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Computation of potentially Barsotti-Tate deformation rings. (Un calcul d’anneaux de déformations potentiellement Barsotti-Tate.) (French. English summary) Zbl 1442.11087
Summary: Let \( F\) be an unramified extension of \( \mathbb{Q}_{p}\). The first aim of this work is to develop a purely local method to compute the potentially Barsotti-Tate deformation rings with tame Galois type of level \( [F : \mathbb{Q}_{p}] \) of irreducible two-dimensional representations of the absolute Galois group of \( F\). We then apply our method in the particular case where \( F\) has degree \( 2\) over \( \mathbb{Q}_{p}\) and determine in this way almost all these deformation rings. In this particular case, we observe a close relationship between the structure of these deformation rings and the geometry of the associated Kisin variety. As a corollary and still assuming that \( F\) has degree \( 2\) over \( \mathbb{Q}_{p}\), we prove, except in two very particular cases, a conjecture of Kisin which predicts that intrinsic Galois multiplicities are all equal to 0 or \( 1\).

MSC:
11F80 Galois representations
11S37 Langlands-Weil conjectures, nonabelian class field theory
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[1] Barthel, L.; Livn\'e, R., Irreducible modular representations of \({\rm GL}_2\) of a local field, Duke Math. J., 75, 2, 261-292, (1994) · Zbl 0826.22019
[2] Berger, Laurent, La correspondance de Langlands locale \(p\)-adique pour \({\rm GL}_2(\textbf{Q}_p)\), Ast\'erisque, 339, Exp. No. 1017, viii, 157-180, (2011) · Zbl 1356.11082
[3] Breuil, Christophe, Une application de corps des normes, Compositio Math., 117, 2, 189-203, (1999) · Zbl 0933.11055
[4] Breuil, Christophe, Sur quelques repr\'esentations modulaires et \(p\)-adiques de \({\rm GL}_2(\mathbf{Q}_p)\). I, Compositio Math., 138, 2, 165-188, (2003) · Zbl 1044.11041
[5] Breuil, Christophe, Sur un probl\`eme de compatibilit\'e local-global modulo \(p\) pour \({\rm GL}_2\), J. Reine Angew. Math., 692, 1-76, (2014) · Zbl 1314.11036
[6] Breuil, Christophe, Correspondance de Langlands \(p\)-adique, compatibilit\'e local-global et applications [d’apr\`es Colmez, Emerton, Kisin, \(…\)], Ast\'erisque, 348, Exp. No. 1031, viii, 119-147, (2012) · Zbl 1280.14006
[7] Breuil, Christophe; Conrad, Brian; Diamond, Fred; Taylor, Richard, On the modularity of elliptic curves over \(\mathbf{Q}\): wild 3-adic exercises, J. Amer. Math. Soc., 14, 4, 843-939 (electronic), (2001) · Zbl 0982.11033
[8] Breuil, Christophe; M\'ezard, Ariane, Multiplicit\'es modulaires et repr\'esentations de \({\rm GL}_2(\textbf{Z}_p)\) et de \({\rm Gal}(\overline\textbf{Q}_p/\textbf{Q}_p)\) en \(l=p\), Duke Math. J., 115, 2, 205-310, (2002) · Zbl 1042.11030
[9] Breuil, Christophe; M\'ezard, Ariane, Multiplicit\'es modulaires raffin\'ees, Bull. Soc. Math. France, 142, 1, 127-175, (2014) · Zbl 1311.11044
[10] Breuil, Christophe; Paskunas, Vytautas, Towards a modulo \(p\) Langlands correspondence for \({\rm GL}_2\), Mem. Amer. Math. Soc., 216, 1016, vi+114 pp., (2012) · Zbl 1245.22010
[11] Buzzard, Kevin; Diamond, Fred; Jarvis, Frazer, On Serre’s conjecture for mod \(ℓ\) Galois representations over totally real fields, Duke Math. J., 155, 1, 105-161, (2010) · Zbl 1227.11070
[12] Chang, Seunghwan; Diamond, Fred, Extensions of rank one \((ϕ,Γ)\)-modules and crystalline representations, Compos. Math., 147, 2, 375-427, (2011) · Zbl 1235.11105
[13] Clozel, Laurent; Harris, Michael; Taylor, Richard, Automorphy for some \(l\)-adic lifts of automorphic mod \(l\) Galois representations, Publ. Math. Inst. Hautes \'Etudes Sci., 108, 1-181, (2008) · Zbl 1169.11020
[14] Colmez, Pierre, Repr\'esentations de \({\rm GL}_2(\mathbf{Q}_p)\) et \((ϕ,Γ)\)-modules, Ast\'erisque, 330, 281-509, (2010) · Zbl 1218.11107
[15] [Dav]Da A. David, <span class=”textit”>P</span>oids de Serre dans la conjecture de Breuil–M\'ezard, preprint, 2013.
[16] [Eme]Em M. Emerton, <span class=”textit”>L</span>ocal-global compatibility in the \(p\)-adic Langlands programme for \(\text GL_2/\mathbb Q\), preprint, 2010.
