J. London Math. Soc. (2) 80 (2009) 567–584 C2009 London Mathematical Society doi:10.1112/jlms/jdp043 Cyclicity of finite Drinfeld modules

Liu 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 φ be a Drinfeld A-module over K of rank r. If φ has good reduction at P, let φ ⊗ FP denote the reduction of φ at P and let φ(FP) denote the A-module (φ ⊗ FP)(FP). If φ is of rank 2 with EndK̄(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 when φ is of rank r 3 a similar result follows. 1