Regularized theta liftings and periods of modular functions

Abstract

Abstract In this paper, we use regularized theta liftings to construct weak Maass forms of weight 1/2 as lifts of weak Maass forms of weight 0. As a special case we give a new proof of some of recent results of Duke, Toth and the third author on cycle integrals of the modular j-invariant and extend these to any congruence subgroup. Moreover, our methods allow us to settle the open question of a geometric interpretation for periods of j along infinite geodesics in the upper half plane. In particular, we give the `central value' of the (non-existent) `L-function' for j. The key to the proofs is the construction of a kind of simultaneous Green function for both the CM points and the geodesic cycles, which is of independent interest.