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Jacobi Maaß forms

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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.

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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 

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Authors and Affiliations

  1. Department of Mathematics, University of Oklahoma, Norman, OK, 73019, USA

    Ameya Pitale

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  1. Ameya Pitale
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Correspondence to Ameya Pitale.

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Communicated by U. Kühn.

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

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