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

Let K be a finitely generated field of transcendence degree 1 over a finite field, and set GK≔Gal(Ksep/K). Let φ be a Drinfeld A-module over K in special characteristic. Set E≔EndK(φ) and let Z be its center. We show that for almost all primes p of A, the image of the group ring Ap[GK] in EndA(Tp(φ)) is the commutant of E. Thus, for almost all p it is a full matrix ring over Z⊗AAp. In the special case E=A it follows that the representation of GK on the p-torsion points φ[p] is absolutely irreducible for almost all p.