Simultaneous Exceptional Primes for Typical Genus-Two Jacobians

Abstract

For a genus-two curve $C/\mathbf Q$ with $\operatorname{End}_{\overline{\mathbf Q}}(J_C)=\mathbf Z$, let $\operatorname{Exc}(C)$ be the set of primes at which the residual Galois representation is not surjective. We conjecture the uniform cardinality bound $|\operatorname{Exc}(C)|\le3$. This bound is attained in the current data, while no example with four exceptional primes is known.

Statement

For a smooth projective genus-two curve $C/\mathbf Q$, define

$$\operatorname{Exc}(C)= \left\{\ell:\operatorname{im}\bar\rho_{J_C,\ell} \ne\operatorname{GSp}_4(\mathbf F_\ell)\right\}.$$

The set records distinct primes, not distinct maximal subgroups or image labels.

Conjecture 1. If

$$\operatorname{End}_{\overline{\mathbf Q}}(J_C)=\mathbf Z,$$

then

$$|\operatorname{Exc}(C)|\le3.$$

Serre’s theorem implies that $\operatorname{Exc}(C)$ is finite for each fixed typical Jacobian. The conjecture asks for a uniform bound on the number of simultaneous failures, rather than a uniform bound on the size of each exceptional prime.

Numerical evidence and sharpness

Among the $63{,}107$ typical LMFDB curves, the distribution of the number of exceptional primes is:

| $|\operatorname{Exc}(C)|$ | 0 | 1 | 2 | 3 | |:-------------------------:|-----------:|-----------:|----------:|-----:| | number of curves | $19{,}586$ | $40{,}192$ | $3{,}306$ | $23$ |

Number of simultaneous exceptional primes in the inspected data.

The three-prime sets are $\{2,3,5\}$ for $20$ curves, $\{2,3,7\}$ for two curves, and $\{2,3,13\}$ for one curve. In particular, the curve with LMFDB label 1116.a.214272.1 realizes the set $\{2,3,13\}$, so the proposed bound is sharp if true.

A larger published computation of $1{,}743{,}737$ typical curves likewise found at most three exceptional primes: $199{,}183$ curves had none, $1{,}394{,}671$ had one, $148{,}606$ had two, and $1{,}277$ had three.

What is known

Open-image and maximal-subgroup results analyze exceptional primes one at a time. Serre proves only curvewise finiteness. Banwait–Brumer–Kim–Klagsbrun–Mayle–Srinivasan–Vogt give fixed-curve algorithms and a GRH-conditional bound for the product of exceptional primes in terms of the conductor. Lombardo gives a large-prime cutoff depending on the height and arithmetic data of the individual surface. None of these results couples exceptional behavior at several distinct primes.

Rational $\ell$-torsion fixes a nonzero vector in $J_C[\ell]$, and a rational $\ell$-isogeny stabilizes a proper subgroup; either forces $\ell\in\operatorname{Exc}(C)$. These observations suggest possible counterexample constructions, but no typical Jacobian with four distinct primes forced in this way is known. Howe’s order-$70$ construction does not qualify because its Jacobian is geometrically isogenous to a product.

The mixed-level problem

A proof of Conjecture 1 would require a genuinely cross-prime uniform theorem. For every four distinct primes $\ell_i$ and every choice of proper maximal subgroups

$$H_i<\operatorname{GSp}_4(\mathbf F_{\ell_i}),$$

one would need to show that every rational point on the associated mixed-level Siegel modular cover either represents a surface with extra geometric endomorphisms or does not arise from a genus-two Jacobian. Existing open-image, isogeny-bound, and maximal-subgroup methods do not control rational points on all such fiber products.

A disproof would require one explicit genus-two curve with geometric endomorphism ring $\mathbf Z$ and four rigorously certified nonsurjective residual images. No such example is presently known.

References

• Barinder S. Banwait, Armand Brumer, Hyun Jong Kim, Zev Klagsbrun, Jacob Mayle, Padmavathi Srinivasan, Isabel Vogt, Computing nonsurjective primes associated to Galois representations of genus 2 curves (2024). https://doi.org/10.1090/conm/796/16000

• Jean-Pierre Serre, Lettre à Marie-France Vignéras du 10/2/1986 / Résumé des cours de 1985–1986 (1986). https://www.college-de-france.fr/sites/default/files/media/document/2023-03/1985-1986_serre.pdf

• Davide Lombardo, Explicit surjectivity of Galois representations for abelian surfaces and $\mathrm{GL}_2$-varieties (2016). https://arxiv.org/abs/1411.1703

• Luis V. Dieulefait, Explicit determination of the images of the Galois representations attached to abelian surfaces with End(A)=Z (2002). https://arxiv.org/abs/math/0110340

• Aaron Landesman, Ashvin Swaminathan, James Tao, Yujie Xu, Hyperelliptic Curves with Maximal Galois Action on the Torsion Points of their Jacobians (2020). https://arxiv.org/abs/1705.08777

• Raymond van Bommel, Shiva Chidambaram, Edgar Costa, Jean Kieffer, Computing isogeny classes of typical principally polarized abelian surfaces over the rationals (2024). https://arxiv.org/abs/2301.10118

• Tyler Genao, Jacob Mayle, Jeremy Rouse, A uniform bound on the smallest surjective prime of an elliptic curve (2026). https://link.springer.com/article/10.1007/s11139-026-01376-8

• The LMFDB Collaboration, LMFDB genus-2 curve 1116.a.214272.1 (2026). https://www.lmfdb.org/Genus2Curve/Q/1116/a/214272/1

• Everett W. Howe, Genus-2 Jacobians with torsion points of large order (2015). https://arxiv.org/abs/1407.2654