After a finite base cover the denominator square root can also be chosen analytically. For $h\notin\{0,-1/4,\infty\}$, both limiting hyperelliptic curves are smooth of genus $g$. Their Jacobians extend as abelian schemes near compact good-base loops. The $R_n$ endomorphisms extend across the smooth limit, preserving the rank-two module structure and self-adjointness.

Put $H=2+1/h$ and change $U$ to $1/(c-2)$ in (9). Multiplication by the square $(c-2)^{n+1}$ gives the two equations

$$
y^2=-h(c-2)(D_n(c)-H),\qquad
y^2=-h(D_n(c)-H).
\tag{13}
$$

The common scalar is a quadratic twist. It disappears after a finite cover of the parameter line and has no effect on connected monodromy. The second curve is $C_1(H/2)$; after $c\mapsto-c$ the first is $C_2(-H/2)$. We work with this sign directly. It interchanges the two finite punctures and does not alter the infinity spectrum.

### 4.2. The joint Lie algebra

We give the calculation for arbitrary odd $n$, extending the prime-degree calculation of [2, Proposition 2.2 and §3]. For $C_1$, the fibers at $H=\pm2$ have $g$ nodes and rational normalization, since

$$
D_n(c)-2=(c-2)R_g(c)^2,\qquad
D_n(c)+2=(c+2)S_g(c)^2.
$$

For $C_2$, one finite fiber has the same description. The other is the square $((c+2)S_g(c))^2$, consisting of two rational components meeting at $g+1$ nodes. At $c=-2$, completing the square in local analytic coordinates gives a total-space equation $uv=t^2$ up to a unit. Resolution inserts a rational chain. Both resulting semistable fibers have dual graph of first Betti number $g$. Thus the two finite monodromies are unipotent with logarithm of rank $g$. Their logarithms have rank at most one at every embedding of $R_n$, so each has rank exactly one at every embedding.

At infinity, scaling $c=H^{1/n}z$ gives smooth limits $w^2=z^n-1$ for $C_1$ and $w^2=z(z^n-1)$ for $C_2$, after base changes of degrees $2n$ and $n$ respectively. Applying the scaling to $c^{j-1}dc/y$ gives infinity eigenvalues $-\zeta_n^j$ for $C_1$ and $\zeta_n^j$ for $C_2$, with $1\le j\le n-1$. Polarization makes the determinant one at each embedding of $R_n$. Consequently the infinity trace elements $\beta_1,\beta_2\in R_n$ have embedding values

$$
\{-2\cos(2\pi j/n):1\le j\le g\},\qquad
\{+2\cos(2\pi j/n):1\le j\le g\}.
$$

The finite fixed lines are distinct at each embedding, because the product of their monodromies has trace $\beta_i\ne2$. Choosing bases over the product algebra $R_n$, the two finite matrices can therefore be written

$$
A=\begin{pmatrix}1&1\\0&1\end{pmatrix},\qquad
B_i=\begin{pmatrix}1&0\\-\alpha_i&1\end{pmatrix},\qquad
\alpha_i=2-\beta_i.
$$

The two lists of $\alpha_i$ embedding values are individually distinct and disjoint from one another. Indeed, if $z+z^{-1}=-(w+w^{-1})$, then $z=-w$ or $-w^{-1}$, impossible when both are $n$-th roots and $n$ is odd. All values are nonzero. The element $\alpha=(\alpha_1,\alpha_2)$ consequently has $2g$ distinct embedding values, so

$$
\mathbb Q[\alpha]=R_n\times R_n.
$$

The Lie algebra of joint monodromy contains $(E,E)$ and $(-\alpha_1F_0,-\alpha_2F_0)$, the logarithms of the finite unipotents. Iterated brackets give all three $\mathfrak{sl}_2$ directions with coefficients in $\mathbb Q[\alpha]$; positive powers suffice because $\alpha$ is invertible. Thus the Lie algebra is $\mathfrak{sl}_2(R_n)\oplus\mathfrak{sl}_2(R_n)$. The self-adjoint $R_n$ action and polarization give the matching upper bound, proving (11) for the limiting pair. Removing the quadratic twists by finite base covers does not change connected monodromy.

### 4.3. Transport to smooth fixed inputs

Take compact loops in the good $h$-base of (13) generating the indicated monodromy. The limiting branch points are distinct along those loops. The compactified hyperelliptic curves consequently form smooth proper families over a neighborhood of the loops including $s=0$. Parallel transport identifies the limiting monodromy with that in sufficiently small nonzero $s$ fibers. Thus a smooth input isogeny near the cusp already has full monodromy (11). It is unnecessary for the elliptic curves or $C_h$ themselves to remain smooth at the cusp.

More explicitly, choose a connected fine modular curve $Y$ with full level divisible by $2n$ and at least three, fixing the Weil-pairing component. It parametrizes the elliptic curves, a chosen cyclic kernel generator, and the two-torsion markings used in §3.3; every smooth marked isogeny has a lift to such a curve. Analytically $Y$ is the upper half-plane modulo a congruence subgroup and is connected. Over $Y$, let $\mathcal P\to Y$ be the target quotient by negation, a projective-line bundle. Remove its four disjoint two-torsion sections and write $\mathcal U\subset\mathcal P$ for the complement. The projection $\mathcal U\to Y$ is locally a topological bundle of four-punctured spheres.

The relative hyperelliptic compactifications of (9) define smooth proper genus-$g$ families $\mathcal B_\pm\to\mathcal U$ locally on this base. The coefficient and square-root choices in these equations can be made along each path in $Y$; their finite ambiguities give only changes of basis or finite character twists of $R^1(\mathcal B_\pm)_*\mathbb Q$. Equivalently one may pass to the corresponding finite covers, which leave connected monodromy unchanged. Thus the local systems and their polarization and $R_n$ actions transport along the entire path. Near the Tate boundary the families being extended are these hyperelliptic compactifications, over a neighborhood of the compact good-$h$ loops in the $(s,h)$ space, rather than the degenerating elliptic family. The noncollision of their branch divisors is exactly what makes those extensions smooth and proper.

Along a path of smooth marked isogenies, the fiber fundamental groups and their cohomological monodromy representations are therefore identified up to conjugation. Full monodromy at one smooth input implies full monodromy at every such input. Removing the two extra surface exclusions only adds punctures and does not decrease the image. This proves Proposition 3.

For very general $h$, connected algebraic monodromy is contained in the Hodge group; see, for example, [10, Lemma 6.1]. The self-adjoint $R_n$ action and polarization bound that group by (11). It follows that every factor $A_{\pm,d}$ is simple with endomorphism field exactly $F_d$ and Hodge group $\operatorname{Res}_{F_d/\mathbb Q}SL_2$.

