Paškūnas, Vytautas The image of Colmez’s Montreal functor. (English) Zbl 1297.22021 Publ. Math., Inst. Hautes Étud. Sci. 118, 1-191 (2013). The subject of the paper is the study of \(p\)-adic representations of \(G =\mathrm{GL}_2(\mathbb Q_p)\). Let \(L\) be a finite extension of \((\mathbb Q_p)\), let \(\mathcal O\) be the ring of integers of \(L\) and \(k\) its residue field. Considered are representations of \(G\) on \(L\)-Banach spaces or on \(\mathcal O\) -modules. In a unitary Banach space representation of \(G\) one may choose a \(G\)-invariant \(\mathcal O\)-lattice \(\Theta\) and reduction gives a \(k\)-representation of \(G\). Let \(\mathcal G\) denote the Galois group of \(\overline{\mathbb Q}_p /\mathbb Q_p\). Colmez’s functor is an exact covariant functor \(\mathbf V\) from the category of smooth \(G\)-representations of finite length with a central character on torsion \(\mathcal O\)-modules to the category of continuous \(\mathcal G\)-representations of finite length on \(\mathcal O\) -modules. The author extends this functor to a functor on \(L\)-Banach space representations. Let \(\Pi\) be an absolutely irreducible admissible unitary \(L\)-Banach representation of \(G\) with a central character. A theorem on the reduction \(\mathrm {mod}-p\) of \(\Pi\), proved here for \(p > 3\), implies that \(\mathrm{dim}_L \mathbf V(\Pi)\leq 2\) and determines for which \(\Pi\) this dimension equals 2. For those \(\Pi\) it is proved that their isomorphism classes correspond bijectively, via the functor \(\mathbf V\), with the absolutely irreducible 2-dimensional continuous \(L\)-representations of \(\mathcal G\) with determinant \(\varepsilon \xi\) (\(\xi\) is the central character of \(\Pi, \varepsilon\) is the cyclotomic character). The final result is an anti-equivalence of categories between the category of admissible unitary \(L\)-Banach representations of \(G\) of finite length, with central character \(\varepsilon\) and all irreducible subquotients isomorphic to \(\Pi\), and the category of modules of finite length over the ring representing the deformation functor of \(V = \mathbf V(\Pi)\) to artinian local \(L\)-algebras. An essential tool for the proof of these results is a deformation theory for \(k\)-representations of \(G\), which is presented in such a way that it can be used for an arbitrary \(p\)-adic analytic group, when certain hypotheses (on Ext groups) are satisfied. In the case of \(G =\mathrm{GL}_2(\mathbb Q_p)\) the category of Banach space (resp. \(\mathcal O\)-module) representations of \(G\) can be decomposed into a product of subcategories, indexed by blocks of irreducible smooth \(k\)-representations of \(G\). There are four types of blocks to be considered and for which the application of the theory mentioned above is different. Also results of deformation theory for 2-dimensional representations of \(\mathrm{Gal}(\mathbb Q_p) / (\mathbb Q_p)\) are needed. They are deduced from work of Böckle. Reviewer: J. G. M. Mars (Utrecht) Cited in 4 ReviewsCited in 71 Documents MSC: 22E50 Representations of Lie and linear algebraic groups over local fields 11S37 Langlands-Weil conjectures, nonabelian class field theory Keywords:\(p\)-adic