### Abstract

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.

Original language | English |
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Pages (from-to) | 505-518 |

Number of pages | 14 |

Journal | Kyoto Journal of Mathematics |

Volume | 57 |

Issue number | 3 |

DOIs | |

Publication status | Published - 2017 Sep |

### All Science Journal Classification (ASJC) codes

- Mathematics(all)

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## Cite this

*Kyoto Journal of Mathematics*,

*57*(3), 505-518. https://doi.org/10.1215/21562261-2017-0004