Skip to main content
Log in

Abstract

In this paper, we give a new definition for the space of non-holomorphic Jacobi Maaß forms (denoted by J nhk,m ) of weight k∈ℤ and index m∈ℕ as eigenfunctions of a degree three differential operator \(\mathcal{C}^{k,m}\) . We show that the three main examples of Jacobi forms known in the literature: holomorphic, skew-holomorphic and real-analytic Eisenstein series, are contained in J nhk,m . We construct new examples of cuspidal Jacobi Maaß forms F f of weight k∈2ℤ and index 1 from weight k−1/2 Maaß forms f with respect to Γ0(4) and show that the map f F f is Hecke equivariant. We also show that the above map is compatible with the well-known representation theory of the Jacobi group. In addition, we show that all of J nhk,m can be “essentially” obtained from scalar or vector valued half integer weight Maaß forms.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Subscribe and save

Springer+ Basic
EUR 32.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or Ebook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Similar content being viewed by others

References

  1. Arakawa, T.: Real Analytic Eisenstein series for the Jacobi Group. Abh. Math. Semin. Univ. Hambg. 60, 131–148 (1990)

    Article  MATH  MathSciNet  Google Scholar 

  2. Arakawa, T.: Saito-Kurokawa lifting for odd weights. Comment. Math. Univ. St. Paul 49(2), 159–176 (2000)

    MATH  MathSciNet  Google Scholar 

  3. Berndt, R.: On the characterization of skew-holomorphic Jacobi forms. Abh. Math. Semin. Univ. Hambg. 65, 301–305 (1995)

    Article  MATH  MathSciNet  Google Scholar 

  4. Berndt, R., Böcherer, S.: Jacobi forms and discrete series representation of the Jacobi group. Math. Z. 204(1), 13–44 (1990)

    Article  MATH  MathSciNet  Google Scholar 

  5. Berndt, R., Schmidt, R.: Elements of the Representation Theory of the Jacobi Group. Progress in Mathematics, vol. 163. Birkhauser, Basel (1998)

    MATH  Google Scholar 

  6. Borho, W.: Primitive and vollprimitive Ideale in Einhüllenden von \(\mathfrak{sl}(5,\mathbb{C})\) . J. Algebra 43, 619–644 (1976)

    Article  MATH  MathSciNet  Google Scholar 

  7. Bringmann, K., Conley, C., Richter, O.: Maaß Jacobi forms over complex quadratic fields. Math. Res. Lett. 14(1), 137–156 (2007)

    MATH  MathSciNet  Google Scholar 

  8. Bruelmann, S., Kuss, M.: On a conjecture of Duke-Imamoglu. Proc. AMS 128, 1595–1604 (2000)

    Article  Google Scholar 

  9. Dulinski, J.: L-functions for Jacobi forms on ℋ×ℂ. Results Math. 31(1–2), 75–94 (1997)

    MATH  MathSciNet  Google Scholar 

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

    MATH  Google Scholar 

  11. Gan, W.-T.: The Saito-Kurokawa space of PGSp4 and its transfer to inner forms. In: Eisenstein Series and Applications. Prog. Math., vol. 258, pp. 87–123. Birkhauser, Boston (2008)

    Chapter  Google Scholar 

  12. Gelbart, S.: Automorphic Forms on Adele Groups. Annals of Mathematics Studies, vol. 83. Princeton University Press, Princeton (1975)

    MATH  Google Scholar 

  13. Ikeda, T.: On the lifting of elliptic cusp forms to Siegelcusp forms of degree 2n. Ann. Math. (2) 154(3), 641–681 (2001)

    Article  MATH  Google Scholar 

  14. Katok, S., Sarnak, P.: Heegner points, cycles and Maaß forms. Isr. Math. J. 84, 237–268 (1993)

    MathSciNet  Google Scholar 

  15. Kohnen, W.: Modular forms of half integer weight on Γ0(4). Math. Ann. 248, 249–266 (1980)

    Article  MATH  MathSciNet  Google Scholar 

  16. Magnus, W., Oberhettinger, F., Soni, R.P.: Formulas and Theorems for the Special Functions of Mathematical Physics, 3rd edn. Die Grundlehren der mathematischen Wissenschaften, Band 52 (1966)

  17. Martin, Y.: A converse theorem for Jacobi forms. J. Number Theory 61(1), 181–193 (1996)

    Article  MATH  MathSciNet  Google Scholar 

  18. Pitale, A.: Lifting from \(\widetilde{\mathrm{SL}}_{2}\) to GSpin(1,4). Int. Math. Res. Not. 63, 3919–3966 (2005)

    Article  MathSciNet  Google Scholar 

  19. Shimura, G.: On modular forms of half integral weight. Ann. Math. (2) 97, 440–481 (1973)

    Article  MathSciNet  Google Scholar 

  20. Skoruppa, N.-P.: Developments in the Theory of Jacobi Forms. Automorphic Functions and Their Applications, Khabarovsk, pp. 167–185 (1990)

  21. Waldspurger, J.-L.: Sur les coefficients de Fourier de formes modulaires de poids demi-entier. J. Math. Pures Appl. 60, 375–484 (1981)

    MATH  MathSciNet  Google Scholar 

  22. Waldspurger, J.-L.: Correspondance de Shimura et quaternions. Forum Math. 3, 219–307 (1991)

    Article  MATH  MathSciNet  Google Scholar 

  23. Yang, J.-H.: A note on Maaß Jacobi forms. Kyungpook Math. J. 43(3), 547–566 (2003)

    MATH  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Ameya Pitale.

Additional information

Communicated by U. Kühn.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Pitale, A. Jacobi Maaß forms. Abh. Math. Semin. Univ. Hambg. 79, 87–111 (2009). https://doi.org/10.1007/s12188-008-0013-9

Download citation

  • Received:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s12188-008-0013-9

Keywords

Mathematics Subject Classification (2000)

Navigation