Langlands correspondence; \(\mathrm{GL}_2(\mathbb Q_p)\); Colmez’s functor × Cite Format Result Cite Review PDF Full Text: DOI arXiv References: [1] L. Barthel and R. Livné, Irreducible modular representations of GL2 of a local field, Duke Math. J., 75 (1994), 261–292. · Zbl 0826.22019 · doi:10.1215/S0012-7094-94-07508-X [2] J. Bellaïche, Pseudodeformations, Math. Z., 270 (2012), 1163–1180. · Zbl 1295.11055 · doi:10.1007/s00209-011-0846-2 [3] J. Bellaïche and G. Chenevier, Families of Galois Representations and Selmer Groups, Astérisque, vol. 324, Soc. Math. France, Paris, 2009. · Zbl 1192.11035 [4] L. Berger, Représentations modulaires de GL2(Q p ) et représentations galoisiennes de dimension 2, Astérisque, 330 (2010), 263–279. [5] L. Berger, La correspondance de Langlands locale p-adique pour GL2(Q p ), in Exposé No 1017 du Séminaire Bourbaki, Astérisque, vol. 339, pp. 157–180, 2011. [6] L. Berger, Central characters for smooth irreducible modular representations of GL2(Q p ), Rendiconti del Seminario Matematico della Università di Padova (F. Baldassarri’s 60th birthday), vol. 127, 2012. · Zbl 1266.22018 [7] L. Berger and C. Breuil, Sur quelques représentations potentiellement cristallines de GL2(Q p ), Astérisque, 330 (2010), 155–211. · Zbl 1243.11063 [8] J.-N. Bernstein (rédigé par P. Deligne), Le ’centre’ de Bernstein, in Représentations des groupes réductifs sur un corps local, pp. 1–32, Herman, Paris, 1984. [9] G. Böckle, Demuškin groups with group actions and applications to deformations of Galois representations, Compositio, 121 (2000), 109–154. · Zbl 0952.11009 · doi:10.1023/A:1001746207573 [10] N. Boston, H. W. Lenstra, and K. A. Ribet, Quotients of group rings arising from two dimensional representations, C.R. Acad. Sci. Paris Sér. I, 312 (1991), 323–328. · Zbl 0718.16018 [11] N. Bourbaki, Algébre, Chapitre 8, Hermann, Paris, 1958. [12] N. Bourbaki, Commutative Algebra, Hermann, Paris, 1972. [13] N. Bourbaki, Algébre Homologique, Masson, Paris, 1980. [14] N. Bourbaki, Algebra I, Chapters 1–3, Springer, Berlin, 1989. · Zbl 0673.00001 [15] C. Breuil and A. Mézard, Multiplicités modulaires et représentations de GL2(Z p ) et de $\(\backslash\)mathrm{Gal}(\(\backslash\)overline{\(\backslash\)mathbf {Q}}_{\(\backslash\),\(\backslash\),p}/\(\backslash\)mathbf {Q}_{\(\backslash\),\(\backslash\),p})$ en l=p, Duke Math. J., 115 (2002), 205–310. · Zbl 1042.11030 · doi:10.1215/S0012-7094-02-11522-1 [16] C. Breuil, Sur quelques représentations modulaires et p-adiques de GL2(Q p ). I, Compositio, 138 (2003), 165–188. · Zbl 1044.11041 · doi:10.1023/A:1026191928449 [17] C. Breuil, Sur quelques représentations modulaires et p-adiques de GL2(Q p ). II, J. Inst. Math. Jussieu, 2 (2003), 1–36. · Zbl 1165.11319 · doi:10.1017/S1474748003000021 [18] C. Breuil, Invariant $\(\backslash\)mathcal{L}$ et série spéciale p-adique, Ann. Sci. Éc. Norm. Super., 37 (2004), 559–610. [19] C. Breuil and M. Emerton, Représentations p-adiques ordinaires de GL2(Q p ) et compatibilité local-global, Astérisque, 331 (2010), 255–315. [20] C. Breuil and V. Paškūnas, Towards a Modulo p Langlands Correspondence for GL2, Memoirs of AMS, vol. 216, 2012. [21] A. Brumer, Pseudo-compact algebras, profinite