$L$-values of harmonic Maass forms
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- by Nikolaos Diamantis and Larry Rolen;
- Trans. Amer. Math. Soc. 377 (2024), 3905-3926
- DOI: https://doi.org/10.1090/tran/9045
- Published electronically: April 3, 2024
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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.References
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Bibliographic Information
- Nikolaos Diamantis
- Affiliation: School of Mathematical Sciences, University of Nottingham, Nottingham, United Kingdom
- MR Author ID: 646563
- ORCID: 0000-0002-3670-278X
- Email: nikolaos.diamantis@nottingham.ac.uk
- Larry Rolen
- Affiliation: Department of Mathematics, Vanderbilt University, Nashville, Tennessee
- MR Author ID: 923990
- ORCID: 0000-0001-8671-8117
- Email: larry.rolen@vanderbilt.edu
- Received by editor(s): June 8, 2022
- Received by editor(s) in revised form: July 10, 2023, and July 14, 2023
- Published electronically: April 3, 2024
- Additional Notes: The first author was partially supported by EPSRC grant EP/S032460/1. This work was supported by a grant from the Simons Foundation (853830, LR)
- © Copyright 2024 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 377 (2024), 3905-3926
- MSC (2020): Primary 11F37, 11M41, 11M35
- DOI: https://doi.org/10.1090/tran/9045
- MathSciNet review: 4748611