The Galois representations associated to a Drinfeld module in special characteristic—III: Image of the group ring

Abstract

Let K be a finitely generated field of transcendence degree 1 over a finite field, and set $G_K≔\mathrm{Gal}(K^{\mathrm{sep}}/K)$. Let $\phi$ be a Drinfeld A-module over K in special characteristic. Set $E≔{\mathrm{End}}_K(\phi )$ and let Z be its center. We show that for almost all primes $\mathfrak{p}$ of A, the image of the group ring $A_{\mathfrak{p}}[G_K]$ in ${\mathrm{End}}_A(T_{\mathfrak{p}}(\phi ))$ is the commutant of E. Thus, for almost all $\mathfrak{p}$ it is a full matrix ring over $Z{\otimes }_A{A}_{\mathfrak{p}}$. In the special case $E=A$ it follows that the representation of $G_K$ on the $\mathfrak{p}$-torsion points $\phi [\mathfrak{p}]$ is absolutely irreducible for almost all $\mathfrak{p}$.