## Duke Mathematical Journal

- Duke Math. J.
- Volume 166, Number 13 (2017), 2447-2519.

### CM values of regularized theta lifts and harmonic weak Maaß forms of weight $1$

#### Abstract

We study special values of regularized theta lifts at complex multiplication (CM) points. In particular, we show that CM values of Borcherds products can be expressed in terms of finitely many Fourier coefficients of certain harmonic weak Maaß forms of weight $1$. As it turns out, these coefficients are logarithms of algebraic integers whose prime ideal factorization is determined by special cycles on an arithmetic curve. Our results imply a conjecture of Duke and Li and give a new proof of the modularity of a certain arithmetic generating series of weight $1$ studied by Kudla, Rapoport, and Yang.

#### Note

The current online version of this article, posted on 15 September 2017, supersedes both the advance publication version posted on 9 June 2017 and the version appearing in print copies of volume 166, number 13. The changes are as follows.

On page 2476, a typographical error in the second display of Lemma 4.1 that appeared in the advance publication version has been corrected. The $1$ appearing in the lower right entry of the first matrix has been changed to $-1$; the entire display appears correctly in the current online version and in print copies of volume 166, number 13 as $$\left(\begin{array}{cc}1& 0\\ -r& -1\end{array}\right)\mapsto N\phantom{\rule{1em}{0ex}}\text{and}\phantom{\rule{1em}{0ex}}\left(\begin{array}{cc}0& \frac{-1}{N}\\ \frac{D-{r}^{2}}{4N}& 0\end{array}\right)\mapsto \frac{r+\sqrt{D}}{2}.$$

In the list of symbols beginning on page 2513, the page reference numbers were not correctly indexed in print copies of volume 166, number 13. These numbers are correct in the current online version.

#### Article information

**Source**

Duke Math. J., Volume 166, Number 13 (2017), 2447-2519.

**Dates**

Received: 22 January 2015

Revised: 6 December 2016

First available in Project Euclid: 9 June 2017

**Permanent link to this document**

https://projecteuclid.org/euclid.dmj/1496995225

**Digital Object Identifier**

doi:10.1215/00127094-2017-0005

**Mathematical Reviews number (MathSciNet)**

MR3703434

**Zentralblatt MATH identifier**

06803183

**Subjects**

Primary: 11G18: Arithmetic aspects of modular and Shimura varieties [See also 14G35]

Secondary: 11F27: Theta series; Weil representation; theta correspondences 11F30: Fourier coefficients of automorphic forms 11F37: Forms of half-integer weight; nonholomorphic modular forms

**Keywords**

harmonic weak Maaß forms CM values singular moduli arithmetic geometry Kudla program regularized theta lifts Borcherds products weight one modular forms theta series

#### Citation

Ehlen, Stephan. CM values of regularized theta lifts and harmonic weak Maaß forms of weight $1$. Duke Math. J. 166 (2017), no. 13, 2447--2519. doi:10.1215/00127094-2017-0005. https://projecteuclid.org/euclid.dmj/1496995225