Regularized theta liftings and periods of modular functions

TL;DR

This paper employs regularized theta liftings to construct weak Maass forms and provides new insights into the periods of modular functions, including a geometric interpretation of the j-invariant's periods along geodesics.

Contribution

It introduces a novel method using regularized theta liftings to study Maass forms and offers a geometric interpretation for periods of the modular j-invariant.

Findings

Constructed weak Maass forms of weight 1/2 from weight 0 forms.

Provided a new proof of results on cycle integrals of the j-invariant.

Established a geometric interpretation for periods of j along geodesics.

Abstract

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.