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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.

Tata Lectures on Theta II: Jacobian theta functions and differential equations

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.