An explicit formula for the Hilbert symbol of a formal group
Annales de l'Institut Fourier, Volume 61 (2011) no. 1, pp. 261-318.

A Brückner-Vostokov formula for the Hilbert symbol of a formal group was established by Abrashkin under the assumption that roots of unity belong to the base field. The main motivation of this work is to remove this hypothesis. It is obtained by combining methods of (ϕ,Γ)-modules and a cohomological interpretation of Abrashkin’s technique. To do this, we build (ϕ,Γ)-modules adapted to the false Tate curve extension and generalize some related tools like the Herr complex with explicit formulas for the cup-product and the Kummer map.

Abrashkin a établi une formule de Brueckner-Vostokov pour le symbole de Hilbert d’un groupe formel sous la condition d’appartenance de racines de l’unité au corps de base. La motivation première de ce travail réside en la suppression de cette hypothèse. On l’obtient en combinant des méthodes de (ϕ,Γ)-modules et une interprétation cohomologique des techniques d’Abrashkin. Pour cela, on construit des (ϕ,Γ)-modules adaptés à l’extension dite de la fausse courbe de Tate et on généralise des outils tels que le complexe de Herr avec des formules explicites pour le cup-produit et l’application de Kummer.

Received:
Revised:
Accepted:
DOI: 10.5802/aif.2602
Classification: 11F80,  11S25,  14L05,  11S31,  11S23,  14F30
Keywords: p-adic representations, (φ,Γ)-modules, formal groups, explicit reciprocity law
Tavares Ribeiro, Floric 1

1 Université de Franche-Comté Département de Mathématiques 16 route de Gray 25000 Besançon (France)
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Tavares Ribeiro, Floric. An explicit formula for the Hilbert symbol of a formal group. Annales de l'Institut Fourier, Volume 61 (2011) no. 1, pp. 261-318. doi : 10.5802/aif.2602. https://aif.centre-mersenne.org/articles/10.5802/aif.2602/

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