Bounds for the Bergman Kernel and the Sup-Norm of Holomorphic Siegel Cusp Forms

Abstract

We prove “polynomial in $k$” bounds on the size of the Bergman kernel for the space of holomorphic Siegel cusp forms of degree $n$ and weight $k$. When $n=1,2$ our bounds agree with the conjectural bounds, while the lower bounds match for all $n \ge 1$. For an $L^{2}$-normalized Siegel cusp form $F$ of degree $2$, our bound for its sup-norm is $O_{\epsilon } (k^{9/4+\epsilon })$. Further, we show that in any compact set $\Omega $ (which does not depend on $k$) contained in the Siegel fundamental domain of $\textrm {Sp}(2,\mathbb {Z})$ on the Siegel upper half space, the sup-norm of $F$ is $O_{\Omega }(k^{3/2 - \eta })$ for some $\eta>0$, going beyond the “generic” bound in this setting.