## Duke Mathematical Journal

### Harmonic Maass forms of weight $1$

W. Duke and Y. Li

#### 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 $p\equiv3(\operatorname{mod}{4})$. 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.

#### Article information

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

Dates
First available in Project Euclid: 9 January 2015

Permanent link to this document
https://projecteuclid.org/euclid.dmj/1420813014

Digital Object Identifier
doi:10.1215/00127094-2838436

Mathematical Reviews number (MathSciNet)
MR3299102

Zentralblatt MATH identifier
06408758

#### Citation

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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