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

Abstract

In this paper, let $\varphi$ be the Drinfeld module over ${\mathbb{F}}_q(T)$ of prime rank $r$ defined by $${\varphi }_T=T+{\tau }^{r-1}+T^{q-1}{\tau }^r.$$ We prove that under certain condition on ${\mathbb{F}}_q$, the adelic Galois representation $${\rho }_{\varphi }:\mathrm{Gal}({\mathbb{F}}_q(T{)}^{\mathrm{sep}}/{\mathbb{F}}_q(T))⟶\underset{\mathfrak{a}}{\underleftarrow{\lim }}\mathrm{Aut}(\varphi [\mathfrak{a}])\cong \mathrm{G}{\mathrm{L}}_{\mathrm{r}}(\hat{A})$$ is surjective.