Endomorphism rings of reductions of Drinfeld modules

Let A=F_q[T] be the polynomial ring over F_q, and F be the field of fractions of A. Let ϕ be a Drinfeld A-module of rank r≥2 over F. For all but finitely many primes p◁A, one can reduce ϕ modulo p to obtain a Drinfeld A-module ϕ⊗F_p of rank r over F_p=A/p. The endomorphism ring E_p=End_Fp (ϕ⊗F_p) is an order in an imaginary field extension K of F of degree r. Let O_p be the integral closure of A in K, and let π_p∈E_p be the Frobenius endomorphism of ϕ⊗F_p. Then we have the inclusion of orders A[π_p]⊂E_p⊂O_p in K. We prove that if End_F^alg (ϕ)=A, then for arbitrary non-zero ideals n,m of A there are infinitely many p such that n divides the index χ(E_p/A[π_p]) and m divides the index χ(O_p/E_p). We show that the index χ(E_p/A[π_p]) is related to a reciprocity law for the extensions of F arising from the division points of ϕ. In the rank r=2 case we describe an algorithm for computing the orders A[π_p]⊂E_p⊂O_p, and give some computational data.