Tame torsion and the tame inverse Galois problem

TL;DR

This paper constructs genus g curves over rationals with tame mod m Jacobian representations and explores the tame inverse Galois problem for symplectic groups, using period matrices of hyperelliptic Mumford curves.

Contribution

It introduces a method to produce genus g curves with tame Jacobian representations and applies it to the tame inverse Galois problem for symplectic groups.

Findings

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

Analysis of period matrices of hyperelliptic Mumford curves.

Insights into the tame inverse Galois problem for symplectic matrix groups.

Abstract

Fix a positive integer $g$ and a squarefree integer $m$. We prove the existence of a genus $g$ curve $C/\mathbb{Q}$ such that the mod $m$ representation of its Jacobian is tame. The method is to analyse the period matrices of hyperelliptic Mumford curves, which could be of independent interest. As an application, we study the tame version of the inverse Galois problem for symplectic matrix groups over finite fields.