  In this paper, we use regularized theta liftings to construct weak Maass
forms 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 Imamoglu 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-existing)
`L-function' for j. The key to the proofs is the construction of some kind of a
simultaneous Green function for both the CM points and the geodesic cycles,
which is of independent interest.
