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

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.