Convergence to the Plancherel measure of Hecke Eigenvalues

2022/01/10 by Peter Sarnak, Sarnak, Peter, Nina Zubrilina +1
Mathematics · #11F11 #11F25 #Advanced Algebra and Geometry #Analytic Number Theory Research #Bounding overwatch #Convergence (economics) #Eigenvalues and eigenvectors #FOS: Mathematics #Graph theory and applications #Hecke operator #Holomorphic function #Mathematical analysis #Mathematics #Measure (data warehouse) #Modular form #Number Theory (math.NT) #Probability (math.PR) #Pure mathematics #Ramanujan's sum #math.NT #math.PR #msc:11F11 #msc:11F25

paper · pdf · doi:10.48550/arxiv.2201.03523

17 pages

arxiv created 2022/01/10 · openalex publication_date 2022/01/10 · arxiv updated 2022/01/11 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/04

Abstract

We give improved uniform estimates for the rate of convergence to Plancherel measure of Hecke eigenvalues of holomorphic forms of weight 2 and level N. These are applied to determine the sharp cutoff for the non-backtracking random walk on arithmetic Ramanujan graphs and to Serre's problem of bounding the multiplicities of modular forms whose coefficients lie in number fields of degree d.

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