Regularized theta liftings and periods of modular functions

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 -invariant and extend these to a...

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Bibliographic Details
Published inJournal für die reine und angewandte Mathematik Vol. 2015; no. 703; pp. 43 - 93
Main Authors Bruinier, Jan H., Funke, Jens, Imamoḡlu, Özlem
Format Journal Article
LanguageEnglish
Published De Gruyter 01.06.2015
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Summary: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 -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 along infinite geodesics in the upper half plane. In particular, we give the `central value' of the (non-existent) ` -function' for . 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.
ISSN:0075-4102
1435-5345
DOI:10.1515/crelle-2013-0035