A basis for the non-archimedean holomorphic theta functions
type: Item
instance of: scholarly article
title: A basis for the non-archimedean holomorphic theta functions
zbMATH Open document ID: 0807.14036
published in: Bulletin of the Belgian Mathematical Society - Simon Stevin
publication date: 1994-07-11T00:00:00Z
full work available at URL: https://eudml.org/doc/229964
review text: Let \(k\) be a complete, non-archimedean, algebraically closed field and \(T = (k^*)^ g/ \Lambda\) an analytic torus. Given a cocycle \(\xi \in Z^ 1 (\Lambda,A)\) \((A\) being the group of nowhere vanishing holomorphic functions on \(G)\) there exists a natural isogeny \(\lambda_{\overline \xi} : T \to \widehat T\) (= dual variety of \(T)\). The author defines an analog of the theta group for abelian varieties for each \(\xi\) and proves that this theta group is a central extension of Ker\((\lambda_{\overline \xi})\) by the multiplicative group. Following the ideas of \textit{D. Mumford} [Invent. Math. 1, 287-354 (1966; Zbl 0219.14024)], the author constructs a basis for the vector space of theta functions in forms of the theta group.
Mathematics Subject Classification ID: 14K25
Mathematics Subject Classification ID: 14M25
Mathematics Subject Classification ID: 14G20
Mathematics Subject Classification ID: 14H30
Mathematics Subject Classification ID: 14K02
zbMATH DE Number: 589777
zbMATH Keywords: non-archimedean groundfield
zbMATH Keywords: isogeny
zbMATH Keywords: theta functions
zbMATH Keywords: theta group
author: Guido Van Steen
reviewed by: J. M. Muñoz Porras
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 589777