Zeroes of Eisenstein series for principal congruence subgroups over rational function fields
type: Item
instance of: scholarly article
title: Zeroes of Eisenstein series for principal congruence subgroups over rational function fields
zbMATH Open document ID: 1268.11062
author: Ernst-Ulrich Gekeler
published in: Journal of Number Theory
publication date: 2011-11-02T00:00:00Z
review text: A classical result of \textit{F. K. C. Rankin} and \textit{H. P. F. Swinnerton-Dyer} [Bull. Lond. Math. Soc. 2, 169--170 (1970; Zbl 0203.35504)] says that Eisenstein series for the full modular group have all their zeros in the standard fundamental domain on the unit circle. An analogue in rational function fields was proven by the reviewer [C. R. Acad. Sci., Paris, Sér. I 321, No. 7, 817--820 (1995; Zbl 0847.11033)]. In this paper, Gekeler studies what happens for Eisenstein series for principal congruence subgroups over a rational function field. He locates the zeros inside certain very specific rational subdomains of a standard fundamental domain inside the Drinfeld upper half space (not necessarily the strict function field analogue of the unit circle). The proofs are based on studying the Newton polygons of some polynomials introduced by \textit{D. Goss} [Compos. Math. 41, 3--38 (1980; Zbl 0422.10020)].
reviewed by: Gunther Cornelissen
Mathematics Subject Classification ID: 11F52
Mathematics Subject Classification ID: 11T55
Mathematics Subject Classification ID: 11F85
zbMATH DE Number: 5966269
zbMATH Keywords: Eisenstein series
zbMATH Keywords: zeros
zbMATH Keywords: function field
MaRDI profile type: Publication
full work available at URL: j.jnt.2011.07.008
OpenAlex ID: W1969107065
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DOI: 10.1016/J.JNT.2011.07.008
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description: scientific article; zbMATH DE number 5966269