\(q\)-adic valuation and additive distribution relation for certain \(q\)-periodic functions
type: Item
instance of: scholarly article
title: \(q\)-adic valuation and additive distribution relation for certain \(q\)-periodic functions
zbMATH Open document ID: 0924.11047
author: A. Bayad
published in: Journal of Number Theory
publication date: 1997-07-24T00:00:00Z
review text: The author gives a local analogue for Tate elliptic curves \(\overline{K}^\times / q^{\mathbb{Z}}\) over a local field \(K\) of the additive distribution law for Siegel functions on classical complex elliptic curves \(\mathbb{C}/\Omega\), that he obtained with \textit{G. Robert} in a previous article [``Une relation de distribution additive satisfaite par une famille de fonctions elliptiques'', Bull. Soc. Math. Fr. (to appear)]. Given a local field \(K\), an element \(q\) of \(K^\times \) with strictly positive valuation, and a torsion point \(\psi\) in \(\overline{K}^\times / q^{\mathbb{Z}}\), he introduces in a quite natural way a local Siegel function associated to the discrete subgroup \(q^{\mathbb{Z}} \psi^{\mathbb{Z}}\) of \(K^\times \) and an ad hoc local discriminant function. Then he is able to compute the \(q\)-valuations of these functions evaluated at certain torsion points (theorem 3.13) and to prove that they satisfy an additive distribution relation (theorem 4.1). In the appendix, the author uses the results above to give a nice simplification of local computations of \(q\)-adic valuations for products of elliptic resolvents which were used to construct quadratic Stickelberger elements in a previous article with \textit{W. Bley} and \textit{Ph. Cassou-Noguès} [J. Algebra 179, 145-190 (1996; Zbl 0863.11074)].
reviewed by: Jean-François Jaulent
Mathematics Subject Classification ID: 11G07
Mathematics Subject Classification ID: 11S80
Mathematics Subject Classification ID: 11R33
zbMATH DE Number: 1037289
zbMATH Keywords: Tate elliptic curves
zbMATH Keywords: local field
zbMATH Keywords: additive distribution law for Siegel functions
zbMATH Keywords: \(q\)-valuations
zbMATH Keywords: products of elliptic resolvents
zbMATH Keywords: quadratic Stickelberger elements
MaRDI profile type: Publication
full work available at URL: jnth.1997.2143
OpenAlex ID: W2066674636
cites work: Arithmetic sums and Stickelberger elements
cites work: Product formulae on elliptic curves
cites work: Q3953865
cites work: Q4143392
cites work: Q5590569
DOI: 10.1006/JNTH.1997.2143
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description: scientific article; zbMATH DE number 1037289