Tata Lectures on Theta II

The second in a series of three volumes that survey the theory of theta functions, this volume emphasizes 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.

It 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.

Auteur
David Mumford was born on June 11, 1937 in England and has been associated with Harvard University continuously from entering as freshman to his present position of Higgins Professor of Mathematics.
Mumford worked in the fields of Algebraic Gemetry in the 60's and 70's, concentrating especially on the theory of moduli spaces: spaces which classify all objects of some type, such as all curves of a given genus or all vector bundles on a fixed curve of given rank and degree. Mumford was awarded the Fields Medal in 1974 for his work on moduli spaces and algebraic surfaces. He is presently working on the mathematics of pattern recognition and artificial intelligence.

Contenu
An Elementary Construction of Hyperelliptic Jacobians.- Review of background in algebraic geometry.- Divisors on hyperelliptic curves.- Algebraic construction of the Jacobian of a hyperelliptic curve.- The translation-invariant vector fields.- Neumann's dynamical system.- Tying together the analytic Jacobian and algebraic Jacobian.- Theta characteristics and the fundamental Vanishing Property.- Frobenius' theta formula.- Thomae's formula and moduli of hyperelliptic curves.- Characterization of hyperelliptic period matrices.- The hyperelliptic p-function.- The Korteweg-deVries dynamical system.- Fay's Trisecant Identity for Jacobian theta functions.- The Prime Form E(x,y)..- Fay's Trisecant Identity.- Corollaries of the identity.- Applications to solutions of differential equations.- The Generalized Jacobian of a Singular Curve and its Theta Function.- Resolution of algebraic equations by theta constants.- Resolution of algebraic equations by theta constants.