The Ramanujan Journal

, Volume 36, Issue 1–2, pp 149–164 | Cite as

A mixed mock modular solution of the Kaneko–Zagier equation

  • P. Guerzhoy


The notion of mixed mock modular forms was recently introduced by Don Zagier. We show that certain solutions of the Kaneko–Zagier differential equation constitute simple yet non-trivial examples of this notion. That allows us to address a question posed by Kaneko and Koike on the (non)-modularity of these solutions.


Mixed mock modular forms Weak harmonic Maass forms Kaneko–Zagier differential equation 

Mathematics Subject Classification (2010)

11F12 11F37 



The author is very grateful to Masanobu Kaneko for enlightening discussions and explanations related to the mathematics around equation (KZ k ). Masanobu Kaneko also read a preliminary version of this note, corrected a bunch of miscalculations and inaccuracies, and filled in some details. The author wants to take this opportunity to express his gratitude to him for doing that. The author thanks the referee for helping to improve the exposition.


  1. 1.
    Bruinier, J.H., Funke, J.: On two geometric theta lifts. Duke Math. J. 125(1), 45–90 (2004) CrossRefMATHMathSciNetGoogle Scholar
  2. 2.
    Bringmann, K., Guerzhoy, P., Kent, Z., Ono, K.: Eichler–Shimura theory for mock modular forms. Math. Ann. 355(3), 1085–1121 (2013) CrossRefMATHMathSciNetGoogle Scholar
  3. 3.
    Dabholkar, A., Murth, S., Zagier, D.: Quantum black holes, wall crossing, and mock modular forms, preprint. arXiv:1208.4074 [hep-th]
  4. 4.
    Guerzhoy, P.: On the Honda–Kaneko congruences, from Fourier analysis and number theory to radon transforms and geometry, in memory of Leon Ehrenpreis. In: Dev. Math. vol. 28, pp. 293–302. Springer, Berlin (2012) Google Scholar
  5. 5.
    Honda, Y., Kaneko, M.: On Fourier coefficients of some meromorphic modular forms, preprint (2012) Google Scholar
  6. 6.
    Kaneko, M., Koike, M.: On modular forms arising from a differential equation of hypergeometric type. Ramanujan J. 7(1–3), 145–164 (2003) CrossRefMATHMathSciNetGoogle Scholar
  7. 7.
    Kaneko, M., Koike, M.: Quasimodular solutions of a differential equation of hypergeometric type. In: Galois Theory and Modular Forms. Dev. Math., vol. 11, pp. 329–336. Kluwer Acad., Boston (2004) CrossRefGoogle Scholar
  8. 8.
    Kaneko, M., Koike, M.: On extremal quasimodular forms. Kyushu J. Math. 60(2), 457–470 (2006) CrossRefMATHMathSciNetGoogle Scholar
  9. 9.
    Kaneko, M.: On modular forms of weight (6n+1)/5 satisfying a certain differential equation. In: Number Theory. Dev. Math., vol. 15, pp. 97–102. Springer, New York (2006) CrossRefGoogle Scholar
  10. 10.
    Kaneko, M., Zagier, D.: Supersingular j-invariants, hypergeometric series, and Atkin’s orthogonal polynomials. In: Computational Perspectives on Number Theory, Chicago, IL, 1995. AMS/IP Stud. Adv. Math., vol. 7, pp. 97–126. Amer. Math. Soc., Providence (1998) Google Scholar
  11. 11.
    Ono, K.: Unearthing the visions of a master: harmonic Maass forms and number theory. In: Proceedings of the 2008 Harvard-MIT Current Developments in Mathematics Conference, pp. 347–454. International Press, Somerville (2009) Google Scholar
  12. 12.
    Weil, A.: Elliptic Functions According to Eisenstein and Kronecker. Classics in Mathematics. Springer, Berlin (1999). Reprint of the 1976 original MATHGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2013

Authors and Affiliations

  1. 1.Department of MathematicsUniversity of HawaiiHonoluluUSA

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