Distribution of the successive minima of the Petersson norm on cusp forms

2023/09/26 by Souparna Purohit, Purohit, Souparna
Mathematics · #Advanced Algebra and Geometry #Algebraic Geometry (math.AG) #Algebraic Geometry and Number Theory #Analytic Number Theory Research #FOS: Mathematics #Number Theory (math.NT)

paper · pdf · doi:10.48550/arxiv.2309.15255

openalex publication_date 2023/09/26 · openalex created_date 2023/09/30 · openalex updated_date 2026/07/28

Abstract

Let Γ⊆ PSL₂(ℤ) be a finite index subgroup. Let \mathscrX(Γ) be a regular proper model of the modular curve associated with Γ, and let \mathscrL^{⊗ k} be the logarithmically singular metrized line bundle on \mathscrX(Γ) associated to modular forms of level Γ and weight 12k, endowed with the Petersson metric. For each k ≥ 1, the sub-lattice \mathscrS_k ⊆ H⁰(\mathscrX(Γ), \mathscrL^{⊗ k}) of integral cusp forms of level Γ and weight 12k is a euclidean lattice with respect to the Petersson norm. In this paper, we describe the distribution of the successive minima of the \mathscrS_k as k → ∞, generalizing the work of Chinburg, Guignard, and Soulé which addressed the case Γ= PSL₂(ℤ).

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