L-values of harmonic Maass forms

Diamantis, Nikolaos; Rolen, Larry

Abstract

Bruinier, Funke, and Imamoglu have proved a formula for what can philosophically be called the "central L-value" of the modular j-invariant. Previously, this had been heuristically suggested by Zagier. Here, we interpret this "L-value" as the value of an actual L-series, and extend it to all integral arguments and to a large class of harmonic Maass forms which includes all weakly holomorphic cusp forms. The context and relation to previously defined L-series for weakly holomorphic and harmonic Maass forms are discussed. These formulas suggest possible arithmetic or geometric meaning of L-values in these situations. The key ingredient of the proof is to apply a recent theory of Lee, Raji, and the authors to describe harmonic Maass L-functions using test functions.

Citation

Diamantis, N., & Rolen, L. (2024). L-values of harmonic Maass forms. Transactions of the American Mathematical Society, 377, 3905-3926. https://doi.org/10.1090/tran/9045

• Journal Article Type: Article
• Acceptance Date: Jul 17, 2023
• Online Publication Date: Apr 3, 2024
• Publication Date: 2024
• Deposit Date: Aug 18, 2023
• Publicly Available Date: Aug 18, 2023
• Journal: Transactions of the American Mathematical Society
• Print ISSN: 0002-9947
• Electronic ISSN: 1088-6850
• Publisher: American Mathematical Society
• Peer Reviewed: Peer Reviewed
• Volume: 377
• Pages: 3905-3926
• DOI: https://doi.org/10.1090/tran/9045
• Public URL: https://nottingham-repository.worktribe.com/output/24422947
• Publisher URL: https://www.ams.org/journals/tran/0000-000-00/S0002-9947-2023-09045-6/