Uniform sup-norm bounds on average for Siegel cusp forms

2023/10/09 by Jürg Kramer, Kramer, Jürg, Antareep Mandal +1 · 1 citation
Mathematics · #Advanced Algebra and Geometry #Algebraic Geometry and Number Theory #Analytic Number Theory Research

paper · pdf · doi:10.48550/arxiv.2310.05334

Abstract

Let Γ\subsetneq Sp_n(ℝ) be an arithmetic subgroup of the symplectic group Sp_n(ℝ) acting on the Siegel upper half-space ℍ_n of degree n. Consider the d-dimensional space of Siegel cusp forms S_κⁿ(Γ) of weight κ for Γ and let \f_j\_{1≤ j≤ d} be a basis of S_κⁿ(Γ) orthonormal with respect to the Petersson inner product. In this paper we show using the heat kernel method that the sup-norm of the quantity S_κ^Γ(Z):=∑_j=1^ddet (Y)^κ\vertf_j(Z)\vert² (Z∈ℍ_n) is bounded above by c_n,Γ κ^n(n+1)/2 when M:=Γ\backslashℍ_n is compact and by c_n,Γ κ^3n(n+1)/4 when M is non-compact of finite volume, where c_n,Γ denotes a positive real constant depending only on the degree n and the group Γ. Furthermore, we show that this bound is uniform in the sense that if we fix a group Γ₀ and take Γ to be a subgroup of Γ₀ of finite index, then the constant c_n,Γ in these bounds depends only on the degree n and the fixed group Γ₀.

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