Tata lectures on theta

Author(s)

• Mumford, David
• Musili, C.
• Nori, Madhav
• Previato, Emma
• Stillman, Martin J.
• Norman, Peter
• Tata Institute of Fundamental Research

Bibliographic Information

Tata lectures on theta

David Mumford with the collaboration of C. Musili, M. Nori, E. Previato, and M. stillman

（Modern Birkhäuser classics）

Birkhäuser, c2007

• 1
• 2
• 3

Note

Includes bibliographical references

"Reprint of the 1983 edtion"--t.p. of v. 1

"Reprint of the 1984 edtion"--t.p. of v. 2

"Reprint of the 1991 edtion"--t.p. of v. 3

"Jacobian theta functions and differential equations"--cover of v. 2

Description and Table of Contents

Volume

2 ISBN 9780817645694

Description

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.

Table of Contents

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.

Volume

3 ISBN 9780817645700

Description

This volume is the third of three in a series surveying the theory of theta functions. Based on lectures given by the author at the Tata Institute of Fundamental Research in Bombay, these volumes constitute a systematic exposition of theta functions, beginning with their historical roots as analytic functions in one variable (Volume I), touching on some of the beautiful ways they can be used to describe moduli spaces (Volume II), and culminating in a methodical comparison of theta functions in analysis, algebraic geometry, and representation theory (Volume III).

Table of Contents

Heisenberg groups in general.- The real Heisenberg groups.- Finite Heisenberg groups and sections of line bundles on abelian varieties.- Adelic Heisenberg groups and towers of abelian varieties.- Algebraic theta functions.- Theta functions with quadratic forms.- Riemann's theta relation.- The metaplectic group and the full functional equation of ?.- Theta Functions in Spherical Harmonics.- The homogeneous coordinate ring of an abelian variety.

Volume

1 ISBN 9780817645724

Description

This volume is the first of three in a series surveying the theory of theta functions. Based on lectures given by the author at the Tata Institute of Fundamental Research in Bombay, these volumes constitute a systematic exposition of theta functions, beginning with their historical roots as analytic functions in one variable (Volume I), touching on some of the beautiful ways they can be used to describe moduli spaces (Volume II), and culminating in a methodical comparison of theta functions in analysis, algebraic geometry, and representation theory (Volume III).

Table of Contents

and motivation: theta functions in one variable.- Basic results on theta functions in several variables.

by "Nielsen BookData"

Details

• NCID: BA80022755
• ISBN:
  • 0817645721
  • 0817645691
  • 0817645705
• Country Code: us
• Title Language Code: eng
• Text Language Code: eng
• Place of Publication: Boston
• Pages/Volumes: v.
• Size: 24 cm
• Classification:
  • LCC : QA345
  • DC19 : 515.9/84
  • NDC8 : 413.5
• Subject Headings:
  • LCSH : Functions, Theta
  • NDLSH : ゼータ関数
• Parent Bibliography ID:
  • BA80210813