Skip to main content

Rankin–Cohen Type Operators for Hilbert–Jacobi Forms

  • 207 Accesses

Part of the Springer Proceedings in Mathematics & Statistics book series (PROMS,volume 340)

Abstract

We construct Rankin–Cohen type differential operators on the space of Hilbert–Jacobi forms. This generalizes a result of Choie and Eholzer (J Number Theory, 68:160–177, 1998) in the case of Jacobi forms to Hilbert–Jacobi forms.

Keywords

  • Hilbert–Jacobi forms
  • Rankin–Cohen brackets

2010 Mathematics Subject Classification

  • 11F41
  • 11F50
  • 11F60

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

Buying options

Chapter
USD   29.95
Price excludes VAT (USA)
  • DOI: 10.1007/978-981-15-8719-1_10
  • Chapter length: 14 pages
  • Instant PDF download
  • Readable on all devices
  • Own it forever
  • Exclusive offer for individuals only
  • Tax calculation will be finalised during checkout
eBook
USD   139.00
Price excludes VAT (USA)
  • ISBN: 978-981-15-8719-1
  • Instant PDF download
  • Readable on all devices
  • Own it forever
  • Exclusive offer for individuals only
  • Tax calculation will be finalised during checkout
Softcover Book
USD   179.99
Price excludes VAT (USA)
Hardcover Book
USD   179.99
Price excludes VAT (USA)

References

  1. Böcherer, S.: Bilinear holomorphic differential operators for the Jacobi group. Comment. Math. Univ. St. Pauli 47, 135–154 (1998)

    MathSciNet  MATH  Google Scholar 

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

    MathSciNet  CrossRef  Google Scholar 

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

    MathSciNet  CrossRef  Google Scholar 

  4. Choie, Y., Eholzer, W.: Rankin-Cohen operators for Jacobi and Siegel forms. J. Number Theory 68, 160–177 (1998)

    MathSciNet  CrossRef  Google Scholar 

  5. Cohen, H.: Sums involving the values at negative integers of \(L\)-functions of quadratic characters. Math. Ann. 217, 271–285 (1975)

    MathSciNet  CrossRef  Google Scholar 

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

    Google Scholar 

  7. Kumari, M.: Rankin-Cohen type operators and some properties of Fourier coefficients of modular forms. Ph.D. Thesis, NISER, Bhubaneswar (HBNI) (2018)

    Google Scholar 

  8. 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 

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

    MathSciNet  CrossRef  Google Scholar 

  10. Skogman, H.: Jacobi forms over totally real number fields. Results Math. 39, 169–182 (2001)

    MathSciNet  CrossRef  Google Scholar 

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

    MathSciNet  CrossRef  Google Scholar 

  12. Zagier, D.: Modular forms whose Fourier coefficients involve zeta-functions of quadratic fields. In: Modular Functions of One Variable VI (Proceedings of 2nd International Conference, University of Bonn, Bonn, 1976), pp. 105–169. Lecture Notes in Mathematics, vol. 627. Springer, Berlin (1977)

    Google Scholar 

Download references

Acknowledgements

Most of the work was carried out when the first author was a graduate student at National Institute of Science Education and Research (NISER), Bhubaneswar. She thanks NISER for all the support. The second author is partly supported by SERB MATRICS Project MTR/2017/000228. The authors would like to thank the referee for the helpful corrections which improved the paper.

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Moni Kumari .

Editor information

Editors and Affiliations

Rights and permissions

Reprints and Permissions

Copyright information

© 2020 The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd.

About this paper

Verify currency and authenticity via CrossMark

Cite this paper

Kumari, M., Sahu, B. (2020). Rankin–Cohen Type Operators for Hilbert–Jacobi Forms. In: Ramakrishnan, B., Heim, B., Sahu, B. (eds) Modular Forms and Related Topics in Number Theory. ICNT 2018. Springer Proceedings in Mathematics & Statistics, vol 340. Springer, Singapore. https://doi.org/10.1007/978-981-15-8719-1_10

Download citation