$L^{\infty}$-norm bounds for Siegel--Jacobi cusp forms

Abstract

In this article, we establish explicit and uniform $L^{\infty}$-norm bounds for $L^{2}$-normalized Siegel--Jacobi cusp forms of integral weight $k$ and index $m$ for the Siegel modular group $\Gamma_{0}=\mathrm{Sp}_{2g}(\mathbb{Z})$ for arbitrary genus $g\geq 1$. Using the generalization of the classical Eichler--Zagier theta decomposition to higher genus, any such Siegel--Jacobi cusp form can be written as a finite linear combination of Siegel cusp forms of half-integral weight $k-1/2$ multiplied by the higher-dimensional analogues of the classical Jacobi theta functions. By building upon the uniform $L^{\infty} $-norm bounds on average for Siegel cusp forms established by J.~Kramer and A.~Mandal~\cite{k1} via the associated Bergman kernels, we prove that for $k\in\mathbb{Z}_{\geq g+1}$, $m\in\mathbb{Z}_{\geq 1}$, and a given $\epsilon>0$, the $L^{\infty}$-norm bound \begin{equation*} \Vert\phi\Vert_{L^{\infty}}=\sup_{(\tau,z)\in\mathbb{H}_{g}\times\mathbb{C}^{g}}\Vert\phi(\tau,z)\Vert_{\mathrm{Pet}}=O_ {\Gamma_{0},\epsilon}\big(k^{(3g^{2}+5g)/8}\,m^{g^{2}+5g/4+\epsilon}\big) \end{equation*} holds for any Siegel--Jacobi cusp form $\phi$ that is $L^{2}$-normalized with respect to the Petersson inner product. These estimates provide the first explicit upper bounds in terms of both parameters $k$ and $m$ for arbitrary genus $g$.