Tame torsion, the tame inverse Galois problem, and endomorphisms

Fix a positive integer $g$ and rational prime $p$. We prove the existence of a genus $g$ curve $C/\mathbb{Q}$ such that the mod $p$ representation of its Jacobian is tame by imposing conditions on the endomorphism ring. As an application, we consider the tame inverse Galois problem and are able to realise general symplectic groups as Galois groups of tame extensions of $\mathbb{Q}$.
Comment: v2: Expanded to include application to tame inverse Galois problem. To appear in Manuscripta Mathematica