Skip to main content

Rankin–Cohen brackets on Jacobi forms and the adjoint of some linear maps

Abstract

Given a fixed Jacobi cusp form, we consider a family of linear maps between the spaces of Jacobi cusp forms using the Rankin–Cohen brackets, and then we compute the adjoint maps of these linear maps with respect to the Petersson scalar product. The Fourier coefficients of the Jacobi cusp forms constructed using this method involve special values of certain Dirichlet series associated to Jacobi cusp forms. This is a generalization of the work due to Kohnen (Math Z, 207:657–660, 1991) and Herrero (Ramanujan J, 10.1007/s11139-013-9536-5, 2014) in case of elliptic modular forms to the case of Jacobi cusp forms which is also considered earlier by Sakata (Proc Japan Acad Ser A, Math Sci 74, 1998) for a special case.

This is a preview of subscription content, access via your institution.

References

  1. Choie, Y.: Jacobi forms and the heat operator. Math. Z. 1, 95–101 (1997)

    MathSciNet  Article  MATH  Google Scholar 

  2. Choie, Y.: Jacobi forms and the heat operator II. Illinois J. Math. 42, 179–186 (1998)

    MathSciNet  MATH  Google Scholar 

  3. Choie, Y., Kohnen, W.: Rankin’s method and Jacobi forms. Abh. Math. Sem. Univ. Hamburg 67, 307–314 (1997)

    MathSciNet  Article  MATH  Google Scholar 

  4. Choie, Y., Kim, H., Knopp, M.: Construction of Jacobi forms. Math. Z. 219, 71–76 (1995)

    MathSciNet  Article  MATH  Google Scholar 

  5. Cohen, H.: Sums involving the values at negative integers of \(L\)-functions of quadratic characters. Math. Ann. 217, 81–94 (1977)

    MathSciNet  Google Scholar 

  6. Eichler, M., Zagier, D.: The Theory of Jacobi Forms. Progress in Mathematics, vol. 55. Birkhäuser, Boston (1985)

    Book  MATH  Google Scholar 

  7. Gross, B., Kohnen, W., Zagier, D.: Heegner points and derivatives of \(L\)-series II. Math. Ann. 278, 497–562 (1987)

    MathSciNet  Article  MATH  Google Scholar 

  8. Herrero, S.D.: The adjoint of some linear maps constructed with the Rankin–Cohen brackets. Ramanujan J. (2014). doi:10.1007/s11139-013-9536-5

  9. Kohnen, W.: Cusp forms and special value of certain Dirichlet Series. Math. Z. 207, 657–660 (1991)

    MathSciNet  Article  MATH  Google Scholar 

  10. Lee, M.H.: Siegel cusp forms and special values of Dirichlet series of Rankin type. Complex Var. Theory Appl. 31(2), 97–103 (1996)

    MathSciNet  Article  Google Scholar 

  11. Lee, M.H., Suh, D.Y.: Fourier coefficients of cusp forms associated to mixed cusp forms. Panam. Math. J. 8(1), 31–38 (1998)

    MathSciNet  MATH  Google Scholar 

  12. Rankin, R.A.: The Construction of automorphic forms from the derivatives of a given form. J. Indian Math. Soc. 20, 103–116 (1956)

    MathSciNet  MATH  Google Scholar 

  13. Rankin, R.A.: The construction of automorphic forms from the derivatives of given forms. Michigan Math. J. 4, 181–186 (1957)

    MathSciNet  Article  MATH  Google Scholar 

  14. Sakata, H.: Construction of Jacobi cusp forms. Proc. Japan. Acad. Ser. A Math. Sci. 74(7), 117–119 (1998)

  15. Wang, X.: Hilbert modular forms and special values of some Dirichlet series. Acta. Math. Sin. 38(3), 336–343 (1995)

    MathSciNet  MATH  Google Scholar 

  16. Zagier, D.: Modular forms whose Fourier coefficients involve zeta-functions of quadratic fields, in Modular functions of one variable, VI (Proc. Second Internat. Conf., Univ. Bonn, Bonn, 1976). Lecture Notes in Mathematics, vol. 627, pp. 105–169. Springer, Berlin (1977)

    Google Scholar 

  17. Zagier, D.: Modular forms and differential operators. Proc. Indian Acad. Sci. Math. Sci. 104, 57–75 (1994)

    MathSciNet  Article  MATH  Google Scholar 

Download references

Acknowledgments

The authors would like to thank Sebastián D. Herrero for providing a copy of his paper [8]. The authors would like to thank B. Ramakrishnan for his helpful comments. Finally, the authors thank the referee for their careful reading of the paper and for many helpful comments.

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Brundaban Sahu.

Additional information

The first author would like to thank the Council of Scientific and Industrial Research (CSIR), India for financial support. The second author is partially funded by SERB Grant SR/FTP/MS-053/2012.

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Jha, A.K., Sahu, B. Rankin–Cohen brackets on Jacobi forms and the adjoint of some linear maps. Ramanujan J 39, 533–544 (2016). https://doi.org/10.1007/s11139-015-9683-y

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11139-015-9683-y

Keywords

  • Jacobi forms
  • Rankin–Cohen brackets
  • Adjoint map

Mathematics Subject Classification

  • Primary 11F50
  • Secondary 11F25
  • 11F66