Galois representations over function fields that are ramified at one prime

TL;DR

This paper constructs specific Galois representations over function fields that are ramified at only one prime, with the image being a finite index subgroup of the general linear group, and explores their ramification properties.

Contribution

It introduces a method to explicitly construct Galois representations over function fields ramified at a single prime with large image.

Findings

Constructed Galois representations with controlled ramification.

Representations have images of finite index in GL_r.

Unramified at all but one prime, and at infinity if degree is 1.

Abstract

Let $\mathbb{F}_q$ be the finite field with $q$ elements, $F:=\mathbb{F}_q(T)$ and $F^{\operatorname{sep}}$ a separable closure of $F$. Set $A$ to denote the polynomial ring $\mathbb{F}_q[T]$. Let $\mathfrak{p}$ be a non-zero prime ideal of $A$, and $\mathscr{O}$ be the completion of $A$ at $\mathfrak{p}$. Given any integer $r\geq 2$, I construct a Galois representation $\rho:\operatorname{Gal}(F^{\operatorname{sep}}/F)\rightarrow \operatorname{GL}_r(\mathscr{O})$ which is unramified at all non-zero primes $\mathfrak{l}\neq \mathfrak{p}$ of $A$, and whose image is a finite index subgroup of $\operatorname{GL}_r(\mathscr{O})$. Moreover, if the degree of $\mathfrak{p}$ is $1$, then $\rho$ is also unramified at $\infty$.