On the size of the Fourier coefficients of Hilbert cusp forms

TL;DR

This paper establishes new upper bounds for Fourier coefficients of Hilbert cusp forms, including infinitely many with improved bounds, and explores implications for elliptic cusp forms and prime power coefficients.

Contribution

It provides the first non-trivial bounds for almost all Fourier coefficients of Hilbert cusp forms, extending previous results and refining bounds for special cases.

Findings

Proved non-trivial upper bounds for almost all Fourier coefficients.

Established existence of infinitely many coefficients with improved bounds.

Derived bounds for Fourier coefficients of elliptic cusp forms beyond the typical size.

Abstract

Let $\bf f$ be a primitive Hilbert cusp form of weight $k$ and level $\mathfrak{n}$ with Fourier coefficients $c_{\bf f}(\mathfrak{m})$. We prove a non-trivial upper bound for almost all Fourier coefficients $c_{\bf f}(\mathfrak{m})$ of $\bf f$. This generalizes the bounds obtained by Luca, Radziwi\l{}\l{} and Shparlinski. We also prove the existence of infinitely many integral ideals $\mathfrak{m}$ for which the Fourier coefficients $c_{\bf f}(\mathfrak{m})$ have the improved upper bound and further we obtain a refinement of these integral ideals in terms of prime powers. In particular, this enable us to deduce the bound for Fourier coefficients of elliptic cusp forms beyond the `typical size'. Moreover, we prove further improvements of the bound under the assumption of Littlewood's conjecture. Finally, We study a lower bound for the Fourier coefficients at prime powers provided the corresponding Hecke eigen angle is badly approximable.