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15 January 2015 Harmonic Maass forms of weight 1
W. Duke, Y. Li
Duke Math. J. 164(1): 39-113 (15 January 2015). DOI: 10.1215/00127094-2838436

Abstract

The object of this paper is to initiate a study of the Fourier coefficients of a weight 1 harmonic Maass form and relate them to the complex Galois representation associated to a weight 1 newform, which is the form’s image under a certain differential operator. In this paper, our focus will be on weight 1 dihedral newforms of prime level p3(mod4). In this case we give properties of the Fourier coefficients that are similar to (and sometimes reduce to) cases of Stark’s conjectures on derivatives of L-functions. We also give a new modular interpretation of certain products of differences of singular moduli studied by Gross and Zagier. Finally, we provide some numerical evidence that the Fourier coefficients of a mock-modular form whose shadow is exotic are similarly related to the associated complex Galois representation.

Citation

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W. Duke. Y. Li. "Harmonic Maass forms of weight 1." Duke Math. J. 164 (1) 39 - 113, 15 January 2015. https://doi.org/10.1215/00127094-2838436

Information

Published: 15 January 2015
First available in Project Euclid: 9 January 2015

zbMATH: 06408758
MathSciNet: MR3299102
Digital Object Identifier: 10.1215/00127094-2838436

Subjects:
Primary: 11Fxx
Secondary: 11SXX

Rights: Copyright © 2015 Duke University Press

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Vol.164 • No. 1 • 15 January 2015
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