Structures for pairs of mock modular forms with the Zagier duality

Choi, Dohoon; Lim, Subong

doi:10.1090/S0002-9947-2014-06284-3

Structures for pairs of mock modular forms with the Zagier duality
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by Dohoon Choi and Subong Lim PDF

Trans. Amer. Math. Soc. 367 (2015), 5831-5861 Request permission

Abstract:

Zagier introduced special bases for weakly holomorphic modular forms to give the new proof of Borcherds’ theorem on the infinite product expansions of integer weight modular forms on $\mathrm {SL}_2(\mathbb {Z})$ with a Heegner divisor. These good bases appear in pairs, and they satisfy a striking duality, which is now called Zagier duality. After the result of Zagier, this type of duality was studied broadly in various viewpoints, including the theory of a mock modular form. In this paper, we consider this problem with Eichler cohomology theory, especially the supplementary function theory developed by Knopp. Using the holomorphic Poincaré series and its supplementary functions, we construct a pair of families of vector-valued harmonic weak Maass forms satisfying the Zagier duality with integer weights $-k$ and $k+2$, respectively, $k>0$, for an $H$-group. We also investigate the structures of them such as the images under the differential operators $D^{k+1}$ and $\xi _{-k}$ and quadric relations of the critical values of their $L$-functions.

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