[17] Emerton, Matthew; Gee, Toby, A geometric perspective on the Breuil-M\'ezard conjecture, J. Inst. Math. Jussieu, 13, 1, 183-223, (2014) · Zbl 1318.11061
[18] Fontaine, Jean-Marc, Repr\'esentations \(l\)-adiques potentiellement semi-stables, Ast\'erisque, 223, 321-347, (1994) · Zbl 0873.14020
[19] Fontaine, Jean-Marc, Repr\'esentations \(p\)-adiques des corps locaux. I. The Grothendieck Festschrift, Vol. II, Progr. Math. 87, 249-309, (1990), Birkh\"auser Boston, Boston, MA · Zbl 0743.11066
[20] Fontaine, Jean-Marc; Laffaille, Guy, Construction de repr\'esentations \(p\)-adiques, Ann. Sci. \'Ecole Norm. Sup. (4), 15, 4, 547-608 (1983), (1982) · Zbl 0579.14037
[21] Gee, Toby; Kisin, Mark, The Breuil-M\'ezard conjecture for potentially Barsotti-Tate representations, Forum Math. Pi, 2, e1, 56 pp., (2014) · Zbl 1408.11033
[22] [Hen]He G. Henniart, <span class=”textit”>S</span>ur l’unicit\'e des types pour \(\rm GL_2\) (appendice \`a [8]), Duke Math. J. 115, 2002, 298–305.
[23] Kim, Wansu, Galois deformation theory for norm fields and flat deformation rings, J. Number Theory, 131, 7, 1258-1275, (2011) · Zbl 1228.11171
[24] Kisin, Mark, Crystalline representations and \(F\)-crystals. Algebraic geometry and number theory, Progr. Math. 253, 459-496, (2006), Birkh\"auser Boston, Boston, MA · Zbl 1184.11052
[25] Kisin, Mark, Moduli of finite flat group schemes, and modularity, Ann. of Math. (2), 170, 3, 1085-1180, (2009) · Zbl 1201.14034
[26] Kisin, Mark, The Fontaine-Mazur conjecture for \({\rm GL}_2\), J. Amer. Math. Soc., 22, 3, 641-690, (2009) · Zbl 1251.11045
[27] Kisin, Mark, Deformations of \(G_{\mathbb{Q}_p}\) and \({\rm GL}_2(\mathbb{Q}_p)\) representations, Ast\'erisque, 330, 511-528, (2010) · Zbl 1233.11126
[28] Kisin, Mark, The structure of potentially semi-stable deformation rings. Proceedings of the International Congress of Mathematicians. Volume II, 294-311, (2010), Hindustan Book Agency, New Delhi · Zbl 1273.11090
[29] [LB]LB J. Le Borgne, <span class=”textit”>R</span>epr\'esentations galoisiennes et \(φ \)-modules : aspects algorithmiques, Th\`ese de doctorat, Universit\'e Rennes 1, 2012.
[30] Liu, Tong, On lattices in semi-stable representations: a proof of a conjecture of Breuil, Compos. Math., 144, 1, 61-88, (2008) · Zbl 1133.14020
[31] Matsumura, Hideyuki, Commutative ring theory, Translated from the Japanese by M. Reid, Cambridge Studies in Advanced Mathematics 8, xiv+320 pp., (1986), Cambridge University Press, Cambridge · Zbl 0603.13001
[32] Mazur, B., Deforming Galois representations. Galois groups over \(\textbf{Q}\), Berkeley, CA, 1987, Math. Sci. Res. Inst. Publ. 16, 385-437, (1989), Springer, New York
[33] Pappas, G.; Rapoport, M., \(Φ\)-modules and coefficient spaces, Mosc. Math. J., 9, 3, 625-663, back matter, (2009) · Zbl 1194.14032
[34] Pa\v sk\=unas, Vytautas, Extensions for supersingular representations of \({\rm GL}_2(\Bbb Q_p)\), Ast\'erisque, 331, 317-353, (2010) · Zbl 1204.22013
[35] Pa\v sk\=unas, Vytautas, The image of Colmez’s Montreal functor, Publ. Math. Inst. Hautes \'Etudes Sci., 118, 1-191, (2013) · Zbl 1297.22021
[36] Savitt, David, On a conjecture of Conrad, Diamond, and Taylor, Duke Math. J., 128, 1, 141-197, (2005) · Zbl 1101.11017
[37] Sander, Fabian, Hilbert-Samuel multiplicities of certain deformation rings, Math. Res. Lett., 21, 3, 605-615, (2014) · Zbl 1320.11050
[38] Taylor, Richard; Wiles, Andrew, Ring-theoretic properties of certain Hecke algebras, Ann. of Math. (2), 141, 3, 553-572, (1995) · Zbl 0823.11030
[39] Wiles, Andrew, Modular elliptic curves and Fermat’s last theorem, Ann. of Math. (2), 141, 3, 443-551, (1995) · Zbl 0823.11029
[40] Wintenberger, Jean-Pierre, Le corps des normes de certaines extensions infinies de corps locaux; applications, Ann. Sci. \'Ecole Norm. Sup. (4), 16, 1, 59-89, (1983) · Zbl 0516.12015
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