The sup-norm problem on the Siegel modular space of rank two

Abstract

Let $F$ be a square integrable Maa{\ss} form on the Siegel upper half space ${\mathcal H}$ of rank $2$ for the Siegel modular group ${\rm Sp}_4({\Bbb Z})$ with Laplace eigenvalue $\lambda$. If, in addition, $F$ is a joint eigenfunction of the Hecke algebra and $\Omega$ is a compact set in ${\rm Sp}_4({\Bbb Z})\backslash{\mathcal H}$, we show the bound $\|F|_{\Omega}\|_{\infty} \ll_{\Omega} (1+\lambda)^{1-\delta}$ for some global constant $\delta0$. As an auxiliary result of independent interest we prove new uniform bounds for spherical functions on semisimple Lie groups.