Surjectivity of the adelic Galois Representation associated to a Drinfeld Module of prime rank

TL;DR

This paper proves the surjectivity of the adelic Galois representation associated with a specific prime rank Drinfeld module over a function field, under certain conditions on the base finite field.

Contribution

It establishes the surjectivity of the adelic Galois representation for a class of Drinfeld modules, extending understanding of their Galois actions.

Findings

The adelic Galois representation is surjective under certain conditions.

Provides explicit conditions on the finite field for surjectivity.

Enhances knowledge of Galois representations in function field arithmetic.

Abstract

In this paper, let $\phi$ be the Drinfeld module over $\mathbb{F}_{q}(T)$ of prime rank $r$ defined by $$\phi_T=T+\tau^{r-1}+T^{q-1}\tau^r.$$ We prove that under certain condition on $\mathbb{F}_q$, the adelic Galois representation $${\rho}_{\phi}:{\rm{Gal}}(\mathbb{F}_q(T)^{{\rm{sep}}}/\mathbb{F}_q(T))\longrightarrow \varprojlim_{\mathfrak{a}}{\rm{Aut}}(\phi[\mathfrak{a}])\cong {\rm{GL_r}}(\widehat{A})$$ is surjective.