Duke Mathematical Journal

Harmonic Maass forms of weight 1

W. Duke and Y. Li

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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.

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Duke Math. J., Volume 164, Number 1 (2015), 39-113.

First available in Project Euclid: 9 January 2015

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Zentralblatt MATH identifier

Primary: 11Fxx: Discontinuous groups and automorphic forms [See also 11R39, 11S37, 14Gxx, 14Kxx, 22E50, 22E55, 30F35, 32Nxx] {For relations with quadratic forms, see 11E45}
Secondary: 11Sxx: Algebraic number theory: local and $p$-adic fields

harmonic modular forms weight 1 mock-modular Galois representations Maass forms Stark’s conjectures


Duke, W.; Li, Y. Harmonic Maass forms of weight $1$. Duke Math. J. 164 (2015), no. 1, 39--113. doi:10.1215/00127094-2838436. https://projecteuclid.org/euclid.dmj/1420813014

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