Potentially crystalline deformation rings in the ordinary case

Levin, Brandon; Morra, Stefano

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 $\overline{\rho }:G_{{\mathbf{Q}}_p}\to {\mathbf{GL}}_3\left({\mathbf{F}}_p\right)$. 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 $\overline{\rho }$ et un seul. Nous montrons que dans cette situation l’espace de déformation potentiellement cristallin est formellement lisse sur ${\mathbf{Z}}_p$ et que tout relèvement potentiellement cristallin de $\overline{\rho }$ 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 ${\mathbf{F}}_p$.

We study potentially crystalline deformation rings for a residual, ordinary Galois representation $\overline{\rho }:G_{{\mathbf{Q}}_p}\to {\mathbf{GL}}_3\left({\mathbf{F}}_p\right).$ We consider deformations with Hodge-Tate weights $(0,1,2)$ and inertial type chosen to contain exactly one Fontaine-Laffaille modular weight for $\overline{\rho }$. We show that, in this setting, the potentially crystalline deformation space is formally smooth over ${\mathbf{Z}}_p$ 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.

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

[1] Breuil, Christophe Représentations semi-stables et modules fortement divisibles, Invent. Math., Volume 136 (1999) no. 1, pp. 89-122 | DOI

[2] Breuil, Christophe Une application de corps des normes, Compositio Math., Volume 117 (1999) no. 2, pp. 189-203 | DOI

[3] Breuil, Christophe Sur quelques représentations modulaires et $p$-adiques de ${\mathrm{GL}}_2\left({\mathbf{Q}}_p\right)$. I, Compositio Math., Volume 138 (2003) no. 2, pp. 165-188 | DOI

[4] Breuil, Christophe Sur un problème de compatibilité local-global modulo $p$ pour ${\mathrm{GL}}_2$, J. Reine Angew. Math., Volume 692 (2014), pp. 1-76 | DOI

[5] Breuil, Christophe; Herzig, Florian Ordinary representations of $G\left({\mathbb{Q}}_p\right)$ and fundamental algebraic representations, Duke Math. J., Volume 164 (2015) no. 7, pp. 1271-1352 | DOI

[6] Breuil, Christophe; Mézard, Ariane Multiplicités modulaires et représentations de ${\mathrm{GL}}_2\left({\mathbf{Z}}_p\right)$ et de $\mathrm{Gal}({\overline{\mathbf{Q}}}_p/{\mathbf{Q}}_p)$ en $l=p$, Duke Math. J., Volume 115 (2002) no. 2, pp. 205-310 (With an appendix by Guy Henniart) | DOI

[7] Breuil, Christophe; Mézard, Ariane Multiplicités modulaires raffinées, Bull. Soc. Math. France, Volume 142 (2014) no. 1, pp. 127-175

[8] Breuil, Christophe; Paškūnas, Vytautas Towards a modulo $p$ Langlands correspondence for ${\mathbf{GL}}_2$, 2016, Memoirs of Amer. Math. Soc., 2013

[9] Buzzard, Kevin; Diamond, Fred; Jarvis, Frazer On Serre’s conjecture for mod $\ell$ Galois representations over totally real fields, Duke Math. J., Volume 155 (2010) no. 1, pp. 105-161 | DOI

[10] Caruso, Xavier; Liu, Tong Quasi-semi-stable representations, Bull. Soc. Math. France, Volume 137 (2009) no. 2, pp. 185-223

[11] Conrad, Brian; Diamond, Fred; Taylor, Richard Modularity of certain potentially Barsotti-Tate Galois representations, J. Amer. Math. Soc., Volume 12 (1999) no. 2, pp. 521-567 | DOI

[12] Emerton, Matthew; Gee, Toby A geometric perspective on the Breuil-Mézard conjecture, J. Inst. Math. Jussieu, Volume 13 (2014) no. 1, pp. 183-223 | DOI

[13] Emerton, Matthew; Gee, Toby; Herzig, Florian Weight cycling and Serre-type conjectures for unitary groups, Duke Math. J., Volume 162 (2013) no. 9, pp. 1649-1722 | DOI

[14] Emerton, Matthew; Gee, Toby; Savitt, David Lattices in the cohomology of Shimura curves, Invent. Math., Volume 200 (2015) no. 1, pp. 1-96 | DOI

[15] Fontaine, Jean-Marc Représentations $p$-adiques des corps locaux. II, The Grothendieck Festschrift, Vol. II (Progr. Math.), Volume 87, Birkhäuser Boston, Boston, MA, 1990, pp. 249-309

[16] Gao, H. Personal communication, email of November 1, 2015

[17] Gee, Toby Automorphic lifts of prescribed types, Math. Ann., Volume 350 (2011) no. 1, pp. 107-144 | DOI

[18] Gee, Toby; Herzig, Florian; Savitt, David General Serre weight conjectures (2015) (preprint, http://arxiv.org/abs/1509.02527)

[19] Gee, Toby; Kisin, Mark The Breuil-Mézard conjecture for potentially Barsotti-Tate representations, Forum Math. Pi, Volume 2 (2014), e1, 56 pages | DOI

[20] Herzig, Florian The weight in a Serre-type conjecture for tame $n$-dimensional Galois representations, Duke Math. J., Volume 149 (2009) no. 1, pp. 37-116 | DOI

[21] Herzig, Florian; Le, D.; Morra, Stefano On local/global compatibility for ${\mathbf{GL}}_3$ in the ordinary case (2015) (in preparation)

[22] Kisin, Mark Crystalline representations and $F$-crystals, Algebraic geometry and number theory (Progr. Math.), Volume 253, Birkhäuser Boston, Boston, MA, 2006, pp. 459-496 | DOI

[23] Kisin, Mark Potentially semi-stable deformation rings, J. Amer. Math. Soc., Volume 21 (2008) no. 2, pp. 513-546 | DOI

[24] Kisin, Mark Moduli of finite flat group schemes, and modularity, Ann. of Math. (2), Volume 170 (2009) no. 3, pp. 1085-1180 | DOI

[25] Le, D.; Le Hung, Bao Viet; Levin, Brandon; Morra, Stefano Potentially crystalline deformation rings and Serre weight conjectures (Shapes and Shadows) (2015) (in preparation)

[26] Liu, Tong On lattices in semi-stable representations: a proof of a conjecture of Breuil, Compos. Math., Volume 144 (2008) no. 1, pp. 61-88 | DOI

[27] Savitt, David On a conjecture of Conrad, Diamond, and Taylor, Duke Math. J., Volume 128 (2005) no. 1, pp. 141-197 | DOI

[28] Serre, Jean-Pierre Sur les représentations modulaires de degré $2$ de $\mathrm{Gal}(\overline{\mathbf{Q}}/\mathbf{Q})$, Duke Math. J., Volume 54 (1987) no. 1, pp. 179-230 | DOI