The variance of arithmetic measures associated to closed geodesics on the modular surface

2008/10/18 by Wenzhi Luo, Zeév Rudnick, Luo, Wenzhi +3 · 1 citation
Mathematics · #11F72 #37D40 #Advanced Algebra and Geometry #Analytic Number Theory Research #Dynamical Systems (math.DS) #FOS: Mathematics #Mathematical Dynamics and Fractals #Number Theory (math.NT)

paper · pdf · doi:10.48550/arxiv.0810.3331

openalex publication_date 2008/10/18 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We determine the variance for the fluctuations of the arithmetic measures obtained by collecting all closed geodesics on the modular surface with the same discriminant and ordering them by the latter. This arithmetic variance differs by subtle factors from the variance that one gets when considering individual closed geodesics when ordered by their length. The arithmetic variance is the same one that appears in the fluctuations of measures associated with quantum states on the modular surface.

Citations

• Heegner points, cycles and Maass forms
• On Modular Forms of Half Integral Weight

Cited by

• Quantum variance for Eisenstein Series

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