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

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