groups and class formations, J. Algebra, 4 (1966), 442–470. · Zbl 0146.04702 · doi:10.1016/0021-8693(66)90034-2 [22] G. Chenevier, The p-adic analytic space of pseudocharacters of a profinite group, and pseudorepresentations over arbitrary rings, arXiv:0809.0415 . · Zbl 1350.11063 [23] P. Colmez, Représentations de GL2(Q p ) et ({\(\phi\)},{\(\Gamma\)})-modules, Astérisque, 330 (2010), 281–509. [24] C. W. Curtis and I. Reiner, Methods of Representation Theory. Volume I, Wiley, New York, 1981. [25] M. Demazure and P. Gabriel, Groupes Algébriques, Tome I, Masson, Paris, 1970. · Zbl 0203.23401 [26] M. Demazure and A. Grothendieck, Schémas en Groupes I, Lect. Notes Math, vol. 151, Springer, Berlin, 1970. · Zbl 0207.51401 [27] G. Dospinescu and B. Schraen, Endomorphism algebras of p-adic representations of p-adic Lie groups, arXiv:1106.2442 . · Zbl 1432.22018 [28] M. Emerton, p-adic L-functions and unitary completions of representations of p-adic reductive groups, Duke Math. J., 130 (2005), 353–392. · Zbl 1092.11024 [29] M. Emerton, A local-global compatibility conjecture in the p-adic Langlands programme for GL2/Q, Pure Appl. Math. Q., 2 (2006), 279–393. · Zbl 1254.11106 · doi:10.4310/PAMQ.2006.v2.n2.a1 [30] M. Emerton, Ordinary parts of admissible representations of p-adic reductive groups I. Definition and first properties, Astérisque, 331 (2010), 335–381. · Zbl 1205.22013 [31] M. Emerton, Ordinary parts of admissible representations of p-adic reductive groups II. Derived functors, Astérisque, 331 (2010), 383–438. · Zbl 1205.22014 [32] M. Emerton, Local-global compatibility in the p-adic Langlands programme for GL2/Q. [33] M. Emerton, Locally analytic vectors in representations of locally p-adic analytic groups, Memoirs of the AMS, to appear. · Zbl 1117.22008 [34] M. Emerton and V. Paškūnas, On effaceability of certain {\(\delta\)}-functors, Astérisque, 331 (2010), 439–447. [35] P. Gabriel, Des catégories abéliennes, Bull. Soc. Math. Fr., 90 (1962), 323–448. [36] E. Ghate and A. Mézard, Filtered modules with coefficients, Trans. Am. Math. Soc., 361 (2009), 2243–2261. · Zbl 1251.11044 · doi:10.1090/S0002-9947-08-04829-0 [37] Y. Hu, Sur quelques représentations supersinguliéres de $\(\backslash\)mathrm{GL}_{2}(\(\backslash\)mathbf {Q}_{\(\backslash\),p\^{f}})$ , J. Algebra, 324 (2010), 1577–1615. · Zbl 1206.22010 · doi:10.1016/j.jalgebra.2010.06.006 [38] I. Kaplansky, Commutative Rings, revised ed., University of Chicago Press, Chicago, 1974. · Zbl 0296.13001 [39] M. Kisin, Moduli of finite flat group schemes and modularity, Ann. Math., 170 (2009), 1085–1180. · Zbl 1201.14034 · doi:10.4007/annals.2009.170.1085 [40] M. Kisin, The Fontaine-Mazur conjecture for GL2, J. Am. Math. Soc., 22 (2009), 641–690. · Zbl 1251.11045 · doi:10.1090/S0894-0347-09-00628-6 [41] M. Kisin, Deformations of $G_{\(\backslash\)mathbf {Q}_{\(\backslash\),\(\backslash\),p}}$ and GL2(Q p ) representations, Astérisque, 330 (2010), 511–528. [42] J. Labute, Classification of Demuškin groups, Can. J. Math., 19 (1967), 106–132. · Zbl 0153.04202 · doi:10.4153/CJM-1967-007-8 [43] T. Y. Lam, A First Course in Noncommutative Rings, Springer GTM, vol. 131, 1991. · Zbl 0728.16001 [44] S. Lang, Algebra, revised 3rd ed., Springer, Berlin, 2002. [45] M. Lazard, Groupes analytiques p-adiques, Publ. Math. IHES 26 (1965). · Zbl 0139.02302 [46] H. Matsumura, Commutative ring theory, CUP. · Zbl 0666.13002 [47] B. Mazur, Deforming Galois representations, in Y. Ihara, K. Ribet, J.