Mathematical Research Letters

Volume 20 (2013)

Number 2

Theta lifts and local Maass forms

Pages: 213 – 234

DOI: http://dx.doi.org/10.4310/MRL.2013.v20.n2.a2

Authors

Kathrin Bringmann (Mathematical Institute, University of Cologne, Germany)

Ben Kane (Mathematical Institute, University of Cologne, Germany)

Maryna Viazovska (Mathematical Institute, University of Cologne, Germany)

Abstract

The first two authors and Kohnen have recently introduced a new class of modular objects called locally harmonic Maass forms, which are annihilated almost everywhere by the hyperbolic Laplacian operator. In this paper, we realize these locally harmonic Maass forms as theta lifts of harmonic weak Maass forms. Using the theory of theta lifts, we then construct examples of (non-harmonic) local Maass forms, which are instead eigenfunctions of the hyperbolic Laplacian almost everywhere.

Keywords

theta lifts, harmonic weak Maass forms, locally harmonic Maass forms, local Maass forms, modular forms

2010 Mathematics Subject Classification

11E16, 11F11, 11F25, 11F27, 11F37

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