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Field name	Details
Dewey Class	620
Title	Manifolds and Modular Forms ([EBook]) / by Friedrich Hirzebruch, Thomas Berger, Rainer Jung.
Author	Hirzebruch, Friedrich. , 1927-2012
Added Personal Name	Berger, Thomas
Added Personal Name	Jung, Rainer
Other name(s)	SpringerLink (Online service)
Edition statement	Second Edition.
Publication	Wiesbaden : Vieweg+Teubner Verlag , 1994.
Physical Details	XI, 212 pages : online resource.
Series	Aspects of mathematics 0179-2156 ; ; 20
ISBN	9783663107262
Summary Note	During the winter term 1987/88 I gave a course at the University of Bonn under the title "Manifolds and Modular Forms". I wanted to develop the theory of "Elliptic Genera" and to learn it myself on this occasion. This theory due to Ochanine, Landweber, Stong and others was relatively new at the time. The word "genus" is meant in the sense of my book "Neue Topologische Methoden in der Algebraischen Geometrie" published in 1956: A genus is a homomorphism of the Thorn cobordism ring of oriented compact manifolds into the complex numbers. Fundamental examples are the signature and the A-genus. The A-genus equals the arithmetic genus of an algebraic manifold, provided the first Chern class of the manifold vanishes. According to Atiyah and Singer it is the index of the Dirac operator on a compact Riemannian manifold with spin structure. The elliptic genera depend on a parameter. For special values of the parameter one obtains the signature and the A-genus. Indeed, the universal elliptic genus can be regarded as a modular form with respect to the subgroup r (2) of the modular group; the two cusps 0 giving the signature and the A-genus. Witten and other physicists have given motivations for the elliptic genus by theoretical physics using the free loop space of a manifold.:
Contents note	1 Background -- 2 Elliptic genera -- 3 A universal addition theorem for genera -- 4 Multiplicativity in fibre bundles -- 5 The Atiyah-Singer index theorem -- 6 Twisted operators and genera -- 7 Riemann-Roch and elliptic genera in the complex case -- 8 A divisibility theorem for elliptic genera -- Appendix I Modular forms -- 1 Fundamental concepts -- 2 Examples of modular forms -- 3 The Weierstraß ?-function as a Jacobi form -- 4 Some special functions and modular forms -- 5 Theta functions, divisors, and elliptic functions -- Appendix II The Dirac operator -- 1 The solution -- 2 The problem -- 1 Zolotarev polynomials -- 2 Interpretation as an algebraic curve -- 3 The differential equation — revisited -- 4 Modular interpretation of Zolotarev polynomials -- 5 The embedding of the modular curve -- 6 Applications to elliptic genera -- Symbols.
System details note	Online access to this digital book is restricted to subscription institutions through IP address (only for SISSA internal users)
Internet Site	http://dx.doi.org/10.1007/978-3-663-10726-2
Links to Related Works	Subject References: Engineering, general . Forms, Modular . Manifolds (Mathematics) . Authors: author . Berger, Thomas . Hirzebruch, Friedrich. 1927-2012 . Jung, Rainer . Corporate Authors: SpringerLink (Online service) . Series: Aspects of mathematics . Classification: 620 . 620 (DDC 21) .

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Manifolds and Modular Forms

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