Large Galois images for Jacobian varieties of genus 3 curves

TL;DR

This paper constructs an infinite family of genus 3 Jacobian varieties over Q with surjective Galois representations on their l-torsion, demonstrating large Galois images in GSp(6, l) for primes l ≥ 5.

Contribution

It provides explicit constructions of genus 3 Jacobians over Q with maximal Galois image in GSp(6, l), extending known results to higher genus cases.

Findings

Constructed infinite families with surjective Galois representations

Ensured prescribed reductions at auxiliary primes to generate Sp(6, l)

Demonstrated the existence of large Galois images for genus 3 Jacobians

Abstract

Given a prime number l greater than or equal to 5, we construct an infinite family of three-dimensional abelian varieties over Q such that, for any A/Q in the family, the Galois representation \rho_{A, l}: Gal_Q -> GSp(6, l) attached to the l-torsion of A is surjective. Any such variety A will be the Jacobian of a genus 3 curve over Q whose respective reductions at two auxiliary primes we prescribe to provide us with generators of Sp(6, l).