%0 Journal Article
%T Tame Galois realizations of GSp_4(F_l) over Q
%A Sara Arias-de-Reyna
%A NšČria Vila
%J Mathematics
%D 2009
%I arXiv
%X 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.
%U http://arxiv.org/abs/0910.1445v1