Spectral fourth moments of Hecke--Maa{\ss} cusp forms

Edgar Assing; Peter Humphries

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arXiv · 2607.13518

Spectral fourth moments of Hecke--Maa{\ss} cusp forms

Edgar Assing·Peter Humphries

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Abstract

In this note, we establish essentially optimal bounds for certain spectral moments of automorphic forms for $\mathrm{GL}(2)$. More precisely, we consider the family of Hecke--Maa{\ss} 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.

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Edgar Assing, Peter Humphries. 2026-07-15. Spectral fourth moments of Hecke--Maa{\ss} cusp forms. https://arxiv.org/abs/2607.13518

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