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. 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

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

The exceptional set is not made effective. It consists of parameters at which one of the higher-dimensional factors in §3.3 is of CM type; we do not assert failure of HC there. 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.

The abelian Hodge results are standard. 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. These 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

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

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 $pI_2$. 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.

The moduli space of cyclic $n$-isogenies with a generator of the kernel and marked two-torsion data is connected: analytically one may use the upper half-plane modulo the corresponding congruence subgroup, adding level structure if needed to have universal families. Over the smooth modular curve, the four distinct critical values form a moving configuration. Their complement is locally a topological bundle of four-punctured spheres. The pair of curve families is smooth and proper over this complement.

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, [8, 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 of the surfaces satisfying the source hypotheses: Hazama's theorem combines all their non-CM real-multiplication factors and all their elliptic factors. Thus finite products of arbitrary powers of such surfaces also satisfy HC. No independence assumption between different members is necessary.

5.1. Arithmetic specialization

Assume the marked input is defined over a number field $K$. Apply the standard fixed-prime specialization theorem [9, 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 the Rosati involution bounds the dimension of a commutative self-adjoint real algebra in $M_m(D)\otimes\mathbb R$ by the number of entries in simultaneous real diagonal Hermitian matrices. In Albert types I, II, III these bounds are respectively $me,2me,me$, where $e$ is the degree of the totally real center. In type IV the bound is $mde$, where the CM center has degree $2e$ and $[D:Z(D)]=d^2$. The corresponding lower bounds on $b$, from the $D$-module $H^1(B)$, are $e,2e,2e,d^2e$. Since the self-adjoint algebra $F\otimes\mathbb R$ has dimension $r=mb$, type III is impossible. Type IV forces $d=1,b=e$, hence CM. The remaining cases have $b=e$ in type I or $b=2e$ in type II: relative dimension one in either case.

In these remaining cases the complex Lefschetz group is a product of copies of $SL_2$, acting in their standard representation with multiplicity one or two. The Hodge group is connected semisimple [5, §1.2]. Its projection to each $SL_2$ is nontrivial, hence surjective; otherwise its endomorphism centralizer would be larger than $D\otimes\mathbb C$. A proper semisimple subdirect subgroup would identify two factors diagonally, introducing off-diagonal equivariant homomorphisms and again enlarging that centralizer. Thus the Hodge and Lefschetz groups agree. The invariant-theory criterion [5, (1.8)] proves stable nondegeneracy, also for $B^m$. $\square$

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 associated Hodge domain is a product of $r_d$ upper half-planes. A special subvariety with this generic Mumford–Tate group has dimension $r_d>1$, so the image curve is not special. André–Oort for $\mathcal A_{r_d}$ [10, Theorem 5.2] implies that it contains only finitely many CM points. Nonconstant maps of algebraic curves have finite fibers, proving the assertion about parameters. 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 [10, §6.1].

Let $\Sigma_\phi$ be the finite union of these CM parameter sets, over both signs and all $d\mid n$ with $r_d>1$. The remaining case $d=3$ has $r_d=1$, so its factors are elliptic and may be CM without being excluded. For $h\notin\Sigma_\phi$, Lemma 4 supplies stable nondegeneracy and absence of type-IV factors for every higher-dimensional factor. Combining these with the elliptic factors by Hazama's theorem and applying the same curve-square transfer proves Theorem 1. The exceptional set may include parameters where HC still holds.

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 [11] 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 and a C++17 compiler, on CPU only. Its README lists the 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. The geometric arguments in §§3–5 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. We do not claim HC at every excluded parameter: CM specializations can introduce mixed Hodge classes not controlled by the non-CM product theorem.

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.

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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