Abstract
Don Zagier introduced and discussed in Zagier [Proc Indian Acad Sci (Math Sci) 104(1) 57–75, 1994] a particular algebraic structure of the graded ring of modular forms. In this note we interpret it in terms of an associative deformation of this graded ring.
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References
Ban K.: On Rankin–Cohen–Ibukiyama operators for automorphic forms of several variables. Comment. Math. Univ. St. Pauli 55(2), 149–171 (2006)
Bayen F., Flato M., Fronsdal C., Lischnerowicz A., Sternheimer D.: Deformation theory and Quantization. Ann. Phys. 111, 61–110 (1978)
Bieliavsky P., Tang X., Yao Y.: Rankin–Cohen brackets and formal quantization. Adv. Math. 212(1), 293–314 (2007)
Choie Y., Mourrain B., Sole P.: Rankin–Cohen brackets and invariant theory. J. Algebr. Comb. 13, 5–13 (2001)
Cohen H.: Sums involving the values at negative integers of L-functions of quadratic characters. Math. Ann. 217, 271–295 (1975)
Cohen, P.B., Manin, Y., Zagier, D.: Automorphic pseudodifferential operators. In: Algebraic Aspects of Integrable Systems. Progr. Nonlinear Differential Equations Appl., vol. 26, pp. 17–47. Birkhäuser, Boston (1997)
van Dijk G., Pevzner M.: Ring structures for holomorphic discrete series and Rankin–Cohen brackets. J. Lie Theory 17, 283–305 (2007)
Eholzer, W., Ibukiyama, T.: Rankin–Cohen type differential operators for Siegel modular forms. Int. J. Math. 9(4), 443–463 (1998); 217, 271–295 (1975)
El Gradechi A.: The Lie theory of the Rankin–Cohen brackets and allied bi-differential operators. Adv. Math. 207(2), 484–531 (2006)
Faraut J.: Distributions sphériques sur les espaces hyperboliques. J. Math. Pures Appl. 58, 369–444 (1979)
Gordan P.: Invariantentheorie. Teubner, Leipzig (1887)
Howe R., Tan E.: Non-Abelian Harmonic Analysis. Applications of \({{SL}(2,\mathbb {R})}\) , Universitext. Springer, New York (1992)
Rankin R.A.: The construction of automorphic forms from the derivatives of a given form. J. Indian Math. Soc. 20, 103–116 (1956)
Molchanov V.F.: Tensor products of unitary representations of the three-dimensional Lorentz group. Math. USSR Izv. 15, 113–143 (1980)
Olver P.J.: Classical Invariant Theory. London Math Society Student Texts, vol. 44. Cambridge University Press, London (1999)
Olver P.J., Sanders J.A.: Transvectants, modular forms, and the Heisenberg algebra. Adv. Appl. Math. 25, 252–283 (2000)
Ovsienko V.: Exotic deformation quantization. J. Differ. Geom. 45, 390–406 (1997)
Repka J.: Tensor products of holomorphic discrete series representations. Can. J. Math. 31, 836–844 (1979)
Strichartz R.S.: Harmonic analysis on hyperboloids. J. Funct. Anal. 12, 218–235 (1973)
Unterberger A., Unterberger J.: Algebras of symbols and modular forms. J. Anal. Math. 68, 121–143 (1996)
Yau, Y.: Autour des déformations de Rankin–Cohen. Ph.D. thesis, École Polytechnique, Paris (2007)
Zagier D.: Modular forms and differential operators. Proc. Indian Acad. Sci. (Math. Sci.) 104(1), 57–75 (1994)
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The author is grateful to Ph. Bonneau, A. Weinstein and D. Zagier for fruitful discussions.
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Pevzner, M. Rankin–Cohen Brackets and Associativity. Lett Math Phys 85, 195–202 (2008). https://doi.org/10.1007/s11005-008-0266-3
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DOI: https://doi.org/10.1007/s11005-008-0266-3