## 5. From the Jacobian to every surface power

Call an abelian variety stably nondegenerate if the Hodge ring of each of its powers is generated by divisor classes. We use the following established facts [3–5]:

1. A simple abelian variety with totally real endomorphism field $F$ and odd relative dimension $\dim A/[F:\mathbb Q]$ is stably nondegenerate.
2. The product of two stably nondegenerate abelian varieties without Albert type-IV simple factors is stably nondegenerate.
3. A stably nondegenerate variety with no type-IV factors can be multiplied by a stably nondegenerate CM variety with the same conclusion.
4. Every product of elliptic curves is stably nondegenerate.

The last three statements are [5, Theorem (3.2) and Corollary (3.9)]. The first applies here with relative dimension one. Alternatively, the first fundamental theorem for $SL_2$ applied at each real embedding directly gives the divisor generators in our case.

For very general $h$, split the elliptic factors in (10) into their non-CM and CM parts. Apply statements 1–2 to all factors $A_{\pm,d}$ and the non-CM elliptic factors, and then statements 3–4 to add the CM elliptic part. Every $A_{\pm,d}$ has relative dimension one. Thus $J(C_h)$ is stably nondegenerate. Lefschetz $(1,1)$ proves HC for all its powers.

For completeness, the transfer through (8) is entirely algebraic. An Abel–Jacobi embedding $C\to J(C)$ induces an isomorphism on $H^1$. Since $H^*(C)$ is generated by $H^1$, the map $H^*(J(C)^n)\to H^*(C^n)$ is surjective in each degree. Surjections of polarizable rational Hodge structures lift rational Hodge classes. Their algebraic representatives pull back, proving HC for every $C^n$.

Resolve (8) by point blowups of $C_h^2$, obtaining $Z\to S_{\phi,h}$ generically finite of degree $N$. The blowup formula adds only Tate summands to the motive of $C_h^2$, so HC holds for every $Z^m$. For a Hodge class $\eta$ on $S_{\phi,h}^m$, algebraic pullback and pushforward give

$$
(f^m)_*(f^m)^*\eta=N^m\eta.
$$

Division by the nonzero integer $N^m$ proves Theorem 1 for very general $h$. Section 5.2 proves the finite-exception assertion.

The same reasoning applies to a finite collection whose factors $A_{\pm,d}$ are non-CM: Hazama's theorem combines all their real-multiplication factors and all their elliptic factors. Thus finite products of arbitrary powers of such surfaces also satisfy HC, with no independence assumption on those non-CM factors. Sections 5.2–5.3 give the additional criteria allowing CM factors.

### 5.1. Arithmetic specialization

Assume the marked input is defined over a number field $K$. Apply the standard fixed-prime specialization theorem [11, Lemma 2.2] to the abelian scheme

$$
\mathcal A=J(B_+)\times_U J(B_-)\longrightarrow U
$$

and one prime $\ell$. It gives a Hilbert subset $\mathcal H\subset U(K)$ on which the compact arithmetic $\ell$-adic image equals that of the family, up to conjugacy. The finite-quotient argument behind the statement is useful here. A compact $\ell$-adic image has an open normal finitely generated pro-$\ell$ subgroup $V$. Its Frattini subgroup $N$ is open and normal in the full image. Any closed subgroup surjecting onto the quotient by $N$ meets $V$ in a subgroup surjecting onto $V/N$, and hence contains $V$. It therefore equals the full image. Hilbert irreducibility applied to this finite quotient gives $\mathcal H$ with thin complement.

Comparison with topological monodromy identifies the connected geometric monodromy with (11) over $\mathbb Q_\ell$. It lies in the connected generic arithmetic monodromy, so the same inclusion holds at each $h\in\mathcal H$. Every geometric endomorphism of either specialized Jacobian is defined over a finite extension of $K$ and therefore commutes with its connected arithmetic monodromy. The centralizer of $G_n$ on the natural $R_n$-rank-two representation is $R_n\otimes\mathbb Q_\ell$. Since the algebraic dihedral correspondences already supply $R_n$, we obtain

$$
\operatorname{End}^0_{\overline K}J(B_+(h))
=\operatorname{End}^0_{\overline K}J(B_-(h))=R_n.
$$

Each factor $A_{\pm,d}$ is consequently absolutely simple of type I and relative dimension one. The Ribet–Hazama argument above proves the arithmetic corollary. This argument needs neither the Mumford–Tate conjecture nor simultaneous control at every prime. It does not identify individual elements of $\mathcal H$; Theorem 2 supplies a separately certified parameter without asserting that it belongs to this subset.

### 5.2. Only finitely many parameters need be excluded

We first record the precise specialization input for abelian varieties. Its proof also covers possible quaternionic endomorphisms at special parameters.

**Lemma 4.** Let $A$ have dimension $r$ and admit a self-adjoint totally real field $F$ of degree $r$, for a polarization. Then $A$ is isotypic. If $A$ is not of CM type, it is stably nondegenerate and has no type-IV simple factors.

**Proof.** Each isotypic component is $F$-stable. At every real embedding of $F$, its cohomology has a nondegenerate alternating form, so its $F$-rank is even. Every nonzero component consequently has dimension at least $r$, proving isotypy.

Write $A\sim B^m$ with $B$ simple of dimension $b$ and $D=\operatorname{End}^0(B)$. Transport the polarization to $B^m$. Positivity of Rosati identifies the real matrix involutions, after a change of basis, with transpose or conjugate transpose. Commuting self-adjoint matrices are simultaneously diagonalizable, with real diagonal entries. This gives the following bounds, where $e$ is the degree of the totally real center in types I–III and half the degree of the CM center in type IV; in the last row $[D:Z(D)]=d^2$.

| Albert type | $M_m(D)\otimes\mathbb R$ | Maximum self-adjoint commutative dimension | Lower bound on $b$ |
|---|---|---:|---:|
| I | $\prod_{1}^{e}M_m(\mathbb R)$ | $me$ | $e$ |
| II | $\prod_{1}^{e}M_{2m}(\mathbb R)$ | $2me$ | $2e$ |
| III | $\prod_{1}^{e}M_m(\mathbb H)$ | $me$ | $2e$ |
| IV | $\prod_{1}^{e}M_{md}(\mathbb C)$ | $mde$ | $d^2e$ |

For the lower bounds, $H^1(B,\mathbb Q)$ is a faithful module over the division algebra $D$, so its dimension is a positive multiple of $[D:\mathbb Q]$. In type I its rank over the real center is additionally even by polarization. Since the embedded self-adjoint algebra $F\otimes\mathbb R$ has dimension $r=mb$, comparison of the last two columns excludes type III, forces $d=1,b=e$ in type IV, and gives $b=e$ or $b=2e$ in types I and II respectively. The type-IV case has a CM field of degree $2b$, hence is CM. The other two cases have relative dimension one.

