Bounds on Fourier coefficients and global sup-norms for Siegel cusp forms of degree 2

TL;DR

This paper establishes bounds on Fourier coefficients and the global sup-norm for certain Siegel cusp forms of degree 2, assuming GRH, advancing understanding of their size and distribution.

Contribution

It provides new bounds on Fourier coefficients and sup-norms of Siegel cusp forms, leveraging recent results on Bessel periods and the refined Gan-Gross-Prasad conjecture.

Findings

Fourier coefficients satisfy a specific upper bound under GRH.

Global sup-norm of the form is bounded by a power of the weight k.

New bounds for local integrals in the Gan-Gross-Prasad framework.

Abstract

Let $F$ be an $L^2$-normalized Siegel cusp form for $\mathrm{Sp}_4(\mathbb{Z})$ of weight $k$ that is a Hecke eigenform and not a Saito--Kurokawa lift. Assuming the Generalized Riemann Hypothesis, we prove that its Fourier coefficients satisfy the bound $|a(F,S)| \ll_\epsilon \frac{k^{1/4+\epsilon} (4\pi)^k}{\Gamma(k)} c(S)^{-\frac12} \det(S)^{\frac{k-1}2+\epsilon}$ where $c(S)$ denotes the gcd of the entries of $S$, and that its global sup-norm satisfies the bound $\|(\det Y)^{\frac{k}2}F\|_\infty \ll_\epsilon k^{\frac54+\epsilon}.$ The former result depends on new bounds that we establish for the relevant local integrals appearing in the refined global Gan-Gross-Prasad conjecture (which is now a theorem due to Furusawa and Morimoto) for Bessel periods.