Uniform sup-norm bounds on average for Siegel cusp forms

TL;DR

This paper establishes uniform upper bounds for the sup-norm of sums of Siegel cusp forms over arithmetic groups, using heat kernel methods, with bounds depending on the weight and the nature of the quotient space.

Contribution

It provides the first uniform sup-norm bounds for Siegel cusp forms that depend only on the degree and a fixed base group, using heat kernel techniques.

Findings

Sup-norm bound for compact quotients: $c_{n, ext{group}}\, ext{const} imes ext{weight}^{n(n+1)/2}$.

Sup-norm bound for non-compact finite volume quotients: $c_{n, ext{group}} ext{const} imes ext{weight}^{3n(n+1)/4}$.

Bounds are uniform across subgroups of a fixed arithmetic group.

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 $\Gamma_0$.