15 September 2017 CM values of regularized theta lifts and harmonic weak Maaß forms of weight 1
Stephan Ehlen
Duke Math. J. 166(13): 2447-2519 (15 September 2017). DOI: 10.1215/00127094-2017-0005


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.

Version Information

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 (10r1)Nand(01NDr24N0)r+D2.

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.


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Stephan Ehlen. "CM values of regularized theta lifts and harmonic weak Maaß forms of weight 1." Duke Math. J. 166 (13) 2447 - 2519, 15 September 2017. https://doi.org/10.1215/00127094-2017-0005


Received: 22 January 2015; Revised: 6 December 2016; Published: 15 September 2017
First available in Project Euclid: 9 June 2017

zbMATH: 06803183
MathSciNet: MR3703434
Digital Object Identifier: 10.1215/00127094-2017-0005

Primary: 11G18
Secondary: 11F27 , 11F30 , 11F37

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

Rights: Copyright © 2017 Duke University Press


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Vol.166 • No. 13 • 15 September 2017
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