Regularized theta liftings and periods of modular functions

Bruinier, Jan; Funke, Jens; Imamoglu, Ozlem

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.

Citation

Bruinier, J., Funke, J., & Imamoglu, O. (2015). Regularized theta liftings and periods of modular functions. Journal für die reine und angewandte Mathematik, 2015(703), 43-93. https://doi.org/10.1515/crelle-2013-0035

• Journal Article Type: Article
• Acceptance Date: Mar 13, 2013
• Online Publication Date: Jun 18, 2013
• Publication Date: Jun 1, 2015
• Deposit Date: Mar 19, 2012
• Publicly Available Date: May 7, 2014
• Journal: Journal für die reine und angewandte Mathematik
• Print ISSN: 0075-4102
• Electronic ISSN: 1435-5345
• Publisher: De Gruyter
• Peer Reviewed: Peer Reviewed
• Volume: 2015
• Issue: 703
• Pages: 43-93
• DOI: https://doi.org/10.1515/crelle-2013-0035
• Public URL: https://durham-repository.worktribe.com/output/1509419