Computing endomorphism rings and Frobenius matrices of Drinfeld modules

Let F_q[T] be the polynomial ring over a finite field F_q. We study the endomorphism rings of Drinfeld F_q[T]-modules of arbitrary rank over finite fields. We compare the endomorphism rings to their subrings generated by the Frobenius endomorphism and deduce from this a refinement of a reciprocity law for division fields of Drinfeld modules proved in our earlier paper. We then use these results to give an efficient algorithm for computing the endomorphism rings and discuss some interesting examples produced by our algorithm.