For clarity, the centralizer argument in the two non-CM cases is as follows. Put $V=H^1(B,\mathbb C)$. In type I,

$$
V=\bigoplus_{\tau:Z(D)\hookrightarrow\mathbb R}W_\tau,
\qquad D\otimes\mathbb C=\bigoplus_\tau\mathbb C;
$$

in type II,

$$
V=\bigoplus_\tau(W_\tau\otimes\mathbb C^2),
\qquad D\otimes\mathbb C=\bigoplus_\tau M_2(\mathbb C).
$$

Each $W_\tau$ has dimension two. The connected Lefschetz group is $\prod_\tau SL(W_\tau)$, acting on the first tensor factor. The Hodge group $H$ is connected semisimple [5, §1.2], and its commutant on $V$ is exactly $D\otimes\mathbb C$, because rational Hodge endomorphisms of $H^1(B)$ are abelian-variety endomorphisms. A zero Lie-algebra projection to one $\mathfrak{sl}(W_\tau)$ would put all $\operatorname{End}(W_\tau)$ in that commutant, contrary to the displayed formulas. Every projection is therefore surjective: a nonzero semisimple Lie subalgebra of $\mathfrak{sl}_2$ is all of $\mathfrak{sl}_2$. A proper semisimple subdirect Lie algebra in a product of simple $\mathfrak{sl}_2$ factors is diagonal on a block of at least two factors. Since all automorphisms of $\mathfrak{sl}_2(\mathbb C)$ are inner, their standard modules become isomorphic. A nonzero map between the corresponding summands of $V$ then commutes with $H$, whereas $D\otimes\mathbb C$ has no such off-diagonal maps. This contradiction proves equality of the Hodge and connected Lefschetz groups. The invariant-theory criterion [5, (1.8)] proves stable nondegeneracy, also for $B^m$. $\square$

We may retain small CM factors under the following explicit criterion. For any finite collection of source Jacobians, suppose every CM simple factor has dimension at most two, and that for each normal closure of a quartic CM endomorphism field there are at most two distinct simple CM surface isogeny types. Multiplicities are unrestricted. A simple CM surface has a faithful irreducible two-dimensional norm-one module whose splitting field is its normal closure, with Galois group $C_4$ or $D_4$ [5, §4]. A product of at most two such surfaces is stably nondegenerate [5, Theorem (0.1)(iv)]. Group the surfaces by their normal closures. Different groups have no common rational torus constituent, since a faithful constituent determines its splitting field. Their joint Hodge group is consequently the full product. The CM elliptic factors have one-dimensional characters and also separate after isogenous copies are combined. Thus the entire CM part is stably nondegenerate. This argument concerns the full collection; pairwise independence alone would not suffice for arbitrary collections of tori.

The restriction for each normal closure has content. For example, put $x=i\sqrt{3+\sqrt2}$ and $y=i\sqrt{3-\sqrt2}$. The quartic CM fields $\mathbb Q(x)$ and $\mathbb Q(x+y)$ have distinct real subfields and the same normal closure $\mathbb Q(x,y)$ with group $D_4$ of order eight. In its natural rational module, the two primitive types of the first field have cocharacters $(1,1)$ and $(1,-1)$. For the second field, the pair representatives $x+y,x-y$ and the intertwiner $\frac12\left(\begin{smallmatrix}1&1\\1&-1\end{smallmatrix}\right)$ give a type with vector $(1,0)$. These three pairwise nonisogenous types have the relation $2(1,0)=(1,1)+(1,-1)$. Their joint Galois orbit spans dimension four, although each pair spans dimension four. Thus one cannot infer a rank-six product torus from the pair calculations. This concerns the proof of independence, not nonalgebraicity of the resulting extra Hodge classes.

Up to isogeny write the product of source Jacobians as $N\times C$, with $N$ non-CM and $C$ CM. Lemma 4 and Hazama's theorem give HC on all powers of $N$, and the preceding criterion gives it for $C$. The group of $N$ is semisimple, whereas that of $C$ is a torus [5, §1.2]. Their product has the full product Hodge group: a proper connected subdirect product would require a nonzero common Lie-algebra quotient, and a semisimple Lie algebra has no nonzero abelian quotient. Hence all Hodge invariants are external products of algebraic classes. Curve-square transfer proves HC for all surface powers and finite mixed products under this criterion.

Apply the lemma to every factor $A_{\pm,d}$ of §3.3. For $r_d=[F_d:\mathbb Q]>1$, only finitely many parameters make this factor CM. To see this, pass to a finite cover so the factor is an abelian scheme with a fixed polarization type. A further finite cover chooses a maximal isotropic subgroup of the polarization kernel; its quotient gives a principally polarized isogenous family. Connected monodromy and the CM condition are unchanged.

The moduli image is a nonconstant algebraic curve in $\mathcal A_{r_d}$, since its connected monodromy is $\operatorname{Res}_{F_d/\mathbb Q}SL_2$. Its generic Hodge group is the same group. The relevant Hilbert modular Shimura datum has group

$$
\mathcal G_F=\{g\in\operatorname{Res}_{F/\mathbb Q}GL_2:
             \det_F(g)\in\mathbb G_m\},\qquad F=F_d,
$$

where $\mathbb G_m$ is embedded diagonally in $\operatorname{Res}_{F/\mathbb Q}\mathbb G_m$. Its symplectic representation is the trace to $\mathbb Q$ of an $F$-alternating form. Its domain has connected components isomorphic to $\mathfrak H^{r_d}$. The full derived generic group and the weight homotheties give this generic Mumford–Tate group, up to isogeny of its center. A special subvariety with that generic group has dimension $r_d>1$, so the image curve is not special. André–Oort for $\mathcal A_{r_d}$ [12, Theorem 5.2] implies that it contains only finitely many CM points. The map from the finite parameter cover to its image curve is nonconstant, hence quasi-finite; the inverse image of that finite CM set is finite and so is its image in the original parameter line. The argument also applies to complex isogeny inputs not defined over a number field: an algebraic curve with infinitely many algebraic CM points would itself be defined over $\overline{\mathbb Q}$; see [12, §6.1].

Let $\Xi_\phi$ consist of the parameters at which some $A_{\pm,d}$ is CM and its simple isogeny factor has dimension at least two. It lies in the finite union just established for $r_d>1$, hence is finite. Outside it all CM simple factors are elliptic, so the preceding criterion proves the cofinite assertion of Theorem 1. Higher-dimensional CM factors that are powers of elliptic curves remain allowed. Set $\Sigma_\phi=\Xi_\phi$ except in degrees five, seven and nine, where the following arguments permit $\Sigma_\phi=\varnothing$.

