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Structures for pairs of mock modular forms with the Zagier duality


Authors: Dohoon Choi and Subong Lim
Journal: Trans. Amer. Math. Soc. 367 (2015), 5831-5861
MSC (2010): Primary 11F11
DOI: https://doi.org/10.1090/S0002-9947-2014-06284-3
Published electronically: November 20, 2014
MathSciNet review: 3347190
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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 $ \textup {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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Additional Information

Dohoon Choi
Affiliation: School of Liberal Arts and Sciences, Korea Aerospace University, 200-1, Hwajeon-dong, Goyang, Gyeonggi 412-791, Republic of Korea
Email: choija@kau.ac.kr

Subong Lim
Affiliation: School of Mathematics, Korea Institute for Advanced Study, Hoegiro 85, Dongdaemun-gu, Seoul 130-722, Republic of Korea
Email: subong@kias.re.kr

DOI: https://doi.org/10.1090/S0002-9947-2014-06284-3
Keywords: Zagier duality, Eichler integral, supplementary function
Received by editor(s): January 24, 2013
Received by editor(s) in revised form: September 12, 2013
Published electronically: November 20, 2014
Additional Notes: The first author was supported by the National Research Foundation of Korea (NRF) grant funded by the Korean government (MSIP) (No. 2014001824).
Article copyright: © Copyright 2014 American Mathematical Society

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