  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 $\Gamma_0$.
