Mock period functions, sesquiharmonic Maass forms, and non-critical values of L-functions

Bringmann, Kathrin; Diamantis, Nikolaos; Raum, Martin

Abstract

We introduce a new technique of completion for 1-cohomology which parallels the corresponding technique in the theory of mock modular forms. This technique is applied in the context of non-critical values of L-functions of GL(2) cusp forms. We prove that a generating series of non-critical values can be interpreted as a mock period function we
define in analogy with period polynomials. Further, we prove that non-critical values can be encoded into a sesquiharmonic Maass form. Finally, we formulate and prove an Eichler-Shimura-type isomorphism for the space of mock period functions.

Citation

Bringmann, K., Diamantis, N., & Raum, M. (2013). Mock period functions, sesquiharmonic Maass forms, and non-critical values of L-functions. Advances in Mathematics, 233(1), 115-134. https://doi.org/10.1016/j.aim.2012.09.025

• Journal Article Type: Article
• Acceptance Date: Sep 27, 2012
• Online Publication Date: Oct 31, 2012
• Publication Date: Jan 30, 2013
• Deposit Date: Apr 17, 2014
• Publicly Available Date: Apr 17, 2014
• Journal: Advances in Mathematics
• Print ISSN: 0001-8708
• Electronic ISSN: 1090-2082
• Publisher: Elsevier
• Peer Reviewed: Peer Reviewed
• Volume: 233
• Issue: 1
• Pages: 115-134
• DOI: https://doi.org/10.1016/j.aim.2012.09.025
• Public URL: https://nottingham-repository.worktribe.com/output/1003445
• Publisher URL: http://dx.doi.org/10.1016/j.aim.2012.09.025
• Additional Information: NOTICE: this is the author’s version of a work that was accepted for publication in Advances in Mathematics. Changes resulting from the publishing process, such as peer review, editing, corrections, structural formatting, and other quality control mechanisms may not be reflected in this document. Changes may have been made to this work since it was submitted for publication. A definitive version was subsequently published in Advances in Mathematics, 233(1) (2013), 115-134. doi: 10.1016/j.aim.2012.09.025