Abstract
We introduce a method in differential geometry to study the derivative operators of Siegel modular forms. By determining the coefficients of the invariant Levi–Civita connection on a Siegel upper half plane, and further by calculating the expressions of the differential forms under this connection, we get a non-holomorphic derivative operator of the Siegel modular forms. In order to get a holomorphic derivative operator, we introduce a weaker notion, called modular connection, on the Siegel upper half plane. Then we show that on a Siegel upper half plane there exists at most one holomorphic \({{\rm Sp}(2g, {\mathbb {Z}})}\)-modular connection in some sense, and get a possible holomorphic derivative operator of Siegel modular forms.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Andrianov A.: Introduction to Siegel Modular Forms and Dirichlet Series. Springer, New York (2009)
Böcherer S.: Bilinear holomorphic differential operators for the Jacobi group. Comm. Math. Univ. Sancti Pauli 47, 135–154 (1998)
Böcherer, S., Das, S.: On holomorphic differential operators equivariant for the inclusion of \({{\rm Sp}(n, \mathbb{R})}\) in U(n, n). In:International Mathematics Research Notices, published online, doi:10.1093/imrn/rns116
Bump D.: Automorphic Forms and Representations, Cambridge Studies in Advanced Mathematics 55. Cambridge University Press, Cambridge (1997)
Chern S.S., Chen W.H., Lam K.S.: Lectures on Differential Geometry. World Scientific, Singapore (1999)
Eholzer W., Ibukiyama T.: Rankin–Cohen type differential operators for Siegel modular forms. Int. J. Math. 9, 443–463 (1998)
Goldfeld D.: Automorphic forms and L-functions for the group \({GL(n, \mathbb{R})}\). Cambridge Studies in Advanced Mathematics 99. Cambridge University Press, Cambridge (2006)
Harris M.: Maass operators and Eisenstein series. Math. Ann. 258, 135–144 (1981)
Ibukiyama T.: On differential operators on automorphic forms and invariant pluri-harmonic polynomials, Comm. Math. Univ. Sancti. Pauli 48, 103–118 (1999)
Jost J.: Riemannian Geometry and Geometric Analysis. Springer (Universitext), Berlin (1988)
Koblitz N.: Introduction to Elliptic curves and Modular Forms. Springer, New York (1984)
Maass, H.: Lectures on Siegel’s Modular Functions, Tate Institute of Fundamental Research, Bombay (1955)
Maass H.: Siegel’s Modular Forms and Dirichlet Series. LNM 216, Berlin (1971)
Paradan, P.-E.: Symmetric Spaces of the Non-compact Type: Lie Groups, http://math.univ-lyon1.fr/~emy/smf_sec_18_02
Shimura G.: Arithmetic of Differential Operators on Symmetric Domains. Duke Math. J. 48, 813–843 (1981)
Geer G. van der: Siegel modular forms and their applications. In: Ranestad, K. (ed.) The 1-2-3 of Modular Forms : Lectures at a Summer School in Nordfjordeid, Norway., pp. 181–245. Universitext, Springer, Berlin (2008)
Weissauer R.: Vektorwertige Siegelsche Modulformen Kleinen Gewichtes. J. Reine Angew. Math. 343, 184–202 (1983)
Weissauer, R.: Divisors of the Siegel Modular Variety. pp. 304–324. LNM 1240, Berlin (1987)
Yang, J., Yin, L.S.: Derivation of Jacobi forms from connetions. arXiv:1301.1156
Zagier D.: Elliptic modular forms and their applications. In: Ranestad, K. (ed.) The 1-2-3 of Modular Forms: Lectures at a Summer School in Nordfjordeid, Norway, Universitext, pp. 1–103. Springer, Berlin (2008)
Author information
Authors and Affiliations
Corresponding author
Additional information
This paper was partially supported by NSFC No. 11271212.
Rights and permissions
About this article
Cite this article
Yang, E., Yin, L. Derivatives of Siegel modular forms and modular connections. manuscripta math. 146, 65–84 (2015). https://doi.org/10.1007/s00229-014-0687-5
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00229-014-0687-5