On the surjectivity of $(T)$-adic Galois Representations of Drinfeld $A$-Modules of Rank 2 and 3: Density results

Abstract

Let $\mathbb{F}_{q}$ be a finite field, and $A:=\mathbb{F}_{q}[T]$. In this article, we give explicit criteria, involving concrete valuations, on the coefficients of the Drinfeld $A$-modules of rank $r$ for $r=2,3$, which ensure the surjectivity of the associated $(T)$-adic Galois representation. As a result, we shall calculate the densities of such Drinfeld $A$-modules.