Quantum unique ergodicity for half-integral weight automorphic forms

2016/06/13 by Stephen Lester, Maksym Radziwiłł · 12 citations
Mathematics · #Advanced Algebra and Geometry #Advanced Mathematical Identities #Analytic Number Theory Research #Automorphic form #Cusp (singularity) #Cusp form #Ergodicity #Holomorphic function #Measure (data warehouse) #Quantum #Riemann hypothesis #math.DS #math.NT #math.SP

paper · pdf · doi:10.1215/00127094-2019-0040

published in Duke Mathematical Journal 169(2) (Duke University Press) · 61 pages

arxiv created 2016/06/13 · openalex created_date 2016/06/24 · openalex publication_date 2019/12/05 · arxiv updated 2020/02/12 · openalex updated_date 2026/08/05

Abstract

We investigate the analogue of the quantum unique ergodicity (QUE) conjecture for half-integral weight automorphic forms. Assuming the generalized Riemann hypothesis (GRH), we establish both QUE for half-integral weight Hecke Maaß cusp forms for Γ0(4) lying in Kohnen’s plus subspace and mass equidistribution for half-integral weight holomorphic Hecke cusp forms for Γ0(4) lying in Kohnen’s plus subspace. By combining the former result along with an argument of Rudnick, it follows that under GRH the zeros of these holomorphic Hecke cusp forms equidistribute with respect to hyperbolic measure on Γ0(4)\H as the weight tends to infinity.

Citations

• Heegner points, cycles and Maass forms
• Modular forms of half-integral weight on ?0(4)
• Distribution of Zeros of Random and Quantum Chaotic Sections of Positive Line Bundles
• Uniform distribution of eigenfunctions on compact hyperbolic surfaces
• Integral moments of L-functions
• Geometric invariants for real quadratic fields
• On Modular Forms of Half Integral Weight
• On Selberg's eigenvalue conjecture