Cyclicity of finite Drinfeld modules

Abstract

Le tA=Fq[T] be the polynomial ring over the finite field Fq,letk=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, by MP the maximal ideal of OP, and by FP the residue field OP/MP. Let φ be a DrinfeldA-module over K of rankr. If φ has good reduction at P, let φ ⊗ FP denote the reduction of φ at P and letφ(FP) denote the A-module (φ⊗FP)(FP). Ifφis of rank 2 with End ̄K(φ)=A, then we obtain an asymptotic formula for the number of primes P of K of degree x for which φ (FP) is cyclic. This result can be viewed as a Drinfeld module analogue of Serre’s cyclicity result on elliptic curves. We also show that whenφis of rankr 3 a similar result follows.