Large Galois images for Jacobian varieties of genus 3 curves
Volume 174 / 2016
Acta Arithmetica 174 (2016), 339-366
MSC: Primary 11F80, 11G30, 11G10; Secondary 12F12.
DOI: 10.4064/aa8250-4-2016
Published online: 5 August 2016
Abstract
Given a prime number $\ell \geq 5$, we construct an infinite family of three-dimensional abelian varieties over $\mathbb{Q}$ such that, for any $A/\mathbb{Q}$ in the family, the Galois representation $\overline{\rho}_{A,\ell} \colon G_{\mathbb{Q}} \to \mathrm{GSp}_6(\mathbb{F}_{\ell})$ attached to the $\ell$-torsion of $A$ is surjective. Any such variety $A$ will be the Jacobian of a genus $3$ curve over $\mathbb{Q}$ whose respective reductions at two auxiliary primes are prescribed to provide us with generators of $\mathrm{Sp}_6(\mathbb{F}_{\ell})$.
