Traces of CM values and cycle integrals of polyharmonic Maass forms

  • Toshiki MatsusakaEmail author


Lagarias and Rhoades generalized harmonic Maass forms by considering forms which are annihilated by a number of iterations of the action of the \(\xi \)-operator. In our previous work, we considered polyharmonic weak Maass forms by allowing the exponential growth at cusps, and constructed a basis of the space of such forms. This paper focuses on the case of half-integral weight. We construct a basis as an analogue of our work, and give arithmetic formulas for the Fourier coefficients in terms of traces of CM values and cycle integrals of polyharmonic weak Maass forms. These results put the known results into a common framework.


Polyharmonic Maass forms Harmonic Modular forms Fourier coefficients 

Mathematics Subject Classification

Primary 11F37 Secondary 11F12 


Author's contributions


The author would like to thank Masanobu Kaneko, Soon-Yi Kang, Chang Heon Kim, and Markus Schwagenscheidt for their helpful comments. He also thanks the referees for their comments on a preliminary version of this paper. This work is supported by Research Fellow (DC) of Japan Society for the Promotion of Science, Grant-in-Aid for JSPS Fellows 18J20590 and JSPS Overseas Challenge Program for Young Researchers.


  1. 1.
    Ahlgren, S., Andersen, N., Samart, D.: A polyharmonic Maass form of depth \(3/2\) for \(\rm SL_2({\mathbb{Z}})\). J. Math. Anal. Appl. 468(2), 1018–1042 (2018)MathSciNetCrossRefGoogle Scholar
  2. 2.
    Alfes-Neumann, C., Schwagenscheidt, M.: On a theta lift related to the Shintani lift. Adv. Math. 328, 858–889 (2018)MathSciNetCrossRefGoogle Scholar
  3. 3.
    Alfes-Neumann, C., Schwagenscheidt, M.: Shintani theta lifts of harmonic Maass forms. arXiv:1712.04491
  4. 4.
    Andersen, N., Lagarias, J.C., Rhoades, R.C.: Shifted polyharmonic Maass forms for \(PSL(2, \mathbb{Z})\). Acta Arith. 185(1), 39–79 (2018)MathSciNetCrossRefGoogle Scholar
  5. 5.
    Bringmann, K., Diamantis, N., Raum, M.: Mock period functions, sesquiharmonic Maass forms, and non-critical values of \(L\)-functions. Adv. Math. 233, 115–134 (2013)MathSciNetCrossRefGoogle Scholar
  6. 6.
    Bringmann, K., Folsom, A., Ono, K., Rolen, L.: Harmonic Maass Forms and Mock Modular Forms: Theory and Applications. American Mathematical Society Colloquium Publications, vol. 64. American Mathematical Society, Providence (2017)Google Scholar
  7. 7.
    Bruinier, J.H., Funke, J.: On two geometric theta lifts. Duke Math. J. 125(1), 45–90 (2004)MathSciNetCrossRefGoogle Scholar
  8. 8.
    Bruinier, J.H., Funke, J.: Traces of CM values of modular functions. J. Reine Angew. Math. 594, 1–33 (2006)MathSciNetCrossRefGoogle Scholar
  9. 9.
    Bruinier, J.H., Funke, J., Imamoḡlu, Ö.: Regularized theta liftings and periods of modular functions. J. Reine Angew. Math. 703, 43–93 (2015)MathSciNetzbMATHGoogle Scholar
  10. 10.
    Duke, W., Jenkins, P.: On the zeros and coefficients of certain weakly holomorphic modular forms. Pure Appl. Math. Q. 4(4), 1327–1340 (2008)MathSciNetCrossRefGoogle Scholar
  11. 11.
    Duke, W., Jenkins, P.: Integral traces of singular values of weak Maass form. Algebra Number Theory 2(5), 573–593 (2008)MathSciNetCrossRefGoogle Scholar
  12. 12.
    Duke, W., Imamoḡlu, Ö., Tóth, Á.: Cycle integrals of the \(j\)-function and mock modular forms. Ann. Math. 173, 947–981 (2011)MathSciNetCrossRefGoogle Scholar
  13. 13.
    Duke, W., Imamoḡlu, Ö., Tóth, Á.: Regularized inner products of modular functions. Ramanujan J. 41, 13–29 (2016)MathSciNetCrossRefGoogle Scholar
  14. 14.
    Duke, W., Imamoḡlu, Ö., Tóth, Á.: Kronecker’s first limit formula, revisited. Res. Math. Sci. 5(2), 1–21 (2018)MathSciNetCrossRefGoogle Scholar
