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Spectral Means of Central Values of Automorphic \(L\)-Functions for GL(2)
Masao Tsuzuki, Sophia University, Tokyo, Japan
Memoirs of the American Mathematical Society
2014; 129 pp; softcover
Volume: 235
ISBN-10: 1-4704-1019-2
ISBN-13: 978-1-4704-1019-3
List Price: US$81
Individual Members: US$48.60
Institutional Members: US$64.80
Order Code: MEMO/235/1110
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Not yet published.
Expected publication date is May 1, 2015.

Starting with Green's functions on adele points of \(\mathrm{GL}(2)\) considered over a totally real number field, the author elaborates an explicit version of the relative trace formula, whose spectral side encodes the informaton on period integrals of cuspidal waveforms along a maximal split torus. As an application, he proves two kinds of asymptotic mean formula for certain central \(L\)-values attached to cuspidal waveforms with square-free level.

Table of Contents

  • Introduction
  • Preliminaries
  • Preliminary analysis
  • Green's functions on \(\mathrm{GL}(2,\mathbb{R})\)
  • Green's functions on \(\mathrm{GL}(2,F_v)\) with \(v\) a non archimedean place
  • Kernel functions
  • Regularized periods
  • Automorphic Green's functions
  • Automorphic smoothed kernels
  • Periods of regularized automorphic smoothed kernels: the spectral side
  • A geometric expression of automorphic smoothed kernels
  • Periods of regularized automorphic smoothed kernels: the geometric side
  • Asymptotic formulas
  • An error term estimate in the Weyl type asymptotic law
  • Appendix
  • Bibliography
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