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

Published online:
DOI: 10.5802/aif.3072
Classification: 11S20, 20G15
Keywords: $p$-adic Hodge theory, deformation rings, algebraic groups
Mot clés : théorie d’Hodge $p$-adique, anneaux de déformations, groupes algébrique
Bellovin, Rebecca 1

1 Imperial College, London Department of Mathematics 180 Queen’s Gate London SW7 2AZ (UK)
     author = {Bellovin, Rebecca},
     title = {Generic smoothness for $G$-valued potentially semi-stable deformation rings},
     journal = {Annales de l'Institut Fourier},
     pages = {2565--2620},
     publisher = {Association des Annales de l{\textquoteright}institut Fourier},
     volume = {66},
     number = {6},
     year = {2016},
     doi = {10.5802/aif.3072},
     language = {en},
     url = {}
AU  - Bellovin, Rebecca
TI  - Generic smoothness for $G$-valued potentially semi-stable deformation rings
JO  - Annales de l'Institut Fourier
PY  - 2016
SP  - 2565
EP  - 2620
VL  - 66
IS  - 6
PB  - Association des Annales de l’institut Fourier
UR  -
DO  - 10.5802/aif.3072
LA  - en
ID  - AIF_2016__66_6_2565_0
ER  - 
%0 Journal Article
%A Bellovin, Rebecca
%T Generic smoothness for $G$-valued potentially semi-stable deformation rings
%J Annales de l'Institut Fourier
%D 2016
%P 2565-2620
%V 66
%N 6
%I Association des Annales de l’institut Fourier
%R 10.5802/aif.3072
%G en
%F 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.

[1] Balaji, Sundeep G-valued potentially semi-stable deformation rings, ProQuest LLC, Ann Arbor, MI, 2012, 71 pages Thesis (Ph.D.)–The University of Chicago

[2] Bellovin, Rebecca p-adic Hodge theory in rigid analytic families, Algebra Number Theory, Volume 9 (2015) no. 2, pp. 371-433 | DOI

[3] Conrad, Brian Generic fibers of deformation rings (

[4] Conrad, Brian Irreducible components of rigid spaces, Ann. Inst. Fourier (Grenoble), Volume 49 (1999) no. 2, pp. 473-541 | DOI

[5] Conrad, Brian Reductive group schemes, Proceedings of the SGA3 Summer School (Luminy, Aug.-Sept. 2011), Volume 1 (Panoramas et Synthèses), Societé Mathematique de France, 2014 (To appear. Available at

[6] Deligne, Pierre; Milne, J.S. Tannakian Categories, Hodge Cycles, Motives, and Shimura Varieties (LNM), Volume 900, Springer, New York, 1982, pp. 101-228

[7] Demazure, Michel; Grothendieck, Alexandre Séminaire de géométrie algébrique du Bois Marie 1962/64; Schémas en groupes, 3, Springer, 1970

[8] Hartl, Urs; Hellmann, Eugen The universal family of semi-stable p-adic Galois representations (Preprint.

[9] Humphreys, James E. Linear algebraic groups, Springer-Verlag, New York, 1975, xiv+247 pages (Graduate Texts in Mathematics, No. 21)

[10] Humphreys, James E. Conjugacy classes in semisimple algebraic groups, Mathematical Surveys and Monographs, 43, American Mathematical Society, Providence, RI, 1995, xviii+196 pages

[11] Jantzen, Jens Carsten Nilpotent orbits in representation theory, Lie theory (Progr. Math.), Volume 228, Birkhäuser Boston, Boston, MA, 2004, pp. 1-211

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

[13] Kisin, Mark The Fontaine-Mazur conjecture for GL 2 , J. Amer. Math. Soc., Volume 22 (2009) no. 3, pp. 641-690 | DOI

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

[15] Kisin, Mark e-mail to Brian Conrad, 2012

[16] Levy, Paul; McNinch, George; Testerman, Donna M. Nilpotent subalgebras of semisimple Lie algebras, C. R. Math. Acad. Sci. Paris, Volume 347 (2009) no. 9-10, pp. 477-482 | DOI

[17] Matsumura, Hideyuki Commutative Ring Theory, Cambridge Studies in Advanced Mathematics, 8, Cambridge University Press, Cambridge, 1989, xiv+320 pages (Translated from the Japanese by M. Reid)

[18] Mazur, Barry An introduction to the deformation theory of Galois representations, Modular forms and Fermat’s last theorem (Boston, MA, 1995), Springer, New York, 1997, pp. 243-311

[19] McNinch, George J. Nilpotent orbits over ground fields of good characteristic, Math. Ann., Volume 329 (2004) no. 1, pp. 49-85 | DOI

[20] Saavedra Rivano, Neantro Catégories Tannakiennes, Lecture Notes in Mathematics, Vol. 265, Springer-Verlag, Berlin, 1972, ii+418 pages

[21] Weyman, Jerzy Cohomology of Vector Bundles and Syzygies, Cambridge Tracts in Mathematics, 149, Cambridge University Press, Cambridge, 2003, xiv+371 pages | DOI

Cited by Sources: