Exceptional cases of adelic surjectivity for Drinfeld modules of rank 2

TL;DR

This paper investigates the conditions under which the adelic Galois representation associated with rank 2 Drinfeld modules over function fields is surjective, focusing on special cases where the base field size is even or equals three.

Contribution

It provides new insights into the adelic surjectivity of Galois representations for rank 2 Drinfeld modules in specific small characteristic cases.

Findings

Identifies cases where adelic Galois representations are surjective

Characterizes exceptions for even q and q=3

Enhances understanding of Galois representations in function field arithmetic

Abstract

In this paper, we study the surjectivity of adelic Galois representation associated to Drinfeld $\mathbb{F}_q[T]$-modules over $\mathbb{F}_q(T)$ of rank $2$ in the cases when $q$ is even or $q=3$.