For $n=5$, both Jacobians in (9) are abelian surfaces, so their source contains at most two simple CM surface isogeny types and every admissible parameter is covered. In fact arbitrary finite mixed products of degree-five members also satisfy the criterion. Their simple CM surface fields contain $F_5=\mathbb Q(\sqrt5)$. Within a $D_4$ normal closure the two conjugacy families of quartic CM fields have different real quadratic subfields; thus only one family can contain $F_5$, and it supplies at most two simple CM isogeny types [5, §4.1]. A cyclic quartic field supplies only one. Distinct normal closures separate as above.

### 5.3. CM threefold factors and the all-parameter cases

We need an all-powers statement about the CM part, including its mixed classes. For one simple threefold $A$ and a CM elliptic curve $E$, HC on every $A^r\times E^s$ is already Milne's Corollary 1.11 [8]. In that proof the elliptic polarization can be rescaled to split the Weil determinant [8, Proposition 1.6], so Markman's earlier split-fourfold theorem suffices. The later unrestricted Weil-fourfold theorem [9, Theorem 1.2] also supplies those classes. The argument below treats any finite collection under explicit joint independence conditions; in particular, every pair of threefold types is allowed.

**Proposition 5.** Let $C$ be isogenous to a finite product of elliptic curves, abelian surfaces, and simple CM threefolds. Suppose its simple CM surface factors satisfy the bound for each normal closure of §5.2. For each normal closure $N$ of a sextic CM endomorphism field, impose the following bounds on the number of distinct simple threefold isogeny types:

| $\operatorname{Gal}(N/\mathbb Q)$ | Allowed number of types with normal closure $N$ |
|---|---:|
| $C_6$ | One, which is automatic |
| $C_2\times S_3$ | At most two |
| $(C_2)^3\rtimes C_3$ or $(C_2)^3\rtimes S_3$ | At most three |

Then HC holds for every power of $C$. Multiplicities and the number of different normal closures are unrestricted. In particular, the conclusion holds for any two simple CM threefold types with arbitrary sextic fields, together with the permitted small factors.

**Proof.**

**Step 1: classification and pair Hodge groups.** We first determine the Hodge group of a pair of threefolds. For a simple CM threefold $A$ with endomorphism field $L$, put $F=L^+$. Its Hodge group is the norm-one torus $U_L$ for $L/F$ [5, §2.3], and its individual powers are stably nondegenerate [5, Theorem (2.7)]. Write $M,N$ for the normal closures of $F,L$, and $P=\operatorname{Gal}(M/\mathbb Q)$, either $C_3$ or $S_3$.

The group $G=\operatorname{Gal}(N/\mathbb Q)$ embeds in the signed-permutation group $(C_2)^3\rtimes S_3$. Its sign kernel $V=\operatorname{Gal}(N/M)$ is stable under $P$ and contains complex conjugation $c=(1,1,1)$. The permutation module over $\mathbb F_2$ splits under $C_3$ into the diagonal line and an irreducible plane. Thus $V$ is the diagonal line or the full sign group.

When $V$ is diagonal and $P=C_3$, the field $L=FK$ is cyclic sextic, for an imaginary quadratic field $K$. When $V$ is diagonal and $P=S_3$, we have $G=C_2\times S_3$. To exclude a nonsplit extension, the order-two field stabilizer $\operatorname{Gal}(N/L)$ supplies an involution $b$ lifting a transposition, since it avoids $c$. If $a$ is a Sylow-three generator, then $bab^{-1}=a^{-1}$: the other lift $ca^{-1}$ has order six. Thus $a,b$ generate a complement. In this case $L=FK$ also, but $N$ has two imaginary quadratic subfields $K$ and $K'=Kk_0$, where $k_0$ is the real discriminant field of $M$. The sextic CM subfields fall into the two conjugacy families $FK$ and $FK'$.

For either split field $FK$, the rational cocharacter module is

$$
X_*(U_L)_{\mathbb Q}=\chi_K\oplus(\chi_K\otimes V_0),
$$

where $V_0$ is the sum-zero plane of the three-dimensional permutation module of $P$. The plane constituent is irreducible and faithful for $\operatorname{Gal}(MK/\mathbb Q)$, so its splitting field has degree six or twelve. Its commutant is $\mathbb Q(\zeta_3)$ for $P=C_3$, and $\mathbb Q$ for $P=S_3$.

When $V$ is full, $G=(C_2)^3\rtimes P$. For $P=C_3$ the norm-one module is the natural signed-permutation module $W$ of dimension three, and all sextic fields with this normal closure are conjugate. Indeed their stabilizers in $V$ are kernels of odd-support characters: the three coordinate characters are conjugate, and the total-sign character gives a cyclic sextic field with smaller normal closure.

For $P=S_3$ there are two field families, with modules $W$ and $W\otimes\operatorname{sgn}(P)$. Here is the full stabilizer calculation. The group $V$ is the maximal normal two-subgroup of $G$, so $M=N^V$ is intrinsic. Fix a real cubic embedding. Its stabilizer is $V\rtimes C_2$, where $C_2$ fixes coordinate one and interchanges coordinates two and three. The quadratic characters taking $c$ to $-1$ are, in additive notation,

$$
s_1+b(s_2+s_3)+d\varepsilon,\qquad b,d\in\mathbb F_2,
$$

where $\varepsilon$ is permutation parity. The $b=1$ characters extend to $G$ and give split fields $FK$ with smaller normal closure. The two $b=0$ families give $W$ and its permutation-sign twist, and the fields within each family are conjugate. Both modules are faithful and absolutely irreducible with commutant $\mathbb Q$: independent sign changes force a commuting matrix to be diagonal, and transitive permutations make it scalar. The two modules are inequivalent. An intertwiner would be diagonal; a transposition fixing a coordinate forces the corresponding entry to vanish.

The Hodge group of a product is a connected subdirect product. Rational finite-Galois representations are semisimple, so a proper subdirect product of tori identifies a nonzero common quotient constituent. Isomorphic faithful constituents have the same splitting field. For two irreducible three-dimensional modules, the classification therefore either separates them or identifies their fields within one family. In the latter case every equivariant map is scalar. Proportional Hodge-circle cocharacters, whose coordinates are all $1$ or $-1$, differ by $1$ or $-1$. Their CM types agree or are conjugate under a field identification, hence the abelian varieties are isogenous [5, §4.1]. Nonisogenous factors consequently have independent three-dimensional modules.

