Shimura lifting on weak Maass forms

Youngju Choie¹ ; Subong Lim²

¹ Department of Mathematics and PMI Pohang University of Science and Technology Pohang, 790-784 Republic of Korea
² Department of Mathematics Education Sungkyunkwan University 25-2, Sungkyunkwan-ro Jongno-gu, Seoul, 03063 Republic of Korea

Acta Arithmetica, Tome 173 (2016) no. 1, pp. 1-18

Résumé

There is a Shimura lifting which sends cusp forms of a half-integral weight to holomorphic modular forms of an even integral weight. Niwa and Cipra studied this lifting using the theta series attached to an indefinite quadratic form; later, Borcherds and Bruinier extended this lifting to weakly holomorphic modular forms and harmonic weak Maass forms of weight ${1/2}$, respectively. We apply Niwa’s theta kernel to weak Maass forms by using a regularized integral. We show that the lifted function satisfies modular transformation properties and is an eigenfunction of the Laplace operator. In particular, this lifting preserves the property of being harmonic. Furthermore, we determine the location of singularities of the lifted function and describe its singularity type.

Détail

DOI : 10.4064/aa7916-12-2015

Keywords: there shimura lifting which sends cusp forms half integral weight holomorphic modular forms even integral weight niwa cipra studied lifting using theta series attached indefinite quadratic form later borcherds bruinier extended lifting weakly holomorphic modular forms harmonic weak maass forms weight respectively apply niwa theta kernel weak maass forms using regularized integral lifted function satisfies modular transformation properties eigenfunction laplace operator particular lifting preserves property being harmonic furthermore determine location singularities lifted function describe its singularity type