Lattice points counting and bounds on periods of maass forms

Andre Reznikov; Feng Su

doi:10.1090/tran/7684

Lattice points counting and bounds on periods of maass forms

Andre Reznikov
, Feng Su

Department of Mathematics

Xi'an Jiaotong-Liverpool University

Research output: Contribution to journal › Article › peer-review

Abstract

We provide a “soft” proof for nontrivial bounds on spherical, hyperbolic, and unipotent Fourier coefficients of a fixed Maass form for a general cofinite lattice Γ in PGL₂(R). We use the amplification method based on the Airy type phenomenon for corresponding matrix coefficients and an effective Selberg type pointwise asymptotic for the lattice points counting in various homogeneous spaces for the group PGL₂(R). This requires only L²-theory. We also show how to use the uniform bound for the L⁴-norm of K-types in a fixed automorphic representation of PGL₂(R) in order to slightly improve these bounds.

Original language	English
Pages (from-to)	2073-2102
Number of pages	30
Journal	Transactions of the American Mathematical Society
Volume	372
Issue number	3
DOIs	https://doi.org/10.1090/tran/7684
State	Published - 1 Aug 2019

Bibliographical note

Publisher Copyright:
© 2019 American Mathematical Society.

Funding

Received by the editors November 7, 2016, and, in revised form, April 8, 2018, and July 30, 2018. 2010 Mathematics Subject Classification. Primary 11M41; Secondary 11M32, 22E55, 11F25, 11F70, 30B40. Key words and phrases. Automorphic representations, periods, subconvexity bounds for L-functions. The research was partially supported by ERC Grant No. 291612, by ISF Grant No. 533/14, and by the National Science Foundation under Grant No. DMS-1638352 during the visit of the first author to IAS. The research was partially supported by ERC Grant No. 291612, by ISF Grant No. 533/14, and by the National Science Foundation under Grant No. DMS-1638352 during the visit of the first author to IAS.

Funders	Funder number
National Science Foundation	DMS-1638352, 1638352
European Commission	291612
Israel Science Foundation	533/14

Keywords

And phrases
Automorphic representations
Periods
Subconvexity bounds for L-functions

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Lattice points counting and bounds on periods of maass forms

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