For the split fields, the one-dimensional constituents coincide exactly when $K$ agrees. Orient each $K$-action to have excess signature $+1$. A primitive sextic type has signature $(2,1)$ or $(1,2)$; constant signs would give a type induced from $K$. A shared norm character then imposes equality of the two norms. The two-dimensional constituents can agree only when the normal closures agree. A fixed cyclic sextic field has one primitive type orbit under its automorphisms: cyclic rotation and simultaneous negation act transitively on the six nonconstant sign vectors. Such factors are isogenous and are combined.

For a common order-twelve normal closure and the same $K$, an intertwiner on the planes is scalar. The primitive cocharacters project to signed permutations of $(2,-1,-1)$. Proportional projections arise only from equal or opposite sign vectors, again giving isogeny. If the quadratic fields are $K$ and $K'=Kk_0$, their plane modules are isomorphic but their Hodge cocharacters cannot be identified. Explicitly,

$$
Jx=(1,1,1)\times x
$$

intertwines the standard plane with its sign twist. Every intertwiner is a scalar multiple of $J$. Its image on a primitive projection has one zero coordinate, whereas every primitive projection has three nonzero coordinates. Thus the two planes are independent also in this case.

It follows that for two nonisogenous simple CM threefolds $A,B$ the group is the full product unless their fields share an imaginary quadratic field $K$. In that case it is

$$
\operatorname{Hg}(A\times B)=U_{L_A}\times_{U_K}U_{L_B},
$$

with the two norm maps and the orientations just chosen. This fiber product is connected: its defining character over a splitting field has coefficients $(1,1,1,-1,-1,-1)$ and is primitive. Thus the Lie-algebra calculation gives equality of groups.

**Step 2: joint independence.** We now verify joint independence for the entire collection. If $W$ is an absolutely irreducible rational Galois module of dimension $d$, the orbit span of $(v_1,\ldots,v_m)\in W^m$ has dimension $d\operatorname{rank}(v_1,\ldots,v_m)$. Indeed the rational group algebra acts as $\operatorname{End}_{\mathbb Q}(W)$, and the span is the image of $M\mapsto(Mv_1,\ldots,Mv_m)$. Within a full-sign module family the fields can be identified, and distinct CM types up to conjugation give distinct cube diagonals in $\mathbb Q^3$. Any three of the four diagonals are independent: their determinants have absolute value four. Thus the stated bound gives independent copies of $W$; the two order-forty-eight module families separate because they are inequivalent. For an order-twelve closure the preceding plane calculation shows that any two distinct types have independent projected cocharacters, including types with different quadratic subfields. The cyclic sextic case has only one primitive isogeny type. Distinct faithful splitting fields give nonisomorphic constituents. Therefore all nonquadratic constituents in the entire collection are independent. The only identifications are the shared one-dimensional norm characters for equal $K$. This is a joint orbit-span argument, not an inference from pairwise independence.

**Step 3: algebraic generation on every power.** For a threefold whose field contains $K$, adjoin an elliptic curve $E_K$ with the opposite excess signature. Its product Hodge group is the inverse-norm graph [5, §5.3]. Milne's result supplies algebraic classes on every power of $A\times E_K$. Over a finite splitting field $\Omega$, the weights of $H^1$ are $\pm e_1,\pm e_2,\pm e_3,\pm f$, with $e_1+e_2+e_3+f=0$. In particular the two Weil determinant lines of these four weights and their negatives are algebraic with $\Omega$ coefficients.

For all threefolds $A_1,\ldots,A_r$ sharing $K$, use the same $E_K$. The complete character relations on their product with $E_K$ are

$$
e_{i1}+e_{i2}+e_{i3}+f=0,\qquad 1\le i\le r.
\tag{N}
$$

The joint calculation gives the rational relation space. These rows generate its full integral lattice: the columns $e_{i1}$ form an identity minor, so the lattice is saturated and its kernel torus is connected, of dimension $2r+1$. For any two members $A,B$ of this group, denote their weights by $e_1,e_2,e_3$ and $g_1,g_2,g_3$. Besides the known four-weight determinants we use a mixed six-weight determinant. Let $\alpha$ be the algebraic Weil class of weights $(e_1,e_2,e_3,f)$ and $\beta$ the class of weights $(-g_1,-g_2,-g_3,-f)$. With the evident projections from $A\times B\times E_K$, form

$$
p_{AB*}\bigl(p_{AE}^*\alpha\smile p_{BE}^*\beta\bigr)
\in H^6(A\times B,\Omega).
$$

Pullback, intersection and proper pushforward preserve algebraicity. The elliptic pairing between the $f$ and $-f$ lines is nonzero, so this class spans the determinant line of weights $(e_1,e_2,e_3,-g_1,-g_2,-g_3)$. The conjugate construction gives its negative. This is an actual codimension-three cycle operation and needs no Weil-sixfold algebraicity theorem.

On every power, torus invariants in the exterior algebra of $H^1$ are spanned by zero-weight monomials. Cancel opposite-weight pairs, which are algebraic by Lefschetz $(1,1)$. By the integral norm relations (N), the remaining signed excess is $p_i$ on each of the three weights of $A_i$, and $s=\sum_i p_i$ on $f$. Match positive units $p_i$ against negative units $-p_j$, using one mixed determinant for each matched unit. This is an integral matching: repeatedly subtract the smaller of the two remaining amounts until one side is exhausted. Every unmatched unit then has the sign of $s$. Use the corresponding four-weight determinant, or its negative, for each unmatched unit; these use exactly the remaining $|s|$ weights on $E_K$. This exhausts the monomial. The argument holds for arbitrary integers $p_i$ and arbitrary $r$, so proves generation in every degree on every power.

Every placement of these determinants is algebraic. Pull back along addition on $A^3\times E_K$, or $A^3\times B^3$ for the mixed class, and project to degree one on each factor. The degree-$j$ projector on an abelian variety is a Lagrange polynomial in $[m]^*$, with value one at $m^j$ and zero at the other eigenvalues $m^i$, for any integer $m>1$. Apply the primitive idempotents of $L_A\otimes\Omega$, $L_B\otimes\Omega$ and $K\otimes\Omega$ to select the embeddings. These are linear combinations of algebraic endomorphism correspondences. Coordinate maps and diagonals then give every placement, including distinct weights in the same copy. In any fixed degree, if $Z$ is the rational span of algebraic classes and $H$ the rational Hodge-class space, we have proved $Z\otimes\Omega=H\otimes\Omega$. Faithful scalar extension gives $Z=H$.

Different $K$ groups and the isolated irreducible-torus factors have a full product Hodge group by the joint constituent calculation. The isolated factors are stably nondegenerate. Auxiliary elliptic factors can be removed by restriction to their origins. Isogenous multiplicities and existing matching CM elliptic factors are included by taking powers of the constructed groups.

