Potential diagonalisability of pseudo-Barsotti–Tate representations

Robin Bartlett

doi:10.5802/jtnb.1248

Robin Bartlett ¹

¹ University of Münster Orléans-Ring 10 48149 Münster, Germany

Journal de théorie des nombres de Bordeaux, Tome 35 (2023) no. 2, pp. 335-371.

Résumé
Abstract

Des travaux antérieurs de Kisin et Gee prouvent la diagonalisabilité potentielle des représentations de Barsotti–Tate de dimension 2 du groupe de Galois d’une extension finie $K / ℚ_{p}$ . Dans cet article, nous nous appuyons sur leur travail en remplaçant la condition de Barsotti–Tate par une condition plus faible que nous appelons pseudo-Barsotti–Tate (ce qui signifie que pour certains plongements $κ : K \to {\bar{ℚ}}_{p}$ les poids de Hodge–Tate relativement à $κ$ sont autorisés à être dans l’intervalle $[0, p]$ plutôt que dans $[0, 1]$ ).

Previous work of Kisin and Gee proves potential diagonalisability of two dimensional Barsotti–Tate representations of the Galois group of a finite extension $K / ℚ_{p}$ . In this paper we build upon their work by relaxing the Barsotti–Tate condition to one we call pseudo-Barsotti–Tate (which means that for certain embeddings $κ : K \to {\bar{ℚ}}_{p}$ we allow the $κ$ -Hodge–Tate weights to be contained in $[0, p]$ rather than $[0, 1]$ ).

Reçu le : 2021-03-27
Révisé le : 2022-06-26
Accepté le : 2023-01-26
Publié le : 2023-10-10

DOI : 10.5802/jtnb.1248

Classification : 11F80, 11F33
Mots clés : Galois representations, integral $p$-adic Hodge theory, deformation rings

Affiliations des auteurs :

Robin Bartlett ¹

¹ University of Münster Orléans-Ring 10 48149 Münster, Germany

Licence :

CC-BY-ND 4.0

Droits d'auteur : Les auteurs conservent leurs droits

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     title = {Potential diagonalisability of {pseudo-Barsotti{\textendash}Tate} representations},
     journal = {Journal de th\'eorie des nombres de Bordeaux},
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     publisher = {Soci\'et\'e Arithm\'etique de Bordeaux},
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Robin Bartlett. Potential diagonalisability of pseudo-Barsotti–Tate representations. Journal de théorie des nombres de Bordeaux, Tome 35 (2023) no. 2, pp. 335-371. doi : 10.5802/jtnb.1248. https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.1248/

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