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Spectral means of central values of automorphic $L$-functions for $\mathrm {GL}(2)$
About this Title
Masao Tsuzuki, Department of Science and Technology, Sophia University, Kioi-cho 7-1 Chiyoda-ku Tokyo, 102-8554, Japan
Publication: Memoirs of the American Mathematical Society
Publication Year:
2015; Volume 235, Number 1110
ISBNs: 978-1-4704-1019-3 (print); 978-1-4704-2228-8 (online)
DOI: https://doi.org/10.1090/memo/1110
Published electronically: October 24, 2014
MSC: Primary 11F72; Secondary 11F67, 11F36
Table of Contents
Chapters
- 1. Introduction
- 2. Preliminaries
- 3. Preliminary analysis
- 4. Green’s functions on $\mathrm {GL}(2,\mathbb {R})$
- 5. Green’s functions on $\mathrm {GL}(2,F_v)$ with $v$ a non archimedean place
- 6. Kernel functions
- 7. Regularized periods
- 8. Automorphic Green’s functions
- 9. Automorphic smoothed kernels
- 10. Periods of regularized automorphic smoothed kernels: the spectral side
- 11. A geometric expression of automorphic smoothed kernels
- 12. Periods of regularized automorphic smoothed kernels: the geometric side
- 13. Asymptotic formulas
- 14. An error term estimate in the Weyl type asymptotic law
- 15. Appendix
Abstract
Starting with Green’s functions on adele points of $\mathrm {GL}(2)$ considered over a totally real number field, we elaborate 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, we prove two kinds of asymptotic mean formula for certain central $L$-values attached to cuspidal waveforms with square-free level.- Ehud Moshe Baruch and Zhengyu Mao, Central value of automorphic $L$-functions, Geom. Funct. Anal. 17 (2007), no. 2, 333–384. MR 2322488, DOI 10.1007/s00039-007-0601-3
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