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. (c) 2019 Elsevier Inc. All rights reserved.

For q an odd prime power, A = F-q[T], and F = F-q(T), let psi : A -> F{tau} be a Drinfeld A-module over F of rank 2 and without complex multiplication, and let p = pA be a prime of A of good reduction for psi, with residue field F-p. We study the growth of the absolute value |Delta(p)| of the discriminant of the F-p-endomorphism ring of the reduction of psi modulo p and prove that, for all p, |Delta(p)| grows with |p|. Moreover, we prove that, for a density 1 of primes p, |Delta(p)| is as close as possible to its upper bound |a(p)(2) - 4 mu(p)p|, where X-2 + a(p)X + mu(p)p is an element of A[X] is the characteristic polynomial of tau(deg p). (c) 2021 Elsevier Inc. All rights reserved.