Divergent Geodesics in Hyperbolic Manifolds and Arithmetic Applications

We give an asymptotic formula for the number of common perpendiculars with length tending to infinity between two divergent geodesics in finite volume real hyperbolic manifolds, presenting a surprising non-purely exponential growth. We apply this result to count ambiguous geodesics in the modular curve, recovering results of Sarnak, and to prove a conjecture of Motohashi on the binary additive divisor problem in imaginary quadratic number fields. This is joint work with Jouni Parkkonen.