Galois representations and Galois groups over Q

TL;DR

This paper extends previous results to hyperelliptic curves of genus n, providing an algorithm to identify primes where the Galois representation of the Jacobian's torsion points is surjective, thus realizing certain symplectic groups as Galois groups over Q.

Contribution

It generalizes existing results to higher genus hyperelliptic curves and offers an explicit algorithm for Galois group realization over Q.

Findings

Identifies primes l for which the Galois representation is surjective.

Realizes GSp(6, l) as a Galois group over Q for primes l in [11, 500000].

Provides an explicit computational method for Galois representations.

Abstract

In this paper we generalize results of P. Le Duff to genus n hyperelliptic curves. More precisely, let C/Q be a hyperelliptic genus n curve and let J(C) be the associated Jacobian variety. Assume that there exists a prime p such that J(C) has semistable reduction with toric dimension 1 at p. We provide an algorithm to compute a list of primes l (if they exist) such that the Galois representation attached to the l-torsion of J(C) is surjective onto the group GSp(2n, l). In particular we realize GSp(6, l) as a Galois group over Q for all primes l in [11, 500000].