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Mock modular forms and quantum modular forms


Authors: Dohoon Choi, Subong Lim and Robert C. Rhoades
Journal: Proc. Amer. Math. Soc. 144 (2016), 2337-2349
MSC (2010): Primary 11F37; Secondary 11F67
DOI: https://doi.org/10.1090/proc/12907
Published electronically: October 20, 2015
MathSciNet review: 3477051
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Abstract: In his last letter to Hardy, Ramanujan introduced mock theta functions. For each of his examples $ f(q)$, Ramanujan claimed that there is a collection $ \{ G_j\}$ of modular forms such that for each root of unity $ \zeta $, there is a $ j$ such that

$\displaystyle \lim _{q \to \zeta }(f(q) - G_j(q)) = O(1).$

Moreover, Ramanujan claimed that this collection must have size larger than $ 1$. In his 2001 PhD thesis, Zwegers showed that the mock theta functions are the holomorphic parts of harmonic weak Maass forms. In this paper, we prove that there must exist such a collection by establishing a more general result for all holomorphic parts of harmonic Maass forms. This complements the result of Griffin, Ono, and Rolen that shows such a collection cannot have size $ 1$. These results arise within the context of Zagier's theory of quantum modular forms. A linear injective map is given from the space of mock modular forms to quantum modular forms. Additionally, we provide expressions for ``Ramanujan's radial limits'' as $ L$-values.

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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: Department of Mathematics Education, Sungkyunkwan University, Jongno-gu, Seoul 110-745, Republic of Korea
Email: subong@skku.edu

Robert C. Rhoades
Affiliation: Center for Communications Research, 805 Bunn Dr., Princeton, New Jersey 08450
Email: rob.rhoades@gmail.com

DOI: https://doi.org/10.1090/proc/12907
Keywords: Quantum modular form, Eichler integral, mock theta function, Ramanujan, radial limit
Received by editor(s): July 9, 2015
Published electronically: October 20, 2015
Additional Notes: The first and second authors were supported by Samsung Science and Technology Foundation under Project SSTF-BA1301-11.
Communicated by: Ken Ono
Article copyright: © Copyright 2015 American Mathematical Society