A simple CM surface has a simple two-dimensional rational torus module with faithful splitting field of degree four or eight [5, §4]. It has no constituent in common with a three-dimensional module, a one-dimensional elliptic character, or a sextic plane constituent whose faithful splitting field has degree six or twelve. This separation persists for arbitrary products of CM surfaces, whose modules are subquotients of sums of their individual modules. The entire small CM part has HC on every power by the criterion of §5.2. Its remaining elliptic characters are distinct from the $K$ already grouped above, so the full product group gives only external products of algebraic invariants. Finally, the non-CM elliptic and surface factors have semisimple Hodge group and are stably nondegenerate [5, §§2.1–2.2]; Hazama [5, Theorem (3.2)] handles all their products and powers. Their group separates from the CM torus as in §5.2. This proves the proposition. $\square$

For $n=7$, the two Jacobians in (9) have dimension three and maximal self-adjoint multiplication by the cyclic cubic $F_7$. Lemma 4 makes each isotypic. A CM one is an elliptic cube or a simple CM threefold. For $n=9$, $R_9=\mathbb Q\times F_9$, so each Jacobian has an elliptic factor and a three-dimensional factor of the same kind. The source (10) therefore has at most two simple CM threefold isogeny types. Proposition 5 handles its CM part. Lemma 4 and Hazama handle its non-CM part, whose semisimple Hodge group separates from that torus. The transfer in §5 proves HC for every power of every admissible surface in degrees seven and nine.

The same proof gives the mixed-product criterion: in a finite collection of source Jacobians, allow CM elliptic factors, simple CM surfaces satisfying the bound for each normal closure of §5.2, and simple CM threefolds satisfying Proposition 5's bounds within each sextic normal closure. Every power of the corresponding surface product satisfies HC. The real cubic fields need not coincide, and there is no bound on the total number of different normal closures or threefold types.

The bound matters. For an irreducible signed-permutation module, four CM types with sign vectors

$$
(1,1,1),\quad(1,-1,-1),\quad(-1,1,-1),\quad(-1,-1,1)
$$

sum to zero. Their joint torus has an extra relation although every triple is independent. Similarly, the three primitive projected cocharacters for one order-twelve field family sum to zero. Thus the bounds within each normal closure have content; this proof does not cover arbitrary mixed CM members.


### 5.4. Fivefolds with cyclic real multiplication

The split Weil-sixfold part of Markman's theorem [9, Theorem 1.2] permits a further extension. Its single-factor input is a direct consequence of existing results. Let a simple CM fivefold $A$ contain an imaginary quadratic field $K$ acting with signature $(3,2)$, and choose the opposite action on a CM elliptic curve $E_K$. Then $A\times E_K$ is of Weil type. Proposition 1.6 of [8] gives a polarization with determinant $-1$ modulo $\operatorname{Nm}_{K/\mathbb Q}(K^\times)$, hence split Weil type in dimension six [9, §11.5]. Its Weil classes are algebraic. Since $A$ is stably nondegenerate [5, Theorem (2.7)], [8, Proposition 1.10 and Theorem 1.8] also give HC on every $A^r\times E_K^s$. The result below assembles this known input with the threefold cycles using a joint classification different from the cubic case.

**Proposition 6.** Start with a product satisfying Proposition 5 and adjoin finitely many simple CM fivefold isogeny types $A$, each with endomorphism field $L$ whose real subfield $L^+$ is cyclic of degree five over $\mathbb Q$. Combine isogenous factors and impose these conditions:

1. For each cyclic degree-ten field $L$, allow at most one simple isogeny type. Write $L=L^+K$. If its $K$-signature is $(4,1)$ up to conjugation, require that no other simple factor contribute the norm character of $K$: there is no CM elliptic factor with field $K$, and no distinct threefold or fivefold factor whose endomorphism field contains $K$.
2. For each normal closure with full sign kernel, require the CM cocharacters to be linearly independent within each of the three module families described below. It suffices to have at most three distinct types in each family. In particular, any two fivefold types with full sign kernel satisfy this condition.

Then HC holds on every power of the enlarged product. Multiplicities and the number of different normal closures are unrestricted.

**Proof.** Put $F=L^+$ and let $N$ be the normal closure of $L$. The action on the five conjugate pairs of embeddings places $G=\operatorname{Gal}(N/\mathbb Q)$ inside $(C_2)^5\rtimes C_5$. Its sign kernel is a $C_5$-invariant subspace of $\mathbb F_2^5$ containing complex conjugation $(1,1,1,1,1)$. This permutation module is the sum of the diagonal line and an irreducible four-dimensional module: the order of $2$ modulo $5$ is four. Thus the sign kernel is diagonal or all of $(C_2)^5$. The extension splits because the quotient has odd order five.

With diagonal kernel, $G=C_{10}$ and $L=N=FK$. Its norm-one cocharacter module is

$$
X_*(U_L)_{\mathbb Q}=\chi_K\oplus(\chi_K\otimes V_4),
$$

where $V_4$ is the rational cyclotomic representation of $C_5$. The four-dimensional constituent is irreducible and faithful, with commutant $\mathbb Q(\zeta_5)$. Cyclic rotation and simultaneous negation give three primitive type orbits: one has signature $(4,1)$ and two have signature $(3,2)$. Constant signs are induced from $K$ and do not give simple fivefolds. Each primitive type has nonzero projections onto both constituents and spans the full norm-one module. Distinct types over one $L$ share the four-dimensional constituent and are excluded by condition 1. Distinct faithful splitting fields give nonisomorphic constituents.

With full sign kernel $V=(C_2)^5$, the group is $V\rtimes C_5$, of order $160$. Its maximal normal two-subgroup is $V$, so $F=N^V$ is intrinsic. A degree-ten CM subfield containing $F$ is fixed by the kernel of a character of $V$ taking complex conjugation to $-1$. Such characters are indexed by odd-support vectors in $\mathbb F_2^5$. The all-ones vector gives a cyclic degree-ten field with smaller normal closure. The other fifteen vectors form three $C_5$-orbits: one of weight one and two of weight three. These give the three field families. For each orbit $\mathcal O$, its norm-one module is the rational representation

$$
W_{\mathcal O}=\operatorname{Ind}_{V}^{V\rtimes C_5}\lambda,
\qquad \lambda\in\mathcal O.
$$

The five distinct sign characters force any commuting matrix to be diagonal; cyclic transitivity makes it scalar. Thus this module is absolutely irreducible. It is faithful: the five odd vectors in the orbit span $\mathbb F_2^5$, because their projections onto both the diagonal and irreducible four-dimensional summands are nonzero. Nontrivial permutations also act nontrivially. The three modules are inequivalent because their restrictions to $V$ have disjoint character sets, and their faithful splitting field is $N$.

