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

TL;DR

This paper establishes precise asymptotic sizes of spaces of Siegel cusp forms of degree n using Bergman kernels, providing new insights into their structure and applications in number theory.

Contribution

It proves conjectures on the sizes of Siegel cusp form spaces in both weight and level aspects, introducing a method based on Fourier expansion of the Bergman kernel.

Findings

Size of Siegel cusp form spaces is asymptotically proportional to k^{3n(n+1)/4}

Method applies to various applications including sup-norms and Poincaré series non-vanishing

Provides an algorithm for practical computation of these sizes

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.