On the zeroes of certain periodic functions over valued fields of positive characteristic
type: Item
instance of: scholarly article
title: On the zeroes of certain periodic functions over valued fields of positive characteristic
zbMATH Open document ID: 1271.11103
author: Ernst-Ulrich Gekeler
author: Philipp Stopp
published in: Journal of Number Theory
publication date: 2013-01-18T00:00:00Z
review text: Let \({\mathbb C}_p\) be an algebraically closed field of finite characteristic \(p\), complete with respect a non-arhimedean absolute value \(|\cdot |\). Consider \(\Lambda \subseteq {\mathbb C}_p\) a discrete \({\mathbb F}_p\)-submodule such that there exists an \({\mathbb F}_p\)--basis \(\{\lambda_0,\lambda_1,\dots\}\) of \(\Lambda\) such that \(0<|\lambda_0|<|\lambda_1|< \cdots \) and \(|\lambda_i|\to\infty\). For a natural number \(k\) define the meromorphic function \[ C_{k,\Lambda}(z):=\sum_{\lambda\in \Lambda}\frac{1} {(z-\lambda)^k} \] on \({\mathbb C}_p\). A weak form of the main result of the paper establishes that all zeroes of \(C_{k,\Lambda}\) lie on critical spheres \(S_{i,\Lambda}=\partial B(0,|\lambda_i|)=\{ y\in {\mathbb C}_p\mid |y|=|\lambda_i|\}\) of \(\Lambda\). The strong form of the result gives an accurate account of the number of zeroes of \(C_{k,\Lambda}\) on \(S_{i,\Lambda}\), which depends only on \(i\) and \(k\), but not on \(\Lambda\). The paper is organized as follows. In Section 2 the authors collect some facts about \(p\)-adic expansions. In Section 3 power sums on elements of finite sublattices are discussed. Goss polynomials attached to \(\Lambda\) are introduced in Section 4. Section 5 is devoted to the study of the fundamental domain \({\mathfrak F}_{\Lambda}\). The proof of the main result, Theorem 6.1, is given in Section 6. The proof is based on properties of power sums and non-archimedean contour integration. In the last section the authors give examples and applications.
Mathematics Subject Classification ID: 11R58
Mathematics Subject Classification ID: 11F52
Mathematics Subject Classification ID: 14G22
zbMATH DE Number: 6126843
zbMATH Keywords: rigid-analytic period functions
zbMATH Keywords: Goss polynomials
zbMATH Keywords: distribution of zeros of \(p\)-adic functions
reviewed by: Gabriel D. Villa Salvador
describes a project that uses: OEIS
Wikidata QID: Q60692234
MaRDI profile type: Publication
full work available at URL: j.jnt.2012.02.006
OpenAlex ID: W2033053344
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DOI: 10.1016/J.JNT.2012.02.006
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description: scientific article; zbMATH DE number 6126843