An explicit bound on reducibility of mod $\mathfrak{l}$ Galois image for Drinfeld modules of arbitrary rank and its application on the uniformity problem

TL;DR

This paper establishes an explicit bound on the reducibility of mod $rak{l}$ Galois representations for Drinfeld modules of any rank, linking it to the degree of $rak{l}$ and applying it to a uniformity problem analogous to Serre's conjecture.

Contribution

It provides the first explicit bound on the reducibility of mod $rak{l}$ Galois images for arbitrary rank Drinfeld modules and applies this to a uniformity problem.

Findings

Bound depends only on rank and minimal good reduction degree

Reduces the uniformity problem to a finite check

Establishes a link between reducibility and prime degree constraints

Abstract

Suppose we are given a Drinfeld Module $\phi$ over $\mathbb{F}_q(t)$ of rank $r$ and a prime ideal $\mathfrak{l}$ of $\mathbb{F}_q[T]$. In this paper, we prove that the reducibility of mod $\mathfrak{l}$ Galois representation $${\rm{Gal}}(\mathbb{F}_q(T)^{\rm{sep}}/\mathbb{F}_q(T))\rightarrow {\rm{Aut}}(\phi[\mathfrak{l}])\cong {\rm{GL}}_r(\mathbb{F}_\mathfrak{l})$$ gives a bound on the degree of $\mathfrak{l}$ which depends only on the rank $r$ of Drinfeld module $\phi$ and the minimal degree of place $\mathcal{P}$ where $\phi$ has good reduction at $\mathcal{P}$. Then, we apply this reducibility bound to study the Drinfeld module analogue of Serre's uniformity problem.