Bounding hyperbolic and spherical coefficients of Maass forms

Blomer, Valentin; Brumley, Farrell; Kontorovich, Alex; Templier, Nicolas

doi:10.5802/jtnb.879

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Cited by 2 publications

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“…The authors would like to express their gratitude to Jens Marklof, Curt McMullen, and Peter Sarnak for many enlightening comments and suggestions. The second author would especially like to thank Valentin Blomer, Farrell Brumley, and Nicolas Templier for many hours of discussion about this problem; see also the related work in [BBKT14].…”

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confidence: 99%

Effective equidistribution of shears and applications

Kelmer

Kontorovich

2017

Math. Ann.

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A "shear" is a unipotent translate of a cuspidal geodesic ray in the quotient of the hyperbolic plane by a non-uniform discrete subgroup of PSL(2, R), possibly of infinite co-volume. We prove the regularized equidistribution of shears under large translates with effective (that is, power saving) rates. We also give applications to weighted second moments of GL(2) automorphic L-functions, and to counting lattice points on locally affine symmetric spaces.

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Section: Acknowledgementsmentioning

confidence: 99%

Effective equidistribution of shears and applications

Kelmer

Kontorovich

2017

Math. Ann.

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Lattice points counting and bounds on periods of Maass forms

Reznikov

2019

Trans. Amer. Math. Soc.

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We provide a "soft" proof for non-trivial bounds on spherical, hyperbolic and unipotent Fourier coefficients of a fixed Maass form for a general co-finite lattice Γ in PGL 2 (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 PGL 2 (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 PGL 2 (R) obtained in [BR2] in order to slightly improve these bounds.

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Bounding hyperbolic and spherical coefficients of Maass forms

Cited by 2 publications

References 13 publications

Effective equidistribution of shears and applications

Effective equidistribution of shears and applications

Lattice points counting and bounds on periods of Maass forms

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