Open Access

Residual representations of semistable principally polarized abelian varieties

Research in Number Theory20162:1

https://doi.org/10.1007/s40993-015-0032-4

Received: 10 August 2015

Accepted: 23 November 2015

Published: 15 February 2016

Abstract

Let \(A/{\mathbb {Q}}\) be a semistable principally polarized abelian variety of dimension d≥1. Let be a prime and let \(\overline {\rho }_{A,\ell }\colon G_{\mathbb {Q}} \rightarrow \text {GSp}_{2d}({\mathbb {F}}_{\ell })\) be the representation giving the action of \(G_{\mathbb {Q}} :=\text {Gal}(\overline {{\mathbb {Q}}}/{\mathbb {Q}})\) on the -torsion group A[]. We show that if ≥ max(5,d+2), and if image of \(\overline {\rho }_{A,\ell }\) contains a transvection then \(\overline {\rho }_{A,\ell }\) is either reducible or surjective.

With the help of this we study surjectivity of \(\overline {\rho }_{A,\ell }\) for semistable polarized abelian threefolds, and give an example of a genus 3 hyperelliptic curve \(C/{\mathbb {Q}}\) such that \(\overline {\rho }_{J,\ell }\) is surjective for all primes ≥3, where J is the Jacobian of C.

Keywords

Galois representationsAbelian varietiesSemistabilitySerre’s uniformity