Intersections of special cycles on Shimura curves and Siegel Maass forms

Abstract

We show that the generating series of the number of pairs of geodesics on a compact Shimura curve with given discriminants and intersection angle are coefficients of a non-holomorphic Siegel modular form, a theta lift of the constant function. This retrieves and generalizes counting results of Rickards via the Siegel-Weil formula. More generally, we study the genus two theta lift of Maass forms on this Shimura curve and prove a Fourier-Taylor expansion in terms of some generalized Whittaker functions. We also provide a geometric interpretation of all Fourier coefficients of these theta lifts in terms of averages of geodesic Taylor coefficients over special cycles.