Algebra, Number theory
Formal deformations of the algebra of Jacobi forms and Rankin–Cohen brackets
Comptes Rendus. Mathématique, Volume 359 (2021) no. 4, pp. 505-521.

This work is devoted to the algebraic and arithmetic properties of Rankin–Cohen brackets allowing to define and study them in several natural situations of number theory. It focuses on the property of these brackets to be formal deformations of the algebras on which they are defined, with related questions on restriction-extension methods. The general algebraic results developed here are applied to the study of formal deformations of the algebra of weak Jacobi forms and their relation with the Rankin–Cohen brackets on modular and quasimodular forms.

Ce travail est consacré aux propriétés algébriques et arithmétiques des crochets de Rankin–Cohen permettant de les définir et de les étudier dans plusieurs situations naturelles de la théorie des nombres. Il se concentre sur la propriété qu’ont ces crochets d’être des déformations formelles des algèbres sur lesquelles ils sont définis, avec des questions connexes sur les méthodes de restriction-extension. Les résultats algébriques généraux développés ici sont appliqués à l’étude des déformations formelles de l’algèbre des formes de Jacobi faibles et leur relation avec les crochets de Rankin–Cohen pour les formes modulaires et quasimodulaires.

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DOI: 10.5802/crmath.193
Classification: 53D55, 17B63, 11F25, 11F11, 16W25
Choie, YoungJu 1; Dumas, François 2; Martin, François 2; Royer, Emmanuel 2

1 Pohang University of Science and Technology, Department of Mathematics, Pohang, Korea
2 Université Clermont Auvergne, CNRS, LMBP, F-63000 Clermont-Ferrand, France
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Choie, YoungJu; Dumas, François; Martin, François; Royer, Emmanuel. Formal deformations of the algebra of Jacobi forms and Rankin–Cohen brackets. Comptes Rendus. Mathématique, Volume 359 (2021) no. 4, pp. 505-521. doi : 10.5802/crmath.193. http://www.numdam.org/articles/10.5802/crmath.193/

[1] Bieliavsky, Pierre; Tang, Xiang; Yao, Yi-Jun Rankin–Cohen brackets and formal quantization, Adv. Math., Volume 212 (2007) no. 1, pp. 293-314 | DOI | MR | Zbl

[2] Choie, YoungJu; Dumas, François; Martin, François; Royer, Emmanuel A derivation on Jacobi forms: Oberdieck derivation (Unpublished note. Available at https://hal.archives-ouvertes.fr/hal-03132764)

[3] Choie, YoungJu; Eholzer, Wolfgang Rankin–Cohen operators for Jacobi and Siegel forms, J. Number Theory, Volume 68 (1998) no. 2, pp. 160-177 | DOI | MR | Zbl

[4] Choie, YoungJu; Eholzer, Wolfgang Jacobi forms and generalized RC-algebras, Rocky Mt. J. Math., Volume 31 (2001) no. 4, pp. 1265-1275 | DOI | MR | Zbl

[5] Cohen, Paula Beazley; Manin, Yuri; Zagier, Don Automorphic pseudodifferential operators, Algebraic aspects of integrable systems (Progress in Nonlinear Differential Equations and their Applications), Volume 26, Birkhäuser, 1997, pp. 17-47 | DOI | MR | Zbl

[6] Connes, Alain; Moscovici, Henri Rankin–Cohen brackets and the Hopf algebra of transverse geometry, Mosc. Math. J., Volume 4 (2004) no. 1, pp. 111-130 | DOI | MR | Zbl

[7] van Dijk, Gerrit; Pevzner, Michael Ring structures for holomorphic discrete series and Rankin–Cohen brackets, J. Lie Theory, Volume 17 (2007) no. 2, pp. 283-305 | MR | Zbl

[8] Dumas, François; Royer, Emmanuel Poisson structures and star products on quasimodular forms, Algebra Number Theory, Volume 8 (2014) no. 5, pp. 1127-1149 | DOI | MR | Zbl

[9] Eichler, Martin; Zagier, Don The theory of Jacobi forms, Progress in Mathematics, 55, Birkhäuser, 1985, v+148 pages | DOI | MR | Zbl

[10] El Gradechi, Amine M. The Lie theory of the Rankin–Cohen brackets and allied bi-differential operators, Adv. Math., Volume 207 (2006) no. 2, pp. 484-531 | DOI | MR | Zbl

[11] Kobayashi, Toshiyuki; Pevzner, Michael Differential symmetry breaking operators: II. Rankin–Cohen operators for symmetric pairs, Sel. Math., New Ser., Volume 22 (2016) no. 2, pp. 847-911 | DOI | MR | Zbl

[12] Laurent-Gengoux, Camille; Pichereau, Anne; Vanhaecke, Pol Poisson structures, Grundlehren der Mathematischen Wissenschaften, 347, Springer, 2013, xxiv+461 pages | DOI | MR | Zbl

[13] Martin, François; Royer, Emmanuel Formes modulaires et périodes, Formes modulaires et transcendance (Séminaires et Congrès), Volume 12, Société Mathématique de France, 2005, pp. 1-117 | MR | Zbl

[14] Oberdieck, Georg A Serre derivative for even weight Jacobi Forms (2014) (https://arxiv.org/abs/1209.5628)

[15] Omori, Hideki; Maeda, Yoshiaki; Miyazaki, Naoya; Yoshioka, Akira Deformation quantizations of the Poisson algebra of Laurent polynomials, Lett. Math. Phys., Volume 46 (1998) no. 2, pp. 171-180 | DOI | MR | Zbl

[16] Ovsienko, Valentin Exotic deformation quantization, J. Differ. Geom., Volume 45 (1997) no. 2, pp. 390-406 | MR | Zbl

[17] Ovsienko, Valentin; Redou, Pascal Generalized transvectants-Rankin–Cohen brackets, Lett. Math. Phys., Volume 63 (2003) no. 1, pp. 19-28 | DOI | MR | Zbl

[18] Pevzner, Michael Rankin–Cohen brackets and associativity, Lett. Math. Phys., Volume 85 (2008) no. 2-3, pp. 195-202 | DOI | MR | Zbl

[19] Pevzner, Michael A generating function for Rankin–Cohen brackets, Lett. Math. Phys., Volume 108 (2018) no. 12, pp. 2627-2633 | DOI | MR | Zbl

[20] Royer, Emmanuel Quasimodular forms: an introduction, Ann. Math. Blaise Pascal, Volume 19 (2012) no. 2, pp. 297-306 | DOI | Numdam | MR | Zbl

[21] Unterberger, André; Unterberger, Julianne Algebras of symbols and modular forms, J. Anal. Math., Volume 68 (1996), pp. 121-143 | DOI | MR | Zbl

[22] Yao, Yi-Jun Autour des déformations de Rankin–Cohen, Ph. D. Thesis, École Polytechnique, Paris (2007) | Zbl

[23] Zagier, Don Modular forms and differential operators, Proc. Indian Acad. Sci., Math. Sci., Volume 104 (1994) no. 1, pp. 57-75 (K. G. Ramanathan memorial issue) | DOI | MR | Zbl

[24] Zagier, Don Elliptic modular forms and their applications, The 1-2-3 of modular forms (Universitext), Springer, 2008, pp. 1-103 | DOI | MR | Zbl

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