Generic smoothness for G-valued potentially semi-stable deformation rings
Annales de l'Institut Fourier, Volume 66 (2016) no. 6, p. 2565-2620

We extend Kisin’s results on the structure of characteristic 0 Galois deformation rings to deformation rings of Galois representations valued in arbitrary connected reductive groups G. In particular, we show that such Galois deformation rings are complete intersections. In addition, we study explicitly the structure of the moduli space X ϕ,N of (framed) (ϕ,N)-modules when G=GL n . We show that when G=GL 3 and K 0 = p , X ϕ,N has a singular irreducible component, and we construct a moduli-theoretic resolution of singularities.

Nous étendons les résultats de Kisin sur la structure des anneaux de déformations de représentations galoisiennes en caractéristique 0 aux anneaux de déformations de représentations galoisiennes à valeurs dans des groupes connexes reductifs G. En particulier, nous prouvons que ces anneaux de déformations de représentations galoisiennes sont d’intersection complète. De plus, nous étudions la structure de l’espace de modules X ϕ,N des (ϕ,N)-modules (cadrés) quand G=GL n . Nous prouvons que X ϕ,N a une composante irréductible singulière quand G=GL 3 , et nous construisons une résolution des singularités avec interprétation modulaire.

Received : 2014-04-21
Revised : 2015-01-20
Accepted : 2015-05-07
Published online : 2016-10-04
DOI : https://doi.org/10.5802/aif.3072
Classification:  11S20,  20G15
Keywords: p-adic Hodge theory, deformation rings, algebraic groups
@article{AIF_2016__66_6_2565_0,
     author = {Bellovin, Rebecca},
     title = {Generic smoothness for $G$-valued potentially semi-stable deformation rings},
     journal = {Annales de l'Institut Fourier},
     publisher = {Association des Annales de l'institut Fourier},
     volume = {66},
     number = {6},
     year = {2016},
     pages = {2565-2620},
     doi = {10.5802/aif.3072},
     language = {en},
     url = {http://www.numdam.org/item/AIF_2016__66_6_2565_0}
}
Bellovin, Rebecca. Generic smoothness for $G$-valued potentially semi-stable deformation rings. Annales de l'Institut Fourier, Volume 66 (2016) no. 6, pp. 2565-2620. doi : 10.5802/aif.3072. http://www.numdam.org/item/AIF_2016__66_6_2565_0/

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