Cyclicity and Titchmarsh divisor problem for Drinfeld modules

Let A = F_q[T], where F_q is a finite field, let Q = F_q(T), and let F be a finite extension of Q. Consider φ a Drinfeld A-module over F of rank r. We writer = hed, where E is the center of D := End _F (φ) ⊗ Q, e = [E : Q], and d = [D : E] ^1/2. If ℘ is a prime of F, we 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.