𝐿-values of harmonic Maass forms

Diamantis, Nikolaos; Rolen, Larry

doi:10.1090/tran/9045

$L$-values of harmonic Maass forms
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by Nikolaos Diamantis and Larry Rolen

Trans. Amer. Math. Soc.

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.

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