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September 2017 Cyclicity and Titchmarsh divisor problem for Drinfeld modules
Cristian Virdol
Kyoto J. Math. 57(3): 505-518 (September 2017). DOI: 10.1215/21562261-2017-0004

Abstract

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 write r=hed, where E is the center of D:=EndF¯(ϕ)Q, e=[E:Q], and d=[D:E]12. 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 rd, we obtain an asymptotic formula for the number of primes of F of degree x for which ϕ¯(F) has at most (r1) 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.

Citation

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Cristian Virdol. "Cyclicity and Titchmarsh divisor problem for Drinfeld modules." Kyoto J. Math. 57 (3) 505 - 518, September 2017. https://doi.org/10.1215/21562261-2017-0004

Information

Received: 1 June 2015; Revised: 17 February 2016; Accepted: 20 April 2016; Published: September 2017
First available in Project Euclid: 14 April 2017

zbMATH: 06774045
MathSciNet: MR3685053
Digital Object Identifier: 10.1215/21562261-2017-0004

Subjects:
Primary: 11G09
Secondary: 11G15

Rights: Copyright © 2017 Kyoto University

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Vol.57 • No. 3 • September 2017
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