Sup-norm bounds for Jacobi cusp forms

2024/04/21 by Anilatmaja Aryasomayajula, Aryasomayajula, Anilatmaja, Jürg Kramer +3
Mathematics · #Advanced Algebra and Geometry #Algebraic Geometry and Number Theory

paper · pdf · doi:10.48550/arxiv.2404.13625

Abstract

In this article, we give L^∞-norm bounds for the natural invariant norm of cusp forms of real weight k and character χ for any cofinite Fuchsian subgroup Γ\subsetSL₂(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 L^∞-norm bounds for the natural invariant norm of these Jacobi cusp forms. More specifically, letting J_k,m^cusp(Γ₀) denote the complex vector space of Jacobi cusp forms under consideration and \Vert⋅\Vert_Pet the pointwise Petersson norm on J_k,m^cusp(Γ_ 0), we prove that for k∈ℤ_{≥ 5} and m∈ℤ_{≥ 1}, and a given ε>0, the L^∞-norm bound \Vertϕ\Vert_L^∞=sup_{(τ,z)∈ℍ×ℂ}\Vertϕ(τ,z)\Vert_Pet=O_Γ₀,ε(k m^\frac 74+ε) holds for any ϕ∈ J_k,m^cusp(Γ₀), which is L²-normalized with respect to the Petersson inner product, where the implied constant depends on Γ₀ and the choice of ε>0.

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