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

Let $F$ be an $L^2$-normalized Siegel cusp form for ${\mathrm{S}\mathrm{p}}_4(\mathbb{Z})$ 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 $\mathrm{\mid }a(F,S)\mathrm{\mid }{\ll }_{\epsilon }\frac{k^{1\mathrm{/}4+\epsilon }(4\pi {)}^k}{\mathrm{\Gamma }(k)}c(S{)}^{-\frac{1}{2}}\det (S{)}^{\frac{k-1}{2}+\epsilon }$ where $c(S)$ denotes the gcd of the entries of $S$, and that its global sup-norm satisfies the bound $\mathrm{\parallel }(\det Y{)}^{\frac{k}{2}}F{\mathrm{\parallel }}_{\mathrm{\infty }}{\ll }_{\epsilon }{k}^{\frac{5}{4}+\epsilon }\mathrm{.}$ 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.