Special 𝐿-values and periods of weakly holomorphic modular forms

Bringmann, Kathrin; Fricke, Karl-Heinz; Kent, Zachary

doi:10.1090/S0002-9939-2014-12092-2

Special $L$-values and periods of weakly holomorphic modular forms
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by Kathrin Bringmann, Karl-Heinz Fricke and Zachary A. Kent PDF

Proc. Amer. Math. Soc. 142 (2014), 3425-3439 Request permission

Abstract:

In this paper, we explore a method for associating $L$-series to weakly holomorphic modular forms and then proceed to study their $L$-values. As our main application, we prove a very curious limiting theorem which relates three “periods” of a mock modular form and its shadow to the ratio of their noncritical $L$-values. Critical $L$-values are then shown to fit nicely within the framework of period polynomials and an extended Eichler-Shimura theory recently studied by Guerzhoy, Ono, and the first and third authors.

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