2018/04/21 by Garai, Sumita, Papikian, Mihran
#11G09 #11R58 #FOS: Mathematics #Number Theory (math.NT)
paper · doi:10.48550/arxiv.1804.07904
Let A=\mathbbFq[T] be the polynomial ring over \mathbbFq, 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 \mathfrakp\lhd A, one can reduce ϕ modulo \mathfrakp to obtain a Drinfeld A-module ϕ⊗\mathbbF_\mathfrakp of rank r over \mathbbF_\mathfrakp=A/\mathfrakp. The endomorphism ring E_\mathfrakp=End_\mathbbF_\mathfrakp(ϕ⊗\mathbbF_\mathfrakp) is an order in an imaginary field extension K of F of degree r. Let O_\mathfrakp be the integral closure of A in K, and let π_\mathfrakp∈ E_\mathfrakp be the Frobenius endomorphism of ϕ⊗\mathbbF_\mathfrakp. Then we have the inclusion of orders A[π_\mathfrakp]⊂ E_\mathfrakp⊂ O_\mathfrakp in K. We prove that if EndFalg(ϕ)=A, then for arbitrary non-zero ideals \mathfrakn, \mathfrakm of A there are infinitely many \mathfrakp such that \mathfrakn divides the index χ(E_\mathfrakp/A[π_\mathfrakp]) and \mathfrakm divides the index χ(O_\mathfrakp/E_\mathfrakp). We show that the index χ(E_\mathfrakp/A[π_\mathfrakp]) 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_\mathfrakp]⊂ E_\mathfrakp⊂ O_\mathfrakp, and give some computational data.