Tata lectures on theta. III
type: Item
instance of: scholarly article
title: Tata lectures on theta. III
zbMATH Open document ID: 0744.14033
author: Madhav V. Nori
author: P. Norman
author: David Mumford
published in: Progress in Mathematics
publication date: 1992-09-18T00:00:00Z
review text: This volume is the third of a three volume series on the theory of theta functions. Like the first volume (1983; Zbl 0509.14049) and the second one (1984; Zbl 0549.14014), it is based on D. Mumford's lectures held at the Tata Institute of Fundamental Research (Bombay) during November 1978 -- March 1979. Whilst Volumes I and II are devoted to the classical, geometric and analytic theory of complex abelian varieties and their associated theta functions, together with their most recent applications to moduli spaces of curves and Jacobian varieties, non-linear partial differential equations and special integrable Hamiltonian systems, this final volume focusses on the purely algebraic theory of theta functions. The aim of the authors is to compare and clarify the deep interrelations between the different ways of viewing (and using) theta functions in analysis, algebraic geometry, and representation theory. To be more precise, theta functions occur (i) as analytic functions depending on complex vectors and Riemann matrices (in the classical setting), (ii) as matrix coefficients of representations of Heisenberg groups and/or metaplectic groups, and (iii) as sections of line bundles on abelian varieties and/or their moduli spaces (in the general setting of algebraic geometry over arbitrary groundfields), and these three conceptually and methodically completely different frameworks for theta functions are shown to be equivalent, on an ultimate and deep level, in the course of the book. In this regard, the present third volume, consisting of the concluding fourth chapter of the whole series, reflects the recent progress that has been made, to a great extent by the authors themselves, in understanding the ubiquity and crucial role of theta functions in various branches of mathematics. - In particular, the algebraic aspect of theta functions, which is the relevant one when viewing them as sections of line bundles on families of polarized abelian varieties over arbitrary base sc…
Mathematics Subject Classification ID: 14K25
Mathematics Subject Classification ID: 14-02
Mathematics Subject Classification ID: 14K10
Mathematics Subject Classification ID: 11F27
zbMATH DE Number: 51158
zbMATH Keywords: metaplectic groups
zbMATH Keywords: algebraic theory of theta functions
zbMATH Keywords: representations of Heisenberg groups
zbMATH Keywords: sections of line bundles on complex abelian varieties
zbMATH Keywords: isogenies
zbMATH Keywords: tower of an abelian variety
zbMATH Keywords: theta relations
zbMATH Keywords: homogeneous coordinate ring of an abelian variety
reviewed by: Werner Kleinert
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 51158