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Tata Lectures on Theta II

Jacobian theta functions and differential equations

  • Book
  • © 2007

Overview

Authors:
  1. David Mumford

    1. Division of Applied Mathematics, Brown University, Providence, USA

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Part of the book series: Modern Birkhäuser Classics (MBC)

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Table of contents (18 chapters)

  1. Front Matter

    Pages i-xiv
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  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
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Keywords

About this book

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.

Authors and Affiliations

  • Division of Applied Mathematics, Brown University, Providence, USA

    David Mumford

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