Sup-norm bounds for Jacobi cusp forms

In this article, we give bounds for the natural invariant norm of cusp forms of real weight k and character χ for any cofinite Fuchsian subgroup Γ₂(R). Using the representation of Jacobi cusp forms of integral weight k and index m for the modular group Γ₀=SL₂(Z) as linear combinations of modular forms of weight k-1/2 for some congruence subgroup of Γ₀ (depending on m) and suitable Jacobi theta functions, we derive bounds for the natural invariant norm of these Jacobi cusp forms. More specifically, letting Jk,mᶜᵘˢᵖ(Γ₀) denote the complex vector space of Jacobi cusp forms under consideration and ·Pet the pointwise Petersson norm on Jk,mᶜᵘˢᵖ(Γ₀), we prove that for given ε>0, the bound {align*} (τ,z) f(τ,z)Pet=Oε(k3/4m3/2+ε) {align*} holds for any f∈ Jk,mᶜᵘˢᵖ(Γ₀), which is normalized with respect to the Petersson inner product, where the implied constant depends only on the choice of ε>0.