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

Let be an ‐normalized Siegel cusp form for of weight 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 where denotes the gcd of the entries of , and that its global sup‐norm satisfies the bound . 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.