Hyperbolic lattice-point counting and modular symbols

TL;DR

This paper investigates the distribution of modular symbols in hyperbolic lattice counting, demonstrating that their normalized values follow a Gaussian distribution under certain restrictions.

Contribution

It establishes the asymptotic Gaussian distribution of normalized modular symbols in hyperbolic lattice counting for cocompact groups.

Findings

Normalized modular symbols are Gaussian distributed

Asymptotic behavior of lattice counting with restrictions

Extension of classical lattice counting results

Abstract

For a cocompact group $\G$ of $\slr$ we fix a real non-zero harmonic 1-form $\alpha$. We study the asymptotics of the hyperbolic lattice-counting problem for $\G$ under restrictions imposed by the modular symbols $\modsym{\gamma}{\a}$. We prove that the normalized values of the modular symbols, when ordered according to this counting, have a Gaussian distribution.