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 ${\mathrm{Sp}}_4\left(Z\right)$ 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 $\left|a(F,S)\right|{\ll }_{\epsilon }\frac{k^{1/4+\epsilon }{\left(4\pi \right)}^k}{\Gamma \left(k\right)}c{\left(S\right)}^{-\frac{1}{2}}det{\left(S\right)}^{\frac{k-1}{2}+\epsilon }$ where $c\left(S\right)$ denotes the gcd of the entries of $S$, and that its global sup-norm satisfies the bound ${\parallel (detY)}^{\frac{k}{2}}{F\parallel }_{\infty }{\ll }_{\epsilon }{k}^{\frac{5}{4}+\epsilon }$. 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.