On fundamental Fourier coefficients of Siegel cusp forms of degree 2

TL;DR

This paper investigates the Fourier coefficients of degree 2 Siegel cusp forms, establishing their sign change frequency, size bounds under GRH and Gan-Gross-Prasad conjecture, and contributing to understanding their oscillatory behavior.

Contribution

It provides new results on the sign changes and size bounds of Fourier coefficients of Siegel cusp forms of degree 2, under certain hypotheses.

Findings

Fourier coefficients have at least X^{1-ε} sign changes.

Fourier coefficients take at least X^{1-ε} large values.

Conditional bounds on Fourier coefficients assuming GRH and Gan-Gross-Prasad conjecture.

Abstract

Let $F$ be a Siegel cusp form of degree 2, even weight $k \geq 2$ and odd squarefree level $N$. We undertake a detailed study of the analytic properties of Fourier coefficients $a(F,S)$ of $F$ at fundamental matrices $S$ (i.e., with $-4 det(S)$ equal to a fundamental discriminant). We prove that as $S$ varies along the equivalence classes of fundamental matrices with $det(S) \asymp X$, the sequence $a(F,S)$ has at least $X^{1-\epsilon}$ sign changes, and takes at least $X^{1-\epsilon}$ "large values". Furthermore, assuming the Generalized Riemann Hypothesis as well as the refined Gan--Gross--Prasad conjecture, we prove the bound $|a(F,S)| \ll_{F, \epsilon} \frac{\det(S)^{\frac{k}2 - \frac{1}{2}}}{ (\log |\det(S)|)^{\frac18 - \epsilon}}$ for fundamental matrices $S$.