Endomorphism rings of reductions of Drinfeld modules

Abstract

Let A=Fq[T] be the polynomial ring over Fq, 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⊗Fp of rank r over Fp=A/p. The endomorphism ring Ep=EndFp(ϕ⊗Fp) is an order in an imaginary field extension K of F of degree r. Let Op be the integral closure of A in K, and let πp∈Ep be the Frobenius endomorphism of ϕ⊗Fp. Then we have the inclusion of orders A[πp]⊂Ep⊂Op in K. We prove that if EndFalg(ϕ)=A, then for arbitrary non-zero ideals n,m of A there are infinitely many p such that n divides the index χ(Ep/A[πp]) and m divides the index χ(Op/Ep). We show that the index χ(Ep/A[πp]) 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[πp]⊂Ep⊂Op, and give some computational data.