A Uniform Surjectivity Bound for Typical Genus-Two Jacobians

Abstract

Let $C/\mathbf Q$ be a genus-two curve whose Jacobian has geometric endomorphism ring $\mathbf Z$. We conjecture that its residual Galois representation is surjective onto $\operatorname{GSp}_4(\mathbf F_\ell)$ for every prime $\ell>31$. The value $31$ is compatible with the current large-scale computations and is known to be a necessary lower boundary, but even the existence of an absolute uniform bound remains open.

Statement

Let $C/\mathbf Q$ be a smooth projective curve of genus two and let

$$\bar\rho_{J_C,\ell}:G_{\mathbf Q}\longrightarrow \operatorname{GSp}_4(\mathbf F_\ell)$$

be the representation on $J_C[\ell]$, defined using the Weil pairing.

Conjecture 1. If

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

then

$$\operatorname{im}\bar\rho_{J_C,\ell} =\operatorname{GSp}_4(\mathbf F_\ell) \qquad\text{for every prime }\ell>31.$$

The endomorphism hypothesis is the usual “typical” condition for an abelian surface. Serre’s open-image theorem gives the same conclusion for all sufficiently large primes, but with a threshold depending on the individual Jacobian. Conjecture 1 replaces this curve-dependent threshold by the universal value $31$.

Computational evidence

Among the $63{,}107$ LMFDB curves in the typical stratum, the exceptional primes and their frequencies are:

$\ell$	2	3	5	7	11	13	17	29
frequency	$42{,}230$	$3{,}558$	$939$	$120$	$8$	$15$	$2$	$1$

Exceptional-prime frequencies in the inspected LMFDB stratum.

The largest exceptional prime in this sample is $29$. A larger published computation found an exceptional prime $31$ and no exceptional prime above $31$ among $1{,}743{,}737$ curves. A certified typical Jacobian with a rational $31$-isogeny shows that no uniform cutoff below $31$ can hold.

Known results

For each fixed principally polarized abelian surface $A/\mathbf Q$ with $\operatorname{End}_{\overline{\mathbf Q}}(A)=\mathbf Z$, Serre proves surjectivity for all sufficiently large $\ell$. Lombardo gives explicit bounds depending on the Faltings height and on arithmetic and reduction data. Banwait–Brumer–Kim–Klagsbrun–Mayle–Srinivasan–Vogt give algorithms for computing the exceptional set of an individual typical Jacobian and a conditional conductor-dependent bound.

These results do not imply a uniform constant. The multiplier of the residual representation is the surjective mod-$\ell$ cyclotomic character, but this does not force the symplectic part to be full. Proper maximal-image possibilities include reducible images, stabilizers of a line or a Lagrangian plane, imprimitive decompositions, and special subgroup types. Present methods eliminate them only with constants depending on the particular surface.

The uniformity gap

A proof of Conjecture 1 must uniformly exclude every proper maximal subgroup of $\operatorname{GSp}_4(\mathbf F_\ell)$ for every typical genus-two Jacobian and every prime $\ell\ge37$. In particular, no theorem uniformly rules out rational $\ell$-isogenies, imprimitive images, or the remaining special-image cases independently of height, conductor, and reduction data.

The distinction in quantifiers is essential:

$$\forall J\ \exists B(J)\quad\text{is known, whereas}\quad \exists B\ \forall J\quad\text{is open.}$$

The conjecture proposes the sharp value $B=31$. Current computations provide substantial evidence but do not settle this universal assertion.

References

• Barinder S. Banwait, Armand Brumer, Hyun Jong Kim, Zev Klagsbrun, Jacob Mayle, Padmavathi Srinivasan, and Isabel Vogt, Computing nonsurjective primes associated to Galois representations of genus 2 curves (2024). https://arxiv.org/abs/2301.02222

• Jean-Pierre Serre, Groupes linéaires modulo p et points d’ordre fini des variétés abéliennes (1986). https://numdam.org/item/CJPS_1986__7_/

• Davide Lombardo, Explicit surjectivity of Galois representations attached to abelian surfaces and GL2-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 (2003). https://arxiv.org/abs/math/0110340

• Raymond van Bommel, Shiva Chidambaram, Edgar Costa, and Jean Kieffer, Computing isogeny classes of typical principally polarized abelian surfaces over the rationals (2023; subsequently revised). https://arxiv.org/abs/2301.10118

• Samuele Anni, Pedro Lemos, and Samir Siksek, Residual Representations of Semistable Principally Polarized Abelian Varieties (2016). https://arxiv.org/abs/1508.00211

• Davide Lombardo and Matteo Verzobio, On the local-global principle for isogenies of abelian surfaces (2023). https://arxiv.org/abs/2206.15240