Regularized theta liftings and periods of modular functions

By Jan H. Bruinier, Jens Funke, Ö. Imamoḡlu. 2011

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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.

Subfields: Algebra and Number Theory, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/9e84994a11f5283919181a887b1be87564d8e348