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Tata Lectures on Theta II

  • David Mumford
Book

Part of the Modern Birkhäuser Classics book series (MBC)

Table of contents

  1. Front Matter
    Pages i-xiv
    PDF

  2. An Elementary Construction of Hyperelliptic Jacobians

    1. Review of background in algebraic geometry
      David Mumford
      Pages 1-11
    2. Divisors on hyperelliptic curves
      David Mumford
      Pages 12-27
    3. Algebraic construction of the Jacobian of a hyperelliptic curve
      David Mumford
      Pages 28-39
    4. The translation-invariant vector fields
      David Mumford
      Pages 40-50
    5. Neumann’s dynamical system
      David Mumford
      Pages 51-74
    6. Tying together the analytic Jacobian and algebraic Jacobian
      David Mumford
      Pages 75-94
    7. Theta characteristics and the fundamental Vanishing Property
      David Mumford
      Pages 95-105
    8. Frobenius’ theta formula
      David Mumford
      Pages 106-119
    9. Thomae’s formula and moduli of hyperelliptic curves
      David Mumford
      Pages 120-136
    10. Characterization of hyperelliptic period matrices
      David Mumford
      Pages 137-154
    11. The hyperelliptic p-function
      David Mumford
      Pages 155-176
    12. The Korteweg-deVries dynamical system
      David Mumford
      Pages 177-206

  3. Fay’s Trisecant Identity for Jacobian theta functions

    1. The Prime Form E(x,y).
      David Mumford
      Pages 207-213
    2. Fay’s Trisecant Identity
      David Mumford
      Pages 214-222
    3. Corollaries of the identity
      David Mumford
      Pages 223-238
    4. Applications to solutions of differential equations
      David Mumford
      Pages 239-242
    5. The Generalized Jacobian of a Singular Curve and its Theta Function
      David Mumford
      Pages 243-260

  4. Resolution of algebraic equations by theta constants

    1. Resolution of algebraic equations by theta constants
      Hiroshi Umemura
      Pages 261-270
  5. Back Matter
    Pages 271-272
    PDF

About this book

Introduction

The second in a series of three volumes surveying the theory of theta functions, this volume gives emphasis to the special properties of the theta functions associated with compact Riemann surfaces and how they lead to solutions of the Korteweg-de-Vries equations as well as other non-linear differential equations of mathematical physics.

This book presents an explicit elementary construction of hyperelliptic Jacobian varieties and is a self-contained introduction to the theory of the Jacobians. It also ties together nineteenth-century discoveries due to Jacobi, Neumann, and Frobenius with recent discoveries of Gelfand, McKean, Moser, John Fay, and others.

A definitive body of information and research on the subject of theta functions, this volume will be a useful addition to individual and mathematics research libraries.

Keywords

Divisor Identity Invariant Riemann surfaces algebra algebraic geometry equation function geometry mathematical physics mathematics

Authors and affiliations

  • David Mumford
    • 1
  1. 1.Division of Applied MathematicsBrown UniversityProvidenceUSA

Bibliographic information