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 ≔ Gal ( K sep / K ) . Let φ be a Drinfeld A-module over K in special characteristic. Set E ≔ End K ( φ ) and let Z be its center. We show that for almost all primes p of A, the image of the group ring A p [ G K ] in End A ( T p ( φ ) ) is the commutant of E. Thus, for almost all p it is a full matrix ring over Z ⊗ A A p . In the special case E = A it follows that the representation of G K on the p -torsion points φ [ p ] is absolutely irreducible for almost all p .