-P. Serre (eds.), Galois Groups over Q, Berkley, CA, 1987, pp. 385–437. Springer, New York, 1989. [48] B. Mazur, An introduction to the deformation theory of Galois representations, in G. Cornell, J. H. Silverman, and G. Stevens (eds.) Modular Forms and Fermat’s Last Theorem, Boston, MA, 1995, pp. 243–311. Springer, New York, 1997. [49] J. Neukirch, A. Schmidt, and K. Wingberg, Cohomology of Number Fields, Springer, Berlin, 2000. · Zbl 0948.11001 [50] L. Nyssen, Pseudo-représentations, Math. Ann., 306 (1996), 257–283. · Zbl 0863.16012 · doi:10.1007/BF01445251 [51] R. Ollivier, Le foncteur des invariants sous l’action du pro-p-Iwahori de GL(2,F), J. Reine Angew. Math., 635 (2009), 149–185. · Zbl 1181.22017 [52] F. Oort, Yoneda extensions in abelian categories, Math. Ann., 153 (1964), 227–235. · Zbl 0126.03401 · doi:10.1007/BF01360318 [53] D. Prasad, Locally algebraic representations of p-adic groups, appendix to [60]. [54] V. Paškūnas, Coefficient Systems and Supersingular Representations of GL2(F), Mémoires de la SMF, vol. 99, 2004. [55] V. Paškūnas, On some crystalline representations of GL2(Q p ), Algebra Number Theory, 3 (2009), 411–421. · Zbl 1173.22015 · doi:10.2140/ant.2009.3.411 [56] V. Paškūnas, Extensions for supersingular representations of GL2(Q p ), Astérisque, 331 (2010), 317–353. [57] V. Paškūnas, Admissible unitary completions of locally Q p -rational representations of GL2(F), Represent. Theory, 14 (2010), 324–354. · Zbl 1192.22009 · doi:10.1090/S1088-4165-10-00373-0 [58] V. Paškūnas, Blocks for mod p representations of GL2(Q p ), preprint (2011), arXiv:1104.5602 . [59] P. Schneider, Nonarchimedean Functional Analysis, Springer, Berlin, 2001. · Zbl 0998.46044 [60] P. Schneider and J. Teitelbaum, $U(\(\backslash\)mathfrak{g})$ -finite locally analytic representations, Represent. Theory, 5 (2001), 111–128. · Zbl 1028.17007 · doi:10.1090/S1088-4165-01-00109-1 [61] P. Schneider and J. Teitelbaum, Banach space representations and Iwasawa theory, Isr. J. Math., 127 (2002), 359–380. · Zbl 1006.46053 · doi:10.1007/BF02784538 [62] J.-P. Serre, Sur la dimension cohomologique des groupes profinis, Topology, 3 (1965), 413–420. · Zbl 0136.27402 · doi:10.1016/0040-9383(65)90006-6 [63] J.-P. Serre, Linear Representation of Finite Groups, Springer, Berlin, 1977. [64] J.-P. Serre, Cohomologie Galoisienne, Cinquième édition, révisée et complétée, Springer, Berlin, 1997. [65] R. Taylor, Galois representations associated to Siegel modular forms of low weight, Duke Math. J., 63 (1991), 281–332. · Zbl 0810.11033 · doi:10.1215/S0012-7094-91-06312-X [66] M. van den Bergh, Blowing up of non-commutative smooth surfaces, Mem. Amer. Math. Soc., 154 (2001), no. 734. · Zbl 0998.14002 [67] M.-F. Vignéras, Representations modulo p of the p-adic group GL(2,F), Compos. Math., 140 (2004), 333–358. · Zbl 1049.22010 · doi:10.1112/S0010437X03000071 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.