Abstract
Since their definition in 2010 by Zagier, quantum modular forms have been connected to numerous topics such as strongly unimodal sequences, ranks, cranks, and asymptotics for mock theta functions near roots of unity. These are functions that are not necessarily defined on the upper half plane but a priori are defined only on a subset of \({\mathbb{Q}}\), and whose obstruction to modularity is some analytically “nice” function. Motivated by Zagier’s example of the quantum modularity of Kontsevich’s “strange” function F(q), we revisit work of Andrews, Jiménez-Urroz, and Ono to construct a natural vector-valued quantum modular form whose components are similarly “strange”.
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Andrews G.: Ramanujan’s “lost” notebook. V. Euler’s partition identity. Adv. in Math. 61, 156–164 (1986)
G. Andrews, J. Jiménez-Urroz, and K. Ono, q-series identities and values of certain L-functions, Duke Math. J. 108 (2001), 395–419.
J. Bryson et al., Unimodal sequences and quantum and mock modular forms. Proceedings of the National Academy of Sciences 109 (2012), 16063–16067.
N. J. Fine, Basic hypergeometric series and applications, Mathematical Surveys and Monographs, vol. 27, American Mathematical Society, Providence, RI, 1988, With a foreword by George E. Andrews.
Kazuhiro H.: q-series and L-functions related to half-derivatives of the Andrews-Gordon identity. Ramanujan J. 11, 175–197 (2006)
R. Lawrence and D. Zagier, Modular forms and quantum invariants of 3-manifolds, Asian J. Math. 3 (1999), 93–107, Sir Michael Atiyah: a great mathematician of the twentieth century.
Zagier D.: Vassiliev invariants and a strange identity related to the Dedekind eta-function. Topology 40, 945–960 (2001)
D. Zagier, Quantum modular forms, Quanta of maths, Clay Math. Proc., vol. 11, Amer. Math. Soc., Providence, RI, 2010, pp. 659–675.
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The authors would like to thank our advisor Ken Ono for his guidance throughout this project. We are also grateful to Robert Rhoades and to the referee for their helpful comments; in particular the referee suggested using Fine’s identity for an alternative expression for \({\theta_1^S(q^{-1})}\) in (4.4).
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Rolen, L., Schneider, R.P. A “strange” vector-valued quantum modular form. Arch. Math. 101, 43–52 (2013). https://doi.org/10.1007/s00013-013-0529-9
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DOI: https://doi.org/10.1007/s00013-013-0529-9