Regularized inner products and errors of modularity

Bringmann, Kathrin; Diamantis, Nikolaos; Ehlen, Stephan

Abstract

© The Author(s) 2016. We develop a regularization for Petersson inner products of arbitrary weakly holomorphic modular forms, generalizing several known regularizations. As one application, we extend work of Duke, Imamoglu, and Toth on regularized inner products of weakly holomorphic modular forms of weights 0 and 3/2. These regularized inner products can be evaluated in terms of the coefficients of holomorphic parts of harmonic Maass forms of dual weights. Moreover, we study the errors of modularity of the holomorphic parts of such a harmonic Maass forms and show that they induce cocyles in the first parabolic cohomology group introduced by Bruggeman, Choie, and the second author. This provides explicit representatives of the cohomology classes constructed abstractly and in a very general setting in their work.

Citation

Bringmann, K., Diamantis, N., & Ehlen, S. (2017). Regularized inner products and errors of modularity. International Mathematics Research Notices, 2017(24), 7420-7458. https://doi.org/10.1093/imrn/rnw225

• Journal Article Type: Article
• Acceptance Date: Sep 21, 2016
• Online Publication Date: Dec 23, 2016
• Publication Date: Dec 1, 2017
• Deposit Date: Mar 6, 2017
• Publicly Available Date: Mar 6, 2017
• Journal: International Mathematics Research Notices
• Print ISSN: 1073-7928
• Electronic ISSN: 1687-0247
• Publisher: Oxford University Press
• Peer Reviewed: Peer Reviewed
• Volume: 2017
• Issue: 24
• Pages: 7420-7458
• DOI: https://doi.org/10.1093/imrn/rnw225
• Public URL: https://nottingham-repository.worktribe.com/output/897783
• Publisher URL: https://academic.oup.com/imrn/article-lookup/doi/10.1093/imrn/rnw225
• Related Public URLs: https://arxiv.org/abs/1603.03056v2
• Additional Information: This is a pre-copyedited, author-produced version of an article accepted for publication in International Mathematics Research Notices following peer review. The version of record Kathrin Bringmann, Nikolaos Diamantis, Stephan Ehlen; Regularized Inner Products and Errors of Modularity. Int Math Res Notices 2016 is available online at: https://academic.oup.com/imrn/article-lookup/doi/10.1093/imrn/rnw225
• Contract Date: Mar 6, 2017