Galois representations over function fields that are ramified at one prime

Abstract

Let ${\mathbb{F}}_q$ be the finite field with $q$ elements, $F:={\mathbb{F}}_q(T)$ and $F^{\mathrm{sep}}$ 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 $\mathcal{O}$ be the completion of $A$ at $\mathfrak{p}$. Given any integer $r\ge 2$, I construct a Galois representation $\rho :\mathrm{Gal}(F^{\mathrm{sep}}/F)\to {\mathrm{GL}}_r(\mathcal{O})$ which is unramified at all non-zero primes $\mathfrak{l}\ne \mathfrak{p}$ of $A$, and whose image is a finite index subgroup of ${\mathrm{GL}}_r(\mathcal{O})$. Moreover, if the degree of $\mathfrak{p}$ is $1$, then $\rho$ is also unramified at $\infty$.

arXiv commentAccepted for publication in the Ramanujan Journal

References in corpus (1)

• Constructing Galois representations ramified at one prime