[Paper Review] The variance of arithmetic measures associated to closed geodesics on the modular surface

Wenzhi Luo, Zeév Rudnick+1 more|ArXiv.org|Oct 18, 2008

Analytic Number Theory Research24 references3 citations

TL;DR

This paper computes the variance of arithmetic measures associated with closed geodesics on the modular surface, grouped by discriminant, using Rankin-Selberg theory and theta lifts. It establishes that the variance is proportional to the standard $L$-function value $L(1/2, f)$ and the symmetric square $L$-function, with explicit constants depending on the form type (holomorphic or Maass).

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.

Motivation & Objective

• To understand the statistical fluctuations of arithmetic measures associated with closed geodesics on the modular surface, grouped by discriminant.
• To compute the variance of these measures in the limit of increasing discriminant, focusing on equidistribution and correlation structure.
• To establish a precise link between the variance and special values of $L$-functions via automorphic forms and theta lifts.
• To resolve the variance for both holomorphic and Maass forms, distinguishing cases based on symmetry and weight.

Proposed method

• Uses the correspondence between closed geodesics and binary quadratic forms to define arithmetic measures $\mu_d$ associated with discriminant $d$.
• Applies theta lifts to relate $\mu_d(f)$ to Fourier coefficients of half-integral weight forms, enabling use of Rankin-Selberg theory.
• Employs the Mellin transform and spectral decomposition to analyze the $L^2$-norm of $\mu_d(f)$, leading to Dirichlet series with poles at $s=1$.
• Computes the residue of the Dirichlet series using Rankin-Selberg theory, linking it to $L(1/2, f)$ and the symmetric square $L$-function.
• Uses representation theory of $\mathrm{SL}_2(\mathbb{R})$ and $\mathbf{r}$-invariance to decompose the space and ensure orthogonality across distinct cusp forms.
• Applies known formulas for Whittaker functions and Barnes integrals to evaluate special values of $\Gamma$-functions in the residue computation.

Experimental results

Research questions

• RQ1 What is the asymptotic variance of the arithmetic measure $\mu_d$ associated with closed geodesics of discriminant $d$?
• RQ2 How does the variance depend on the spectral type (holomorphic vs. Maass) and weight of the automorphic form?
• RQ3 Can the variance be expressed in terms of special values of $L$-functions, and if so, which ones?
• RQ4 What role does the theta lift play in connecting geodesic measures to Fourier coefficients of half-integral weight forms?
• RQ5 Why do the variance formulas differ by a factor of $6/\pi$ for Maass forms and $1/\pi$ for holomorphic forms?

Key findings

• For holomorphic cusp forms of even weight $2k$, the variance is $\frac{1}{\pi} V^{\mathrm{sym}}(f,f) L(1/2, f) \langle f, f \rangle$, with $V^{\mathrm{sym}}$ the symmetric square $L$-function.
• For even Maass forms, the variance is $\frac{6}{\pi} V^{\mathrm{sym}}(\phi,\phi) L(1/2, \phi) \langle \phi, \phi \rangle$, with the same $V^{\mathrm{sym}}$ structure.
• The residue of the Dirichlet series $\sum_d |\mu_d(f)|^2 / d^{1/2}$ is shown to be proportional to $L(1/2, f)$ and the symmetric square $L$-function.
• The variance vanishes for holomorphic forms of weight $\equiv 2 \mod 4$ and for odd Maass forms, due to symmetry and $\mathbf{r}$-invariance.
• Orthogonality of distinct cusp forms in the sum is established via Rankin-Selberg theory, ensuring no cross-terms in the variance.
• The explicit constants $1/\pi$ and $6/\pi$ arise from $\Gamma$-function identities and normalization of the theta lift.