Endomorphism rings of reductions of Drinfeld modules

摘要

Let A=F-q[T] be the polynomial ring over F-q, and F be the field of fractions of A. Let phi be a Drinfeld A-module of rank r >= 2 over F. For all but finitely many primes p (sic) A, one can reduce phi modulo p to obtain a Drinfeld A-module phi circle times F-p of rank r over F-p = A/p. The endomorphism ring epsilon(p) = EndF(p) (phi circle times Fp) is an order in an imaginary field extension K of F of degree tau. Let O-p be the integral closure of A in K, and let pi(p) is an element of epsilon(p) be the Frobenius endomorphism of phi circle times F-p. Then we have the inclusion of orders A [pi(p)] subset of epsilon(p) subset of Op in K. We prove that if End(Falg)(phi) = A, then for arbitrary non-zero ideals n, m of A there are infinitely many p such that n divides the index chi(epsilon(p)/A[pi(p)]) and m divides the index chi(O-p/epsilon(p)). We show that the index chi(epsilon(p)/A[pi(p)]) is related to a reciprocity law for the extensions of F arising from the division points of phi. In the rank r = 2 case we describe an algorithm for computing the orders A[pi(p)] subset of epsilon(p )subset of O-p, and give some computational data. (C) 2019 Elsevier Inc. All rights reserved.