Tame Galois Realizations of GS p4 (𝔽 ℓ ) over ℚ

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 $\mathbb Q$. 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 i...