Cyclicity and Titchmarsh divisor problem for Drinfeld modules

Abstract

where Fq is a finite field, let Q = Fq(T ), and let F be a finite extension of Q. Consider φ a Drinfeld A-module over F of rank r. We write r = hed, where E is the center ofwe denote by F℘ the residue field at ℘. If φ has good reduction at ℘, let φ denote the reduction of φ at ℘. In this article, in particular, when r = d, we obtain an asymptotic formula for the number of primes ℘ of F of degree x for which φ(F℘) has at most (r − 1) cyclic components. This result answers an old question of Serre on the cyclicity of general Drinfeld A-modules. We also prove an analogue of the Titchmarsh divisor problem for Drinfeld modules. f φ,F (x) = ℘ ∈ P φ | deg F ℘ = x, φ(F ℘ ) has at most (r − 1) cyclic components ,