Sup-norm bounds for Jacobi cusp forms

Abstract

In this article, we give $L^{\infty}$-norm bounds for the natural invariant norm of cusp forms of real weight $k$ and character $\chi$ for any cofinite Fuchsian subgroup $\Gamma\subset\mathrm{SL}_{2}(\mathbb{R})$. Using the representation of Jacobi cusp forms of integral weight $k$ and index $m$ for the modular group $\Gamma_{0}=\mathrm{SL}_{2}(\mathbb{Z})$ as linear combinations of modular forms of weight $k-\frac{1}{2}$ for some congruence subgroup of $\Gamma_{0}$ (depending on $m$) and suitable Jacobi theta functions, we derive $L^{\infty}$-norm bounds for the natural invariant norm of these Jacobi cusp forms. More specifically, letting $J_{k,m}^{\mathrm{cusp}}(\Gamma_{0})$ denote the complex vector space of Jacobi cusp forms under consideration and $\Vert\cdot\Vert_{\mathrm{Pet}}$ the pointwise Petersson norm on $J_{k,m}^{\mathrm{cusp}}(\Gamma_ {0})$, we prove that for $k\in\mathbb{Z}_{\ge 5}$ and $m\in\mathbb{Z}_{\ge 1}$, and a given $\epsilon>0$, the $L^{\infty}$-norm bound \begin{align*} \Vert\phi\Vert_{L^{\infty}}=\sup_{(\tau,z)\in\mathbb{H}\times\mathbb{C}}\Vert\phi(\tau,z)\Vert_{\mathrm{Pet}}=O_{\Gamma_{0},\epsilon}\big(k\,m^{\frac {7}{4}+\epsilon}\big) \end{align*} holds for any $\phi\in J_{k,m}^{\mathrm{cusp}}(\Gamma_{0})$, which is $L^{2}$-normalized with respect to the Petersson inner product, where the implied constant depends on $\Gamma_{0}$ and the choice of $\epsilon>0$.