Author	Mumford, David. author
Title	Tata Lectures on Theta II [electronic resource] / by David Mumford
Imprint	Boston, MA : Birkhรคuser Boston : Imprint: Birkhรคuser, 2007
Connect to	http://dx.doi.org/10.1007/978-0-8176-4578-6
Descript	XIV, 272 p. 21 illus. online resource

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

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 -- Neumannโs dynamical system -- Tying together the analytic Jacobian and algebraic Jacobian -- Theta characteristics and the fundamental Vanishing Property -- Frobeniusโ theta formula -- Thomaeโs formula and moduli of hyperelliptic curves -- Characterization of hyperelliptic period matrices -- The hyperelliptic p-function -- The Korteweg-deVries dynamical system -- Fayโs Trisecant Identity for Jacobian theta functions -- The Prime Form E(x,y). -- Fayโs 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

1. Mathematics
2. Geometry
3. algebraic
4. Functions of complex variables
5. Differential equations
6. partial
7. Functions
8. special
9. Algebraic topology
10. Mathematical physics
11. Mathematics
12. Special Functions
13. Algebraic Geometry
14. Mathematical Methods in Physics
15. Functions of a Complex Variable
16. Algebraic Topology
17. Partial Differential Equations