Tame torsion, the tame inverse Galois problem, and endomorphisms

TL;DR

This paper constructs genus g curves over rationals with tame mod p Jacobian representations and applies these results to realize symplectic groups as Galois groups of tame extensions.

Contribution

It demonstrates the existence of specific algebraic curves with tame Jacobian representations and uses this to solve cases of the tame inverse Galois problem.

Findings

Existence of genus g curves with tame mod p Jacobian representations.

Realization of symplectic groups as Galois groups of tame extensions.

Conditions on endomorphism rings enable these constructions.

Abstract

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 $\mathbb{Q}$.