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

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.