$L^\infty$-sizes of the spaces Siegel cusp forms of degree $n$ via Poincar\'e series

Abstract

We prove the conjectures on the ($L^{\infty}$)-sizes of the spaces of Siegel cusp forms of degree $n$, weight $k$, for any congruence subgroup in the weight aspect as well as for all principal congruence subgroups in the level aspect, in particular. This size is measured by the size of the Bergman kernel of the space. More precisely we show that the aforementioned size is $\asymp_{n} k^{3n(n+1)/4}$. Our method uses the Fourier expansion of the Bergman kernel, and has wide applicability. We illustrate this by a simple algorithm. We also include some of the applications of our method, including individual sup-norms of small weights and non-vanishing of Poincar\'e series.