The Heat Kernel and Theta Inversion on SL2(C) / Edition 1

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The present monograph devlops the fundamental ideas and relults surronding heat kenels, spectral theory, and regularized traces associated the full modular group action on SL2(C) through spherical transform, from which one manifestation of the heat kernel on quotient spaces is obtained through group perodization. Fron a different point of view, one constructs the heat kernel on the group space using an eignfuction, or spectral, expansion, which then leads to a theta function and a theta inbversion 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 Selgerg trace formula and uncover the second step in their envisioned "ladder" of geometrically defined zet functions, where each conjectured step would include lower level zeta functions as factors in functional equations.

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Editorial Reviews

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"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)

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Product Details

  • ISBN-13: 9780387380315
  • Publisher: Springer New York
  • Publication date: 10/15/2008
  • Series: Springer Monographs in Mathematics Series
  • Edition description: 2008
  • Edition number: 1
  • Pages: 319
  • Product dimensions: 6.30 (w) x 9.30 (h) x 0.80 (d)

Table of Contents

Pt. I Gaussians, Spherical Inversion, and the Heat Kernel
1 Spherical Inversion on SL[subscript 2](C) 13
2 The Heat Gaussian and Heat Kernel 45
3 QED, LEG, Transpose, and Casimir 67 Pt. II Enter [Gamma]: The General Trace Formula
4 Convergence and Divergence of the Selberg Trace 85
5 The Cuspidal and Noncuspidal Traces 97 Pt. III The Heat Kernel on [Gamma]\G/K
6 The Fundamental Domain 117
7 [Gamma]-Periodization of the Heat Kernel 135
8 Heat Kernel Convolution on L[superscript 2][subscript cusp] ([Gamma]\G/K) 151 Pt. IV Fourier-Eisenstein Eigenfunction Expansions
9 The Tube Domain for [Gamma][infinity] 167
10 The [Gamma][subscript U]\U-Fourier Expansion of Eisenstein Series 191
11 Adjointness Formula and the [Gamma]\G-Eigenfunction Expansion 223 Pt. V The Eisenstein - Cuspidal Affiar
12 The Eisenstein Y-Asymptotics 243
13 The Cuspidal Trace Y-Asymptotics 261
14 Analytic Evaluations 287 References 311 Index 317
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