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

27 pages; to appear in JLMS