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Riemann Surfaces and Generalized Theta Functions

  • Robert C. Gunning

Part of the Ergebnisse der Mathematik und ihrer Grenzgebiete book series (MATHE2, volume 91)

Table of contents

  1. Front Matter
    Pages I-XII
  2. Robert C. Gunning
    Pages 1-16
  3. Robert C. Gunning
    Pages 17-38
  4. Robert C. Gunning
    Pages 39-90
  5. Robert C. Gunning
    Pages 91-129
  6. Back Matter
    Pages 130-168

About this book

Introduction

The investigation of the relationships between compact Riemann surfaces (al­ gebraic curves) and their associated complex tori (Jacobi varieties) has long been basic to the study both of Riemann surfaces and of complex tori. A Riemann surface is naturally imbedded as an analytic submanifold in its associated torus; and various spaces of linear equivalence elasses of divisors on the surface (or equivalently spaces of analytic equivalence elasses of complex line bundies over the surface), elassified according to the dimensions of the associated linear series (or the dimensions of the spaces of analytic cross-sections), are naturally realized as analytic subvarieties of the associated torus. One of the most fruitful of the elassical approaches to this investigation has been by way of theta functions. The space of linear equivalence elasses of positive divisors of order g -1 on a compact connected Riemann surface M of genus g is realized by an irreducible (g -1)-dimensional analytic subvariety, an irreducible hypersurface, of the associated g-dimensional complex torus J(M); this hyper­ 1 surface W- r;;;, J(M) is the image of the natural mapping Mg- -+J(M), and is g 1 1 birationally equivalent to the (g -1)-fold symmetric product Mg- jSg-l of the Riemann surface M.

Keywords

Division Equivalence Jacobi Natural Riemann surface Riemannsche Fläche Theta Functions Theta function Thetafunktion function functions manifold mapping

Authors and affiliations

  • Robert C. Gunning
    • 1
  1. 1.Dept. of MathematicsPrinceton UniversityPrincetonUSA

Bibliographic information

  • DOI https://doi.org/10.1007/978-3-642-66382-6
  • Copyright Information Springer-Verlag Berlin Heidelberg 1976
  • Publisher Name Springer, Berlin, Heidelberg
  • eBook Packages Springer Book Archive
  • Print ISBN 978-3-642-66384-0
  • Online ISBN 978-3-642-66382-6
  • Series Print ISSN 0071-1136
  • Buy this book on publisher's site