Lattice points counting and bounds on periods of Maass forms

Reznikov, Andre; Su, Feng

doi:10.1090/tran/7684

Lattice points counting and bounds on periods of Maass forms
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by Andre Reznikov and Feng Su PDF

Trans. Amer. Math. Soc. 372 (2019), 2073-2102 Request permission

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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