Sup-norm bounds for Jacobi cusp forms

TL;DR

This work establishes sharp sup-norm bounds for Jacobi cusp forms and related Bergman-kernel norms, extending known results from integer-weight cusp forms to real weights and general characters. By expressing Jacobi cusp forms as linear combinations of half-integral weight modular forms via the Eichler–Zagier decomposition and Jacobi theta functions, the authors transfer $L^{\infty}$-norm control to the Jacobi setting, obtaining $\Vert\phi\Vert_{\infty}=O_{Γ_0,\varepsilon}(k\,m^{7/4+\varepsilon})$ for $k\ge5$, $m\ge1$ and $L^{2}$-normalized $φ$. They also prove $L^{\infty}$-norm bounds for the Bergman kernel of cusp forms $B_{k,χ}$ on cocompact and cofinite Fuchsian groups, showing $\Vert B_{k,χ}\Vert_{L^{\infty}}=O_{Γ}(k)$ or $O_{Γ}(k^{3/2})$ respectively, and establish uniformity in the subgroup $Γ$. The results recover and strengthen average bounds from prior work while improving the dependence on the index $m$ for Jacobi forms, and they illuminate the interaction between half-integral weight forms and Jacobi theta components in sup-norm problems. These findings have implications for quantitative analysis of Jacobi forms and related automorphic objects across real weights and general arithmetic groups.

Abstract

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

Theorems & Definitions (20)

• Proposition 3.1
• proof
• Proposition 3.2
• proof
• Theorem 3.3
• proof
• Corollary 3.4
• proof
• Theorem 3.5
• proof