The Heat Kernel and Theta Inversion on SL2(C)

  • Jay Jorgenson
  • Serge Lang
Part of the Springer Monographs in Mathematics book series (SMM)

Table of contents

  1. Front Matter
    Pages i-x
  2. Gaussians, Spherical Inversion, and the Heat Kernel

  3. Enter Γ: The General Trace Formula

  4. The Heat Kernel on Γ\G/K

  5. Fourier-Eisenstein Eigenfunction Expansions

  6. The Eisenstein-Cuspidal Affair

  7. Back Matter
    Pages 311-319

About this book

Introduction

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.

Keywords

Division Invariant convergence convolution development evaluation form function functions kernel zeta function

Authors and affiliations

  • Jay Jorgenson
    • 1
  • Serge Lang
  1. 1.Department of MathematicsCity College of New YorkNew YorkUSA

Bibliographic information

  • DOI https://doi.org/10.1007/978-0-387-38032-2
  • Copyright Information Springer-Verlag New York 2008
  • Publisher Name Springer, New York, NY
  • eBook Packages Mathematics and Statistics
  • Print ISBN 978-0-387-38031-5
  • Online ISBN 978-0-387-38032-2
  • Series Print ISSN 1439-7382
  • About this book