### 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 |
---|---|

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 1 |

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### All Science Journal Classification (ASJC) codes

- Mathematics(all)

### Cite this

*Kyoto Journal of Mathematics*,

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

}

*Kyoto Journal of Mathematics*, vol. 57, no. 3, pp. 505-518. https://doi.org/10.1215/21562261-2017-0004

**Cyclicity and Titchmarsh divisor problem for Drinfeld modules.** / Virdol, Cristian.

Research output: Contribution to journal › Article

TY - JOUR

T1 - Cyclicity and Titchmarsh divisor problem for Drinfeld modules

AU - Virdol, Cristian

PY - 2017/9/1

Y1 - 2017/9/1

N2 - 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.

AB - 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.

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U2 - 10.1215/21562261-2017-0004

DO - 10.1215/21562261-2017-0004

M3 - Article

VL - 57

SP - 505

EP - 518

JO - Kyoto Journal of Mathematics

JF - Kyoto Journal of Mathematics

SN - 0023-608X

IS - 3

ER -