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

Abstract

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

The proof of Proposition 3.1 is wrong, which makes Main Theorem 1 groundless