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

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 Serres uniformity problem.