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# memo_has_moved_text();Spectral means of central values of automorphic $L$-functions for $\mathrm {GL}(2)$

Masao Tsuzuki

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: http://dx.doi.org/10.1090/memo/1110
Published electronically: October 24, 2014

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Chapters

• Chapter 1. Introduction
• Chapter 2. Preliminaries
• Chapter 3. Preliminary analysis
• Chapter 4. Green’s functions on $\mathrm {GL}(2,\mathbb {R})$
• Chapter 5. Green’s functions on $\mathrm {GL}(2,F_v)$ with $v$ a non archimedean place
• Chapter 6. Kernel functions
• Chapter 7. Regularized periods
• Chapter 8. Automorphic Green’s functions
• Chapter 9. Automorphic smoothed kernels
• Chapter 10. Periods of regularized automorphic smoothed kernels: the spectral side
• Chapter 11. A geometric expression of automorphic smoothed kernels
• Chapter 12. Periods of regularized automorphic smoothed kernels: the geometric side
• Chapter 13. Asymptotic formulas
• Chapter 14. An error term estimate in the Weyl type asymptotic law
• Chapter 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.