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Spectral fourth moments of Hecke--Maaß cusp forms
Abstract
In this note, we establish essentially optimal bounds for certain spectral moments of automorphic forms for GL(2). More precisely, we consider the family of Hecke--Maaß cusp forms with spectral parameter in a dyadic interval and study the fourth moment of these forms evaluated at a Heegner point. We additionally present applications of our main result to the shifted convolution problem involving the sum of two squares function r(n) as well as to pointwise Diophantine exponents.
Versions
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BibTeX
@misc{assing2026spectral,
title = {{Spectral fourth moments of Hecke--Maaß cusp forms}},
author = {Edgar Assing and Peter Humphries},
howpublished = {arXiv},
year = {2026},
month = jul,
doi = {10.48550/arxiv.2607.13518},
eprint = {2607.13518},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2607.13518},
}
RIS
TY - UNPB
TI - Spectral fourth moments of Hecke--Maaß cusp forms
AU - Edgar Assing
AU - Peter Humphries
PY - 2026
DA - 2026/07/15
PB - arXiv
DO - 10.48550/arxiv.2607.13518
UR - https://arxiv.org/abs/2607.13518
AB - In this note, we establish essentially optimal bounds for certain spectral moments of automorphic forms for GL(2). More precisely, we consider the family of Hecke--Maaß cusp forms with spectral parameter in a dyadic interval and study the fourth moment of these forms evaluated at a Heegner point. We additionally present applications of our main result to the shifted convolution problem involving the sum of two squares function r(n) as well as to pointwise Diophantine exponents.
ER -