Overview
- Contains new research and results
- Simplifies the development of the trace formula and theta inversion by using the heat kernel
- One of the co-authors, Serge Lang, was the most prolific author of the 20th century
Part of the book series: Springer Monographs in Mathematics (SMM)
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Table of contents (15 chapters)
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Front Matter
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Gaussians, Spherical Inversion, and the Heat Kernel
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Enter Γ: The General Trace Formula
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Fourier-Eisenstein Eigenfunction Expansions
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The Eisenstein-Cuspidal Affair
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Back Matter
Keywords
About this book
The present monograph develops the fundamental ideas and results surrounding heat kernels, spectral theory, and regularized traces associated to the full modular group acting on SL2(C). The authors begin with the realization of the heat kernel on SL2(C) through spherical transform, from which one manifestation of the heat kernel on quotient spaces is obtained through group periodization. From a different point of view, one constructs the heat kernel on the group space using an eigenfunction, or spectral, expansion, which then leads to a theta function and a theta inversion formula by equating the two realizations of the heat kernel on the quotient space. The trace of the heat kernel diverges, which naturally leads to a regularization of the trace by studying Eisenstein series on the eigenfunction side and the cuspidal elements on the group periodization side. By focusing on the case of SL2(Z[i]) acting on SL2(C), the authors are able to emphasize the importance of specific examples of the general theory of the general Selberg trace formula and uncover the second step in their envisioned "ladder" of geometrically defined zeta functions, where each conjectured step would include lower level zeta functions as factors in functional equations.
Reviews
From the reviews:
"The book under review … provides an introduction to the general theory of semisimple or reductive groups G, with symmetric space G/K (K maximal compact). … It is … meant for experienced insiders, even as the presentation of the material is excellent and accessible. A well-prepared graduate student would do well with this book. More experienced analytic number theorists will find it enjoyable and spellbinding. … I heartily recommend to other analytic number theorists of a similar disposition." (Michael Berg, MAA Online, December, 2008)
“This book is part of a program of the authors to develop a systematic theory of theta and zeta functions on homogeneous spaces, using techniques of harmonic analysis and, in particular, heat kernels. … the book includes many details that would likely have been left for the reader to work out by her/himself in a more streamlined monograph. … all in all, an enjoyable book to read.” (Fredrik Strömberg, Zentralblatt MATH, Vol. 1192, 2010)
Authors and Affiliations
Bibliographic Information
Book Title: The Heat Kernel and Theta Inversion on SL2(C)
Authors: Jay Jorgenson, Serge Lang
Series Title: Springer Monographs in Mathematics
DOI: https://doi.org/10.1007/978-0-387-38032-2
Publisher: Springer New York, NY
eBook Packages: Mathematics and Statistics, Mathematics and Statistics (R0)
Copyright Information: Springer-Verlag New York 2008
Hardcover ISBN: 978-0-387-38031-5Published: 15 October 2008
Softcover ISBN: 978-1-4419-2282-3Published: 19 November 2010
eBook ISBN: 978-0-387-38032-2Published: 20 February 2009
Series ISSN: 1439-7382
Series E-ISSN: 2196-9922
Edition Number: 1
Number of Pages: X, 319
Topics: Number Theory, Group Theory and Generalizations, Algebra, Analysis