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Regularized theta liftings and periods of modular functions

2013; De Gruyter; Volume: 2015; Issue: 703 Linguagem: Inglês

10.1515/crelle-2013-0035

1435-5345

Jan Hendrik Bruinier, Jens Funke, Özlem Imamoğlu,

Analytic Number Theory Research

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.