Correlations of multiplicities in length spectra for congruence subgroups

TL;DR

This paper extends asymptotic formulas for correlations of length spectrum multiplicities from the modular surface to all congruence subgroups, revealing deeper connections with eigenvalue statistics and prime number Euler products.

Contribution

It generalizes previous asymptotic formulas for length spectrum multiplicity correlations to all congruence subgroups of the modular group.

Findings

Extended asymptotic formulas to higher level correlations

Connected length spectrum correlations with eigenvalue statistics

Described coefficients via Euler products over primes

Abstract

Bogomolny-Leyvraz-Schmit (1996) and Peter (2002) proposed an asymptotic formula for the correlation of the multiplicities in length spectrum on the modular surface, and Lukianov (2007) extended its asymptotic formula to the Riemann surfaces derived from the congruence subgroup $\Gam_0(n)$ and the quaternion type co-compact arithmetic groups. The coefficients of the leading terms in these asymptotic formulas are described in terms of Euler products over prime numbers, and they appear in eigenvalue statistic formulas found by Rudnick (2005) and Lukianov (2007) for the Laplace-Beltrami operators on the corresponding Riemann surfaces. In the present paper, we further extend their asymptotic formulas to the higher level correlations of the multiplicities for any congruence subgroup of the modular group.