Sup-norm bounds for Jacobi cusp forms

TL;DR

This paper establishes bounds on the maximum size of Jacobi cusp forms of real weight for cofinite Fuchsian groups, extending understanding of their growth and providing explicit estimates depending on weight and index.

Contribution

The paper derives new $L^{ abla}$-norm bounds for Jacobi cusp forms of integral weight and index, using their representation as linear combinations of modular forms and theta functions.

Findings

Proves $L^{ abla}$-norm bounds of order $k m^{7/4+ ext{epsilon}}$ for Jacobi cusp forms.

Provides explicit bounds depending on weight $k$, index $m$, and group $ ext{Gamma}_0$.

Extends previous bounds to a broader class of Fuchsian groups and 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 $\epsilon>0$.