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arXiv · 0910.1445

Tame Galois realizations of GSp_4(F_l) over Q

Sara Arias-de-Reyna·Núria Vila
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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 $ρ_{\ell}$ 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 $ρ_{\ell}$ 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.

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Sara Arias-de-Reyna, Núria Vila. 2009-10-08. Tame Galois realizations of GSp_4(F_l) over Q. https://arxiv.org/abs/0910.1445

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