Residual Representations of Semistable Principally Polarized Abelian Varieties

Abstract

Let $A$ be a semistable principally polarized abelian variety of dimension $d$ defined over the rationals. Let $\ell$ be a prime and let ${\bar{\rho }}_{A,\ell }:G_{\mathbb{Q}}\to {\mathrm{GSp}}_{2d}({\mathbb{F}}_{\ell })$ 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 $C$.

Paper has appeared in Research in Number Theory

References in corpus (2)

• Arithmetic of hyperelliptic curves over local fields
• An explicit Jacobian of dimension 3 with maximal Galois action

Cited by in corpus (5)

• Surjectivity of Galois Representations in Rational Families of Abelian Varieties
• An explicit Jacobian of dimension 3 with maximal Galois action
• Hyperelliptic Curves with Maximal Galois Action on the Torsion Points of their Jacobians
• Large 2-adic Galois image and non-existence of certain abelian surfaces over Q
• Large Galois images for Jacobian varieties of genus 3 curves