1202.2603 (Yasufumi Hashimoto)

Correlations of multiplicities in length spectra for congruence
subgroups    [PDF]

Yasufumi Hashimoto

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.

View original: http://arxiv.org/abs/1202.2603