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

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.