  15. 15.
    Fay, J.: Fourier coefficients of the resolvent for a Fuchsian group. J. Reine Angew. Math. 294, 143–203 (1977)MathSciNetzbMATHGoogle Scholar
  16. 16.
    Gradshteyn, I.S., Ryzhik, I.M.: Table of Integrals, Series, and Products, 6th edn. Translated from the Russian. Translation and edited with a preface by A. Jeffrey and D. Zwillinger. Academic Press, Inc., San Diego (2000)Google Scholar
  17. 17.
    Gross, B., Kohnen, W., Zagier, D.: Heegner points and derivatives of \(L\)-series, II. Math. Ann. 278, 497–562 (1987)MathSciNetCrossRefGoogle Scholar
  18. 18.
    Hirzebruch, F., Zagier, D.: Intersection numbers of curves on Hilbert modular surfaces and modular forms of Nebentypus. Invent. Math. 36, 57–113 (1976)MathSciNetCrossRefGoogle Scholar
  19. 19.
    Ibukiyama, T., Saito, H.: On zeta functions associated to symmetric matrices, II: functional equations and special values. Nagoya Math. J. 208, 265–316 (2012)MathSciNetCrossRefGoogle Scholar
  20. 20.
    Jeon, D., Kang, S.-Y., Kim, C.H.: Weak Maass-Poincaré series and weight \(3/2\) mock modular forms. J. Number Theory 133, 2567–2587 (2013)MathSciNetCrossRefGoogle Scholar
  21. 21.
    Jeon, D., Kang, S.-Y., Kim, C.H.: Cycle integrals of a sesqui-harmonic Maass form of weight zero. J. Number Theory 141, 92–108 (2014)MathSciNetCrossRefGoogle Scholar
  22. 22.
    Jeon, D., Kang, S.-Y., Kim, C.H.: Zagier-lift type arithmetic in harmonic weak Maass forms. J. Number Theory 169, 227–249 (2016)MathSciNetCrossRefGoogle Scholar
  23. 23.
    Jeon, D., Kang, S.-Y., Kim, C.H.: Bases on spaces of harmonic weak Maass forms and Shintani lifts on harmonic weak Maass forms (submitted for publication)Google Scholar
  24. 24.
    Koblitz, N.: Introduction to Elliptic Curves and Modular Forms. Graduate Texts in Mathematics, vol. 97, 2nd edn. Springer, New York (1993)Google Scholar
  25. 25.
    Kohnen, W.: Fourier coefficients of modular forms of half-integral weight. Math. Ann. 271, 237–268 (1985)MathSciNetCrossRefGoogle Scholar
  26. 26.
    Lagarias, J.C., Rhoades, R.C.: Polyharmonic Maass forms for \(\rm PSL(2,{mathbb Z\rm })\). Ramanujan J. 41, 191–232 (2016)MathSciNetCrossRefGoogle Scholar
  27. 27.
    Magnus, W., Oberhettinger, F., Soni, R.: Formulas and Theorems for the Special Functions of Mathematical Physics, Third enlarged edition. Die Grundlehren der mathematischen Wissenschaften, Band 52. Springer, New York (1966)Google Scholar
  28. 28.
    Matsusaka, T.: Polyharmonic weak Maass forms of higher depth for \(\text{SL}_2({\mathbb{Z}}).\) Ramanujan J. arXiv:1801.02146
  29. 29.
    Mertens, M.H.: Mock modular forms and class number relations. Res. Math. Sci. 1, 1–16 (2014)MathSciNetCrossRefGoogle Scholar
  30. 30.
    Niebur, D.: A class of nonanalytic automorphic functions. Nagoya Math. J. 52, 133–145 (1973)MathSciNetCrossRefGoogle Scholar
  31. 31.
    Petersson, H.: Konstruktion der Modulformen und der zu gewissen Grenzkreisgruppen gehörigen automorphen Formen von positiver reeller Dimension und die vollsts̈ndige Bestimmung ihrer Fourierkoeffizienten, S.-B. Heidelberger Akad. Wiss. Math. Nat. Kl. 417–494 (1950)Google Scholar
  32. 32.
    Zagier, D.: Nombres de classes et formes modulaires de poids \(3/2.\) C. R. Acad. Sci. Paris Sér. A-B 281(21, Ai), 883–886 (1975)Google Scholar
  33. 33.
    Zagier, D.:Traces of singular moduli, Motives, polylogarithms and Hodge theory, Part I (Irvine, CA, 1998), Int. Press Lect. Ser., vol. 3, pp. 211–244. Int. Press. Somerville MA (2002)Google Scholar

Copyright information

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Authors and Affiliations

  1. 1.Graduate School of MathematicsKyushu UniversityFukuokaJapan

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