Residual Representations of Semistable Principally Polarized Abelian Varieties

TL;DR

This paper investigates the properties of residual Galois representations associated with semistable principally polarized abelian varieties, establishing conditions for their reducibility or surjectivity, and provides explicit examples in genus 3.

Contribution

It proves that under certain conditions, the residual Galois representation is either reducible or surjective, and constructs explicit examples of surjective representations for genus 3 Jacobians.

Findings

If the image contains a transvection, the representation is reducible or surjective.

For genus 3 Jacobians, there exist cases where the residual representation is surjective for all primes.

Provides explicit example of a genus 3 hyperelliptic curve with surjective residual representations.

Abstract

Let $A$ be a semistable principally polarized abelian variety of dimension $d$ defined over the rationals. Let $\ell$ be a prime and let $\bar{\rho}_{A,\ell} : G_{\mathbb{Q}} \rightarrow \mathrm{GSp}_{2d}(\mathbb{F}_\ell)$ be the representation giving the action of $G_{\mathrm{Q}} :=\mathrm{Gal}(\bar{\mathrm{Q}}/\mathrm{Q})$ on the $\ell$-torsion group $A[\ell]$. We show that if $\ell \ge \max(5,d+2)$, and if image of $\bar{\rho}_{A,\ell}$ contains a transvection then $\bar{\rho}_{A,\ell}$ is either reducible or surjective. With the help of this we study surjectivity of $\bar{\rho}_{A,\ell}$ for semistable principally polarized abelian threefolds, and give an example of a genus $3$ hyperelliptic curve $C/\mathbb{Q}$ such that $\bar{\rho}_{J,\ell}$ is surjective for all primes $\ell \ge 3$, where $J$ is the Jacobian of $C$.