no. 5
Potentially crystalline deformation rings in the ordinary case
[Anneaux de déformation potentiellement cristallins dans le cas ordinaire]
Levin, Brandon1 ; Morra, Stefano2
1 The University of Chicago 5734 S. University Avenue Chicago, Illinois 60637 (USA)
2 Institut Montpelliérain A. Grothendieck Université de Montpellier Cc 051, Place E. Bataillon 34095 Montpellier Cedex (France)
Annales de l'Institut Fourier, Tome 66 (2016) no. 5, pp. 1923-1964.

Nous étudions les anneaux de déformation potentiellement cristallins pour une représentation Galoisienne ordinaire ρ ¯:G Q p →GL 3 (F p ). Nous considérons des déformations à poids de Hodge-Tate (0,1,2) et type inertiel choisi de telle sorte qu’il contient un poids Fontaine-Laffaille pour ρ ¯ et un seul. Nous montrons que dans cette situation l’espace de déformation potentiellement cristallin est formellement lisse sur Z p et que tout relèvement potentiellement cristallin de ρ ¯ est ordinaire. La preuve nécessite une étude fine des conditions imposées par l’opérateur de monodromie sur les modules de Breuil avec donnée de descente, en particulier que la fibre spéciale du lieu de monodromie est formellement lisse sur F p .

We study potentially crystalline deformation rings for a residual, ordinary Galois representation ρ ¯:G Q p →GL 3 (F p ). We consider deformations with Hodge-Tate weights (0,1,2) and inertial type chosen to contain exactly one Fontaine-Laffaille modular weight for ρ ¯. We show that, in this setting, the potentially crystalline deformation space is formally smooth over Z p and any potentially crystalline lift is ordinary. The proof requires an understanding of the condition imposed by the monodromy operator on Breuil modules with descent datum, in particular, that this locus mod p is formally smooth.

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DOI : 10.5802/aif.3053
Classification : 11F33
Keywords: potentially crystalline deformation rings, Serre-type conjectures, integral $p$-adic Hodge theory
Mot clés : Anneaux de déformation potentiellement cristallins, conjectures de type Serre, théorie de Hodge $p$-adique entière

Levin, Brandon 1 ; Morra, Stefano 2

1 The University of Chicago 5734 S. University Avenue Chicago, Illinois 60637 (USA)
2 Institut Montpelliérain A. Grothendieck Université de Montpellier Cc 051, Place E. Bataillon 34095 Montpellier Cedex (France) 
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Levin, Brandon; Morra, Stefano. Potentially crystalline deformation rings in the ordinary case. Annales de l'Institut Fourier, Tome 66 (2016) no. 5, pp. 1923-1964. doi : 10.5802/aif.3053. https://aif.centre-mersenne.org/articles/10.5802/aif.3053/

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