Abstract
For any positive integers n and m, H n,m := H n ×C(m,n) is called the Siegel-Jacobi space, with the Jacobi group acting on it. The Jacobi forms are defined on this space. We compute the Chern connection of the Siegel-Jacobi space and use it to obtain derivations of Jacobi forms. Using these results, we construct a series of invariant differential operators for Siegel-Jacobi forms. Also two kinds of Maass-Shimura type differential operators for H n,m are obtained.
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References
Berceanu S. A convenient coordinatization of Siegel-Jacobi domains. Rev Math Phys, 2012, 24: 10
Berndt R, Schmidt R. Elements of the Representation Theory of the Jacobi Group. Basel: Birkhauser, 1998
Böcherer S. Bilinear holomorphic differential operators for the Jacobi group. Comm Math Univ St Pauli, 1998, 47: 135–154
Böcherer S, Das S. On holomorphic differential operators equivariant for the inclusion of Sp(n, R) in U(n, n). Int Math Res Not, 2012, doi:10.1093/imrn/rns116
Böcherer S, Nagaoka S. On mod p properties of Siegel modular forms. Math Ann, 2007, 338: 421–433
Chern S S, Chen W H, Lam K S. Lectures on Differential Geometry. Singapore: World Scientific, 2000
Choie Y, Eholzer W. Rankin-Cohen operators for Jacobi and Siegel forms. J Number Theory, 1998, 68: 160–177
Conley C, Raum M. Harmonic Maass-Jacobi forms of degree 1 with higher rank indices. Int J Number Theory, 2015, doi: 10.1142/S1793042116501165
Eholzer W, Ibukiyama T. Rankin-Cohen type differential operators for Siegel modular forms. Int J Math, 1988, 9: 443–463
Eichler M, Zagier D. The Theory of Jacobi Forms. Boston: Birkhauser, 1985
Ibukiyama T. On differential operators on automorphic forms and invariant pluriharmonic polynomials. Comment Math Univ St Pauli, 1999, 48: 103–118
Kobayashi S, Nomizu K. Foundations of Differential Geometry, vol. 2. New York: Wiley, 1969
Maass H. Siegel’s Modular Forms and Dirichlet Series. Berlin-New York: Springer-Verlag, 1971
Shimura G. Arithmeticity in the Theory of Automorphic Forms. Providence, RI: Amer Math Soc, 2000
Yang E, Yin L. Derivatives of Siegel modular forms and modular connections. Manuscripta Math, 2015, 146: 65–84
Yang J-H. Invariant metrics and Laplacians on Siegel-Jacobi space. J Number Thoery, 2007, 127: 83–102
Yang J-H. Invariant differential operators on the Siegel-Jacobi space. ArXiv:1107.0509v1, 2011
Ziegler C. Jacobi forms of higher degree. Abh Math Sem Hamburg, 1989, 59: 191–224
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Yang, J., Yin, L. Differential operators for Siegel-Jacobi forms. Sci. China Math. 59, 1029–1050 (2016). https://doi.org/10.1007/s11425-015-5111-4
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DOI: https://doi.org/10.1007/s11425-015-5111-4