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Transactions of the American Mathematical Society

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Lattice points counting and bounds on periods of Maass forms


Authors: Andre Reznikov and Feng Su
Journal: Trans. Amer. Math. Soc. 372 (2019), 2073-2102
MSC (2010): Primary 11M41; Secondary 11M32, 22E55, 11F25, 11F70, 30B40
DOI: https://doi.org/10.1090/tran/7684
Published electronically: January 16, 2019
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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 $ \Gamma $ in $ {\operatorname {PGL}_2(\mathbb{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 $ {\operatorname {PGL}_2(\mathbb{R})}$. This requires only $ L^2$-theory. We also show how to use the uniform bound for the $ L^4$-norm of $ K$-types in a fixed automorphic representation of $ {\operatorname {PGL}_2(\mathbb{R})}$ in order to slightly improve these bounds.


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Additional Information

Andre Reznikov
Affiliation: Department of Mathematics, Bar-Ilan University, Ramat Gan 52900, Israel
Email: reznikov@math.biu.ac.il

Feng Su
Affiliation: Department of Mathematics, Bar-Ilan University, Ramat Gan 52900, Israel
Address at time of publication: Department of Mathematical Sciences, Xi’an Jiaotong-Liverpool University, 111 Ren’ai Road, Suzhou Industrial Park, Suzhou 215123, People’s Republic of China
Email: fsuxyz@126.com

DOI: https://doi.org/10.1090/tran/7684
Keywords: Automorphic representations, periods, subconvexity bounds for $L$-functions
Received by editor(s): November 7, 2016
Received by editor(s) in revised form: April 8, 2018, and July 30, 2018
Published electronically: January 16, 2019
Additional Notes: 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
Article copyright: © Copyright 2019 American Mathematical Society