Tame Galois realizations of GSp_4(F_l) over Q

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.

29 pages

References in corpus (1)

• On Modular Forms and the Inverse Galois Problem

Cited by in corpus (1)

• Formal groups, supersingular abelian varieties and tame ramification