Within a family identify the conjugate fields. Types up to conjugation are the sixteen cube diagonals in $\mathbb Q^5$, and Proposition 5's joint orbit-span formula applies. Any three distinct diagonals are independent. Orient their first coordinates to be one. If $v_3=a v_1+b v_2$, then $a+b=1$. A coordinate where $v_1,v_2$ disagree gives $a-b\in\{1,-1\}$, so one coefficient is zero. Condition 2 therefore gives independent copies of each $W_{\mathcal O}$. Different families and faithful normal closures separate. The new nonquadratic constituents have dimensions four and five, and cannot identify with the constituents of dimensions two and three in Proposition 5. Only the one-dimensional norm characters $\chi_K$ can be shared.

A permitted signature-$(4,1)$ type has no shared constituent, including $\chi_K$. Its powers are stably nondegenerate [5, Theorem (2.7)], and it separates from the remaining product. For each other shared $K$, orient the threefold and fivefold actions to have excess signature $+1$ and use one oppositely oriented auxiliary elliptic curve $E_K$. Writing their dimensions as $g_i\in\{3,5\}$, the complete integral weight relations are

$$
\sum_{j=1}^{g_i}e_{ij}+f=0,\qquad 1\le i\le r.
\tag{N5}
$$

Their row lattice is saturated because the columns $e_{i1}$ give an identity minor. The rational calculation therefore identifies the whole connected Hodge group, of dimension $1+\sum_i(g_i-1)$.

The known fourfold and sixfold theorems supply algebraic determinant classes $D_i$ of the $g_i+1$ weights in (N5), and their negatives. Pull $D_i$ and the opposite $D_j$ to $A_i\times A_j\times E_K$, intersect, and push forward along $E_K$. The nonzero elliptic pairing gives the mixed determinant of weights

$$
(e_{i1},\ldots,e_{ig_i},-e_{j1},\ldots,-e_{jg_j}),
$$

an algebraic class of codimension $(g_i+g_j)/2$. This includes the threefold–fivefold and fivefold–fivefold contractions and uses no general Weil-eightfold or Weil-tenfold theorem.

Proposition 5's generation argument applies with these variable block sizes. After opposite-weight cancellation, a zero-weight monomial has equal signed excess $p_i$ on the $g_i$ weights of $A_i$ and excess $\sum_i p_i$ on $f$. Match positive and negative units using mixed determinants; all unmatched units have the sign of the remaining elliptic excess and are supplied by $D_i$ or its negative. Addition maps, degree projectors and CM idempotents give every placement on every power. Faithful scalar extension gives rational algebraicity. Remove auxiliary elliptic factors by restriction to their origins. The isolated constituents, allowed surface configurations and non-CM factors separate as in Proposition 5. $\square$

For $n=11$, each $B_\pm$ has dimension five and maximal self-adjoint multiplication by the cyclic real quintic $F_{11}$. Lemma 4 makes a CM specialization either an elliptic fifth power or a simple CM fivefold. Thus the source contains at most two simple fivefold types. The non-CM source factors are handled by Lemma 4 and Hazama as before. Proposition 6 covers every configuration in which all simple CM fivefold types present have full sign kernel, and cyclic cases satisfying condition 1 and its restrictions relative to the other source factors. This is an additional CM-specialization criterion, not an all-parameter degree-eleven theorem. Two distinct types over one cyclic degree-ten field, or a signature-$(4,1)$ factor sharing $K$, remain outside this argument. The latter requires a determinant on $A\times E_K^3$, a Weil eightfold, and is not covered by the sixfold input.

## 6. A specified algebraic member with geometric genus five

### 6.1. Equations over a number field

Let $e$ be a root of $e^3-264e+1694=0$, and put $L=\mathbb Q(e)$. Begin with

$$
E_s:y^2=x^3-264x+1694
$$

and the kernel polynomial

$$
K_0=x^5-44x^4+220x^3+6776x^2-71632x+166496.
$$

For $f_0=x^3-264x+1694$, define

$$
P_0=(11x-88)K_0^2+4f_0(K_0'^2-K_0K_0'')-2f_0'K_0'K_0.
\tag{14}
$$

This gives the normalized cyclic-11 isogeny to

$$
E_t:y^2=w^3-31944w-2254714.
$$

The supplement verifies the rational-function curve identity exactly; equivalently $(x,y)\mapsto(P_0/K_0^2,y(P_0/K_0^2)')$. Its degree is 11, and it fixes the origin, so it is an isogeny with cyclic kernel. Its target two-torsion values are $-11$ times the source values.

Choose the marked source point $x=e$, so $e_3=-11e$ on the target. Put $a=3e$, $b=3e^2-264$. On the source two-isogenous curve,

$$
f(U)=U^3-2aU^2+(a^2-4b)U.
$$

Define the monic polynomial

$$
M(U)=\operatorname{monic}\operatorname{Res}_x
\left(K_0(x+e),x^2-(U-a)x+b\right).
$$

With $S_1=-[U^4]M$, set

$$
\begin{aligned}
N&=(11U-2S_1)M^2+4f(M'^2-MM'')-2f'M'M,\\
P&=N-24eM^2,\qquad G=P-M^2.
\end{aligned}
\tag{15}
$$

The constant subtraction puts the target marked dual two-torsion at zero. Over $L$, the verifier checks $P=UH_5^2$ and

$$
f(U)(P'M-2PM')^2
=P^3+66eP^2M^2+121(1056-3e^2)PM^4.
\tag{16}
$$

Thus our specified curves are $B_+:y^2=G$ and $B_-:y^2=UG$, with $h=1$ in this normalization.

To specify the surface itself, put $a_t=-33e$, $b_t=121b$, and choose an order of the two roots $r_1,r_2$ of $r^2+a_tr+b_t=0$. Set

$$
e_i=r_i-11e,\qquad
k=\frac{4r_2}{127777+66e-363e^2},\qquad
 d=x+e_2/11.
$$

Insert these data and (14) into (1). This defines $S_1$ over $L(r_1,r_2)$. Either fixed ordering gives a surface to which the argument applies. Admissibility follows already modulo 7: with $e=0$, the duplication coordinate is $r=5$, which is nonzero and is not a root of $r^2+4$. Smoothness of the two source curves follows from their squarefree reductions recorded next.

