Cyclicity of finite Drinfeld modules

Abstract

Let A = Fq[T] be the polynomial ring over the finite field Fq, let k = Fq(T) be the rational function field, and let K be a finite extension of k. For a prime P of K, we denote by OP the valuation ring of P, byMP the maximal ideal of OP, and by FP the residue field OP/MP. Let �O be a Drinfeld A-module over K of rank r. If �O has good reduction at P, let �O . FP denote the reduction of �O at P and let �O(FP) denote the A-module (�O . FP)(FP). If �O is of rank 2 with End �PK (�O) = A, then we obtain an asymptotic formula for the number of primes P of K of degree x for which �O(FP) is cyclic. This result can be viewed as a Drinfeld module analogue of Serre�fs cyclicity result on elliptic curves. We also show that when �O is of rank r  3 a similar result follows.