Galois representations over function fields that are ramified at one prime

TL;DR

The paper addresses constructing Galois representations over the function field $F=\mathbb{F}_q(T)$ ramified at exactly one prime $\mathfrak{p}$, yielding an open image in $\operatorname{GL}_r(\mathcal{O})$ with controlled inertia. It builds the representation from a rank $r$ Drinfeld module $\phi$ with $\operatorname{End}_{\bar{F}}(\phi)=A$, and leverages the Pink–R"utsche open-image theorem to explain why the image is of finite index. It also provides a detailed description of the ramification: inertia at $\mathfrak{p}$ has a totally ramified one-dimensional component and an unramified $(r-1)$-dimensional quotient, while, for $\deg \mathfrak{p}=1$, the representation is unramified at $\infty$. The construction offers a function-field analogue of Greenberg-type results over $\mathbb{Q}$ and highlights a geometric origin via Drinfeld modules, including a stable reduction at $\mathfrak{p}$ and a good reduction away from $\{\mathfrak{p},\infty\}$.

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 $ρ:\operatorname{Gal}(F^{\operatorname{sep}}/F)\rightarrow \operatorname{GL}_r(\mathscr{O})$ 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 $ρ$ is also unramified at $\infty$.

Theorems & Definitions (25)

• Theorem 1.2
• Definition 2.1
• Definition 2.2
• Theorem 2.3: Pink-Rütsche
• Lemma 3.1
• proof
• Proposition 3.2
• proof
• Proposition 3.3
• proof