Tame Galois Realizations of over

In this paper, we obtain realizations of the 4-dimensional general symplectic group over a prime field of characteristic ℓ > 3 as the Galois group of a tamely ramified Galois extension of ⁠. The strategy is to consider the Galois representation ρℓ attached to the Tate module at ℓ of a suitable abelian surface. We need to choose the abelian surfaces carefully in order to ensure that the image of ρℓ 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 ℓ-torsion points of their Jacobian varieties provides tame Galois realizations of the desired symplectic groups.