Intersections of geodesics on Shimura curves

Rickards recently noted that a certain generating series of intersection numbers of geodesics on a compact Shimura curve defines a weight 2 elliptic modular form.
This turns out to be a diagonal restriction of a very specific Hilbert modular form. I will explain how this is a special case of a more general story with ties to the
work of Gross and Zagier on differences of singular moduli, geometric theta lifts, and recent work of Darmon-Pozzi-Vonk.