Divergent geodesics, ambiguous closed geodesics and the binary additive divisor problem

TL;DR

This paper derives an asymptotic formula for counting common perpendiculars between divergent geodesics in negatively curved spaces, with applications to ambiguous geodesics and the binary additive divisor problem in number theory.

Contribution

It introduces a new asymptotic counting method for geodesic intersections and applies it to solve problems in modular orbifolds and number theory, extending previous results.

Findings

Asymptotic formula with non-purely exponential growth for common perpendiculars

Counting of ambiguous geodesics in the modular orbifold

Extension of Motohashi's conjecture on the binary additive divisor problem

Abstract

We give an asymptotic formula as $t\to+\infty$ for the number of common perpendiculars of length at most $t$ between two divergent geodesics or a divergent geodesic and a compact locally convex subset in negatively curved locally symmetric spaces with exponentially mixing geodesic flow, presenting a surprising non-purely exponential growth. We apply this result to count ambiguous geodesics in the modular orbifold recovering results of Sarnak, and to confirm and extend a conjecture of Motohashi on the binary additive divisor problem in imaginary quadratic number fields.