Formal deformations of the algebra of Jacobi forms and Rankin–Cohen brackets

YoungJu Choie; François Dumas; François Martin; Emmanuel Royer

doi:10.5802/crmath.193

Algèbre, Théorie des nombres

Formal deformations of the algebra of Jacobi forms and Rankin–Cohen brackets
[Déformations formelles de l’algèbre des formes de Jacobi et crochets de Rankin–Cohen]

YoungJu Choie ¹ ; François Dumas ² ; François Martin ² ; Emmanuel Royer ²

¹ Pohang University of Science and Technology, Department of Mathematics, Pohang, Korea
² Université Clermont Auvergne, CNRS, LMBP, F-63000 Clermont-Ferrand, France

Comptes Rendus. Mathématique, Volume 359 (2021) no. 4, pp. 505-521.

Résumé
Abstract

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.

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.

Reçu le : 2020-12-03
Révisé le : 2021-02-18
Accepté le : 2021-02-18
Publié le : 2021-06-17

Zbl

DOI : 10.5802/crmath.193

Classification : 53D55, 17B63, 11F25, 11F11, 16W25

Affiliations des auteurs :

YoungJu Choie ¹ ; François Dumas ² ; François Martin ² ; Emmanuel Royer ²

¹ Pohang University of Science and Technology, Department of Mathematics, Pohang, Korea
² Université Clermont Auvergne, CNRS, LMBP, F-63000 Clermont-Ferrand, France

Licence :

CC-BY 4.0

Droits d'auteur : Les auteurs conservent leurs droits

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     author = {YoungJu Choie and Fran\c{c}ois Dumas and Fran\c{c}ois Martin and Emmanuel Royer},
     title = {Formal deformations of the algebra of {Jacobi} forms and {Rankin{\textendash}Cohen} brackets},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {505--521},
     publisher = {Acad\'emie des sciences, Paris},
     volume = {359},
     number = {4},
     year = {2021},
     doi = {10.5802/crmath.193},
     zbl = {07362171},
     language = {en},
}

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YoungJu Choie; François Dumas; François Martin; Emmanuel Royer. 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. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.5802/crmath.193/

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