Uniform sup-norm bounds on average for Siegel cusp forms

Abstract

Let $\Gamma\subsetneq \mathrm{Sp}_n(\mathbb{R})$ be an arithmetic subgroup of the symplectic group $\mathrm{Sp}_n(\mathbb{R})$ acting on the Siegel upper half-space $\mathbb{H}_n$ of degree $n$. Consider the $d$-dimensional space of Siegel cusp forms $\mathcal{S}_{\kappa}^n(\Gamma)$ of weight $\kappa$ for $\Gamma$ and let $\{f_j\}_{1\leq j\leq d}$ be a basis of $\mathcal{S}_{\kappa}^n(\Gamma)$ orthonormal with respect to the Petersson inner product. In this paper we show using the heat kernel method that the sup-norm of the quantity $S_{\kappa}^{\Gamma}(Z):=\sum_{j=1}^{d}\det (Y)^{\kappa}\vert{f_j(Z)}\vert^2\,(Z\in\mathbb{H}_n)$ is bounded above by $c_{n,\Gamma} {\kappa}^{n(n+1)/2}$ when $M:=\Gamma\backslash\mathbb{H}_n$ is compact and by $c_{n,\Gamma} {\kappa}^{3n(n+1)/4}$ when $M$ is non-compact of finite volume, where $c_{n,\Gamma}$ denotes a positive real constant depending only on the degree $n$ and the group $\Gamma$. Furthermore, we show that this bound is uniform in the sense that if we fix a group $\Gamma_0$ and take $\Gamma$ to be a subgroup of $\Gamma_0$ of finite index, then the constant $c_{n,\Gamma}$ in these bounds depends only on the degree $n$ and the fixed group $\Gamma_0$.