Let A = Fq[T], 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 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.
Bibliographical noteFunding Information:
The author's work supported by the Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education (2015R1D1A1A01056643).
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