### 6.2. Exact finite-field evidence

There are linear primes above 7 and 13 with residues $e=0$ and $e=4$, respectively. Both $G$ and $UG$ are squarefree at these primes. Exact point counts recover the following Frobenius polynomials:

| Prime | $B_+$ | $B_-$ |
|---|---|---|
| $7$ | $T^{10}+88T^5+16807$ | $T^{10}-72T^5+16807$ |
| $13$ | $T^{10}-146T^5+371293$ | $T^{10}+398T^5+371293$ |

For extension degrees $j=1,2,3,4$, each curve has $p^j+1$ points. At degree five the counts are $(17248,16448)$ for $p=7$ and $(370564,373284)$ for $p=13$, in the order $(B_+,B_-)$. The first five coefficients follow from Newton identities; the functional equation determines the remaining coefficients. The sparse form is an output of this calculation, not an assumption.

The counter uses explicitly verified irreducible moduli for each finite field and enumerates squares to evaluate quadratic characters. The odd-degree curve has one point at infinity; the even-degree monic curve has two. Independent prime-field counts use modular exponentiation. A sixth-extension count at 7, not used to reconstruct the Frobenius polynomials, gives $117650$ for both curves, agreeing with the predicted value. This is an additional consistency check using the same extension-field counter, not a second independent implementation of it.

### 6.3. Excluding CM without claiming absolute simplicity

Write a polynomial from the table as $T^{10}+A T^5+p^5$. Since $p\nmid A$, the reduction is ordinary. The fifth power of Frobenius has characteristic polynomial $(T^2+A T+p^5)^5$. Honda–Tate theory [13] therefore identifies the geometric reduction with the fifth power of an ordinary elliptic curve whose endomorphism field is

$$
K_p=\mathbb Q\left(\sqrt{A^2-4p^5}\right).
$$

The corresponding squarefree radicands are

| Jacobian | At 7 | At 13 |
|---|---:|---:|
| $J(B_+)$ | $-14871$ | $-91491$ |
| $J(B_-)$ | $-15511$ | $-82923$ |

Let $A_0$ be either characteristic-zero Jacobian, with its algebraic action of the cyclic real quintic field $F=\mathbb Q(\zeta_{11}+\zeta_{11}^{-1})$. This action makes $A_0$ isotypic: its restriction to every isotypic component is faithful, so the rational $H^1$ dimension of each component is divisible by 5 and, being even, by 10. As $\dim A_0=5$, there is only one component. Consequently $A_0$ is either simple or isogenous to an elliptic fifth power.

If $A_0$ is simple, its division endomorphism algebra has rational dimension dividing $\dim_{\mathbb Q}H^1(A_0)=10$. A division algebra's degree over its center is a square, so this algebra is a field. Containing $F$, it is either $F$ or a degree-ten CM field. If $A_0$ is an elliptic fifth power, its elliptic factor is either CM or non-CM.

Specialization injects the characteristic-zero endomorphism algebra into the endomorphism algebra of each good geometric reduction. Endomorphisms may be defined over a finite extension; using geometric reductions accommodates this without requiring them to be defined over $\mathbb F_p$. That target algebra is $M_5(K_p)$.

Since $F\otimes K_p=FK_p$ is a degree-five field extension of $K_p$, its action on $K_p^5$ has rank one. Thus the centralizer of $F$ in $M_5(K_p)$ is $FK_p$. If $A_0$ were CM, its characteristic-zero centralizer of $F$ would have degree ten, both in the simple CM case and in the CM elliptic-power case. Specialization would identify it with $FK_7$ and with $FK_{13}$.

This is impossible. Each compositum is cyclic of degree ten and has a unique quadratic subfield, namely $K_p$, whereas the two displayed quadratic fields for either curve differ. Hence $A_0$ is non-CM. It is either simple with endomorphism field $F$, or a fifth power of a non-CM elliptic curve. Both are stably nondegenerate with no type-IV factors.

This proves the specialization criterion stated after Theorem 2, and Hazama's theorem now applies to (10). The algebraic transfer in §5 proves Theorem 2. The argument does not establish, or require, absolute simplicity. As an additional consequence of the different quadratic fields at 7, the two Jacobians are not geometrically isogenous.

## 7. Reproduction, scope, and remaining questions

The supplement contains exact inputs, complete point-count outputs, and the verification code. It uses Python with SymPy 1.14.0 and a C++17 compiler, on CPU only. Its requirements file pins the tested SymPy release, and its README lists installation and execution commands in dependency order. The arithmetic checks verify the source isogenies, the nodal equations, parameter conversion, finite-field smoothness, Frobenius polynomials, and quadratic-field comparisons. Five additional audits check the CM orbit ranks, stabilizers, commutants, integral character relations and examples of the arbitrary-block matching argument, including the cyclic-real-quintic module families and mixed threefold/fivefold contractions. The geometric arguments in §§3–5, including the algebraic mixed determinant and generation on every power, are mathematical proofs, not claims of formal verification by those scripts.

The finite-exception theorem applies separately to every smooth fixed isogeny input and every odd degree. André–Oort makes the higher-dimensional CM parameter set finite but does not make it effective. For number-field inputs, standard Hilbert specialization additionally gives the generic endomorphism algebras on a Hilbert subset. The named algebraic member is certified by a separate endomorphism argument. The degree-five, degree-seven and degree-nine families have no exceptional admissible parameters for the individual all-powers assertion. The finite mixed-product criterion allows CM elliptic factors, at most two simple CM surface isogeny types for each quartic normal closure, and the threefold configurations in Proposition 5. It includes arbitrary finite mixed products of degree-five members, and permits arbitrarily many threefold types across distinct sextic normal closures. Its proof uses Milne's known one-threefold-plus-elliptic theorem, joint orbit-span independence, and actual contractions of Weil-fourfold classes. Proposition 6 further allows fivefolds with cyclic real quintic field: full-sign types satisfy a joint cocharacter condition, and split types satisfy the stated shared-norm restrictions. The additional algebraic input is Markman's split Weil-sixfold theorem. The bounds within a common normal closure remain necessary for this proof. We do not claim HC at every excluded parameter in other degrees: larger CM configurations can introduce mixed Hodge classes not controlled by these arguments.

The numerical bound on transcendental rank is $\operatorname{rank}T(S)\le2n-2$. We have not used equality, nor identified a particular surface with an independently specified CM Hodge structure. Constructing such a correspondence remains a different problem.

A related use of elliptic contractions appears in our earlier paper [14, §7] for Fermat double covers and explicit CM factors. That work constructs half-twist correspondences from concrete elliptic fibrations. Here the input cycles are algebraic Weil-fourfold and split Weil-sixfold classes, and the new step is their joint generation across arbitrary shared-norm blocks under Propositions 5 and 6. The two papers treat different surface families.

The new cases illustrate how actual domination by a curve square can control all higher powers after its entire Jacobian has been analyzed. The arithmetic example also shows why two reductions can suffice without proving absolute simplicity: excluding the CM centralizer already supplies the property needed by the product theorem.

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