Tame Galois realizations of GSp_4(F_l) over Q

TL;DR

This paper constructs explicit tamely ramified Galois extensions of the rationals with Galois group GSp_4(F_l) by using Galois representations from Jacobians of genus 2 curves, advancing the realization problem in inverse Galois theory.

Contribution

It provides explicit families of genus 2 curves whose Jacobians' Galois representations realize GSp_4(F_l) as Galois groups over Q with tame ramification.

Findings

Explicit Galois realizations of GSp_4(F_l) over Q.

Construction of genus 2 curves with desired Galois properties.

Control over ramification in Galois extensions.

Abstract

In this paper we obtain realizations of the 4-dimensional general symplectic group over a prime field of characteristic $\ell>3$ as the Galois group of a tamely ramified Galois extension of $\mathbb{Q}$. The strategy is to consider the Galois representation $\rho_{\ell}$ attached to the Tate module at $\ell$ of a suitable abelian surface. We need to choose the abelian varieties carefully in order to ensure that the image of $\rho_{\ell}$ is large and simultaneously maintain a control on the ramification of the corresponding Galois extension. We obtain an explicit family of curves of genus 2 such that the Galois representation attached to the $\ell$-torsion points of their Jacobian varieties provide tame Galois realizations of the desired symplectic groups.