Quantum modular forms and Hecke operators

  • Seewoo Lee


It is known that there are one-to-one correspondences among the space of cusp forms, the space of homogeneous period polynomials and the space of Dedekind symbols with polynomial reciprocity laws. We add one more space, the space of quantum modular forms with polynomial period functions, to extend results from Fukuhara. Also, we consider Hecke operators on the space of quantum modular forms and construct new quantum modular forms.


Modular form Quantum modular form Dedekind symbol Period polynomial Hecke operator 



This is part of the author’s undergraduate thesis paper. The author is grateful to the advisor Y. Choie for her helpful advice. The author is also grateful to S. Fukuhara, D. Choi and K. Ono for their comments via emails and J. Baek for his help about SAGE codes.


  1. 1.
    Andrews, G.E.: Ramanujan’s “Lost” Notebook V: Euler’s partition identity. Adv. Math. 61, 156–184 (1986)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Choi, D., Lim, S., Rhoades, R.C.: Mock modular forms and quantum modular forms. Am. Math. Soc. 144(6), 2337–2349 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Choie, Y., Zagier, D.: Rational Period Functions for PSL(2, Z). A Tribute to Emil Gross-wald: Number Theory and Related Analysis, Contemporary Mathematics, vol. 143, pp. 89–107. AMS, Providence (1993)Google Scholar
  4. 4.
    Cohen, H.: q-identities for Maass wave forms. Invent. Math. 91, 409–422 (1988)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Diamond, F., Shurman, J.: A First Course in Modular Forms, Graduate Texts in Mathematics. Springer, Berlin (2005)zbMATHGoogle Scholar
  6. 6.
    Fukuhara, S.: Modular forms, generalized Dedekind symbols and period polynomials. Math. Ann. 310, 83–101 (1998)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Fukuhara, S.: Dedekind symbols with polynomial reciprocity laws. Math. Ann. 329, 315–334 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Fukuhara, S.: Hecke operators on weighted Dedekind symbols. J. Reine Angew. Math. 593, 1–29 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Fukuhara, S.: Explicit formulas for Hecke operators on cusp forms, Dedekind symbols and period polynomials. J. Reine Angew. Math. 607, 163–216 (2007)MathSciNetzbMATHGoogle Scholar
  10. 10.
    Kohnen, W., Zagier, D.: Modular forms with rational periods. In: Rankin, R.A. (ed.) Modular Forms, pp. 197–249. Ellis Horwood, Chicheseter (1984)Google Scholar
  11. 11.
    Zagier, D.: Hecke operators and periods of modular forms. In: Israel Mathematical Conference Proceedings, vol. 3 (1990)Google Scholar
  12. 12.
    Zagier, D.: Quantum modular forms. In: Clay Mathematics Proceedings, vol. 12 (2010)Google Scholar

Copyright information

© SpringerNature 2018

Authors and Affiliations

  1. 1.Department of MathematicsPohang University of Science and TechnologyPohangRepublic of Korea

Personalised recommendations