Bounds for the Bergman kernel and the sup-norm of holomorphic Siegel cusp forms

TL;DR

This paper establishes polynomial bounds on the Bergman kernel for holomorphic Siegel cusp forms and derives new sup-norm bounds for these forms, improving understanding of their size and growth in relation to the weight and degree.

Contribution

It provides the first polynomial bounds on the Bergman kernel for Siegel cusp forms of arbitrary degree and improves sup-norm bounds for degree 2 forms beyond generic estimates.

Findings

Bounds match conjectural estimates for degrees 1 and 2.

Sup-norm of degree 2 forms is bounded by O(k^{9/4+ε}).

Sup-norm in compact sets is O(k^{3/2 - η}) for some η>0.

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 on the aforementioned size, while the lower bounds match for all $n \ge 1$. For an $L^2$-normalised 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 $\mathrm{Sp}(2, \mathbb Z)$ 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.