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An Approach to the Selberg Trace Formula via the Selberg Zeta-Function

  • Book
  • © 1987

Overview

Authors:
  1. Jürgen Fischer
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Part of the book series: Lecture Notes in Mathematics (LNM, volume 1253)

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Table of contents (5 chapters)

  1. Front Matter

    Pages I-III
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  2. Introduction

    • Jürgen Fischer
    Pages 1-13

  3. Basic facts

    • Jürgen Fischer
    Pages 14-39

  4. The trace of the iterated resolvent kernel

    • Jürgen Fischer
    Pages 40-112

  5. The entire function Ξ associated with the selberg zeta-function

    • Jürgen Fischer
    Pages 113-161

  6. The general selberg trace formula

    • Jürgen Fischer
    Pages 162-175

  7. Back Matter

    Pages 176-184
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Keywords

About this book

The Notes give a direct approach to the Selberg zeta-function for cofinite discrete subgroups of SL (2,#3) acting on the upper half-plane. The basic idea is to compute the trace of the iterated resolvent kernel of the hyperbolic Laplacian in order to arrive at the logarithmic derivative of the Selberg zeta-function. Previous knowledge of the Selberg trace formula is not assumed. The theory is developed for arbitrary real weights and for arbitrary multiplier systems permitting an approach to known results on classical automorphic forms without the Riemann-Roch theorem. The author's discussion of the Selberg trace formula stresses the analogy with the Riemann zeta-function. For example, the canonical factorization theorem involves an analogue of the Euler constant. Finally the general Selberg trace formula is deduced easily from the properties of the Selberg zeta-function: this is similar to the procedure in analytic number theory where the explicit formulae are deduced from the properties of the Riemann zeta-function. Apart from the basic spectral theory of the Laplacian for cofinite groups the book is self-contained and will be useful as a quick approach to the Selberg zeta-function and the Selberg trace formula.

Bibliographic Information

  • Book Title: An Approach to the Selberg Trace Formula via the Selberg Zeta-Function

  • Authors: Jürgen Fischer

  • Series Title: Lecture Notes in Mathematics

  • DOI: https://doi.org/10.1007/BFb0077696

  • Publisher: Springer Berlin, Heidelberg

  • eBook Packages: Springer Book Archive

  • Copyright Information: Springer-Verlag Berlin Heidelberg 1987

  • Softcover ISBN: 978-3-540-15208-8Published: 23 April 1987

  • eBook ISBN: 978-3-540-39331-3Published: 15 November 2006

  • Series ISSN: 0075-8434

  • Series E-ISSN: 1617-9692

  • Edition Number: 1

  • Number of Pages: IV, 188

  • Topics: Number Theory

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