Pair correlation of angles between reciprocal geodesics on the modular surface

TL;DR

This paper proves the existence of a limiting pair correlation for angles between reciprocal geodesics on the modular surface, providing an explicit formula that links geometric and arithmetic data, and highlights a key difference from Euclidean cases.

Contribution

It establishes the limiting pair correlation for reciprocal geodesic angles on the modular surface and derives an explicit formula connecting geometry and arithmetic.

Findings

No gap beyond zero in the limiting distribution

Explicit formula captures geometric and arithmetic information

Contrasts with Euclidean analogs

Abstract

The existence of the limiting pair correlation for angles between reciprocal geodesics on the modular surface is established. An explicit formula is provided, which captures geometric information about the length of reciprocal geodesics, as well as arithmetic information about the associated reciprocal classes of binary quadratic forms. One striking feature is the absence of a gap beyond zero in the limiting distribution, contrasting with the analog Euclidean situation.