PreprintCC BY 4.0arXiv
Low moments of Hecke eigenvalue sums
Abstract
We show that partial sums of the Sato--Tate random multiplicative functions introduced by Cogdell and Michel exhibit better-than-square-root cancellation. The proof proceeds via a connection to multiplicative chaos, following Harper's seminal work. By a non-trivial adaptation of Harper's derandomization argument for character sums, we also obtain upper bounds for low moments of Hecke eigenvalue sums and of Hecke eigenforms near the cusp $0$; to our knowledge, this is the first appearance of multiplicative chaos in the context of automorphic forms on GL(2). A novel ingredient is the introduction of Hecke s-norms.
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BibTeX
@misc{hamdan2026moments,
title = {{Low moments of Hecke eigenvalue sums}},
author = {Jad Hamdan and Sun-Kai Leung and Mo Dick Wong},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.15937},
eprint = {2609.15937},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.15937},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Low moments of Hecke eigenvalue sums
AU - Jad Hamdan
AU - Sun-Kai Leung
AU - Mo Dick Wong
PY - 2026
DA - 2026/09/14
PB - arXiv
DO - 10.48550/arxiv.2609.15937
UR - https://arxiv.org/abs/2609.15937
AB - We show that partial sums of the Sato--Tate random multiplicative functions introduced by Cogdell and Michel exhibit better-than-square-root cancellation. The proof proceeds via a connection to multiplicative chaos, following Harper's seminal work. By a non-trivial adaptation of Harper's derandomization argument for character sums, we also obtain upper bounds for low moments of Hecke eigenvalue sums and of Hecke eigenforms near the cusp $0$; to our knowledge, this is the first appearance of multiplicative chaos in the context of automorphic forms on GL(2). A novel ingredient is the introduction of Hecke s-norms.
ER -