Intersections of special cycles on Shimura curves and Siegel Maass forms

TL;DR

This paper establishes a connection between intersection counts of geodesics on Shimura curves and Siegel modular forms, generalizing previous counting results through advanced theta lift techniques.

Contribution

It introduces a novel geometric interpretation of Fourier coefficients of Siegel Maass forms via intersections on Shimura curves and extends counting formulas using the Siegel-Weil formula.

Findings

Generating series of geodesic intersections form non-holomorphic Siegel modular forms.

Fourier-Taylor expansion of genus two theta lifts expressed with generalized Whittaker functions.

Geometric interpretation of Fourier coefficients as averages over special cycles.

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.