The image of Colmez’s Montreal functor
Publications Mathématiques de l'IHÉS, Tome 118 (2013), pp. 1-191.

We prove a conjecture of Colmez concerning the reduction modulo p of invariant lattices in irreducible admissible unitary p -adic Banach space representations of GL 2 (𝐐 p ) with p 5 . This enables us to restate nicely the with p -adic local Langlands correspondence for GL 2 ( 𝐐 p ) and deduce a conjecture of Breuil on irreducible admissible unitary completions of locally algebraic representations.

DOI : 10.1007/s10240-013-0049-y
Paškūnas, Vytautas 1

1 Fakultät für Mathematik, Universität Duisburg-Essen 45127, Essen Germany
@article{PMIHES_2013__118__1_0,
     author = {Pa\v{s}k\={u}nas, Vytautas},
     title = {The image of {Colmez{\textquoteright}s} {Montreal} functor},
     journal = {Publications Math\'ematiques de l'IH\'ES},
     pages = {1--191},
     publisher = {Springer Berlin Heidelberg},
     address = {Berlin/Heidelberg},
     volume = {118},
     year = {2013},
     doi = {10.1007/s10240-013-0049-y},
     mrnumber = {3150248},
     zbl = {1297.22021},
     language = {en},
     url = {http://archive.numdam.org/articles/10.1007/s10240-013-0049-y/}
}
TY  - JOUR
AU  - Paškūnas, Vytautas
TI  - The image of Colmez’s Montreal functor
JO  - Publications Mathématiques de l'IHÉS
PY  - 2013
SP  - 1
EP  - 191
VL  - 118
PB  - Springer Berlin Heidelberg
PP  - Berlin/Heidelberg
UR  - http://archive.numdam.org/articles/10.1007/s10240-013-0049-y/
DO  - 10.1007/s10240-013-0049-y
LA  - en
ID  - PMIHES_2013__118__1_0
ER  - 
%0 Journal Article
%A Paškūnas, Vytautas
%T The image of Colmez’s Montreal functor
%J Publications Mathématiques de l'IHÉS
%D 2013
%P 1-191
%V 118
%I Springer Berlin Heidelberg
%C Berlin/Heidelberg
%U http://archive.numdam.org/articles/10.1007/s10240-013-0049-y/
%R 10.1007/s10240-013-0049-y
%G en
%F PMIHES_2013__118__1_0
Paškūnas, Vytautas. The image of Colmez’s Montreal functor. Publications Mathématiques de l'IHÉS, Tome 118 (2013), pp. 1-191. doi : 10.1007/s10240-013-0049-y. http://archive.numdam.org/articles/10.1007/s10240-013-0049-y/

[1.] Barthel, L.; Livné, R. Irreducible modular representations of GL2 of a local field, Duke Math. J., Volume 75 (1994), pp. 261-292 | DOI | MR | Zbl

[2.] Bellaïche, J. Pseudodeformations, Math. Z., Volume 270 (2012), pp. 1163-1180 | DOI | MR | Zbl

[3.] Bellaïche, J.; Chenevier, G. Families of Galois Representations and Selmer Groups, Astérisque, 324, Soc. Math. France, Paris, 2009 | Numdam | MR | Zbl

[4.] Berger, L. Représentations modulaires de GL2(Q    p ) et représentations galoisiennes de dimension 2, Astérisque, Volume 330 (2010), pp. 263-279 | Numdam | MR | Zbl

[5.] Berger, L. La correspondance de Langlands locale p-adique pour GL2(Q    p ), Exposé No 1017 du Séminaire Bourbaki (Astérisque, 339) (2011), pp. 157-180 | MR | Zbl

[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. | Numdam | MR | Zbl

[7.] Berger, L.; Breuil, C. Sur quelques représentations potentiellement cristallines de GL2(Q    p ), Astérisque, Volume 330 (2010), pp. 155-211 | Numdam | MR | Zbl

[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. | MR | Zbl

[9.] Böckle, G. Demuškin groups with group actions and applications to deformations of Galois representations, Compositio, Volume 121 (2000), pp. 109-154 | DOI | MR | Zbl

[10.] Boston, N.; Lenstra, H. W.; Ribet, K. A. Quotients of group rings arising from two dimensional representations, C.R. Acad. Sci. Paris Sér. I, Volume 312 (1991), pp. 323-328 | MR | Zbl

[11.] Bourbaki, N. Algébre, Chapitre 8, Hermann, Paris, 1958 | MR

[12.] Bourbaki, N. Commutative Algebra, Hermann, Paris, 1972 | Zbl

[13.] Bourbaki, N. Algébre Homologique, Masson, Paris, 1980 | Zbl

[14.] Bourbaki, N. Algebra I, Chapters 1–3, Springer, Berlin, 1989 | MR | Zbl

[15.] Breuil, C.; Mézard, A. Multiplicités modulaires et représentations de GL2(Z p ) et de Gal (𝐐 ¯ p /𝐐 p ) en l=p , Duke Math. J., Volume 115 (2002), pp. 205-310 | DOI | MR | Zbl

[16.] Breuil, C. Sur quelques représentations modulaires et p-adiques de GL2(Q    p ). I, Compositio, Volume 138 (2003), pp. 165-188 | DOI | MR | Zbl

[17.] Breuil, C. Sur quelques représentations modulaires et p-adiques de GL2(Q    p ). II, J. Inst. Math. Jussieu, Volume 2 (2003), pp. 1-36 | DOI | MR | Zbl

[18.] Breuil, C. Invariant et série spéciale p-adique, Ann. Sci. Éc. Norm. Super., Volume 37 (2004), pp. 559-610 | MR | Zbl

[19.] Breuil, C.; Emerton, M. Représentations p-adiques ordinaires de GL2(Q    p ) et compatibilité local-global, Astérisque, Volume 331 (2010), pp. 255-315 | Numdam | MR | Zbl

[20.] Breuil, C.; Paškūnas, V. Towards a Modulo p Langlands Correspondence for GL2 , Memoirs of AMS, 216, 2012 | MR | Zbl

[21.] Brumer, A. Pseudo-compact algebras, profinite groups and class formations, J. Algebra, Volume 4 (1966), pp. 442-470 | DOI | MR | Zbl

[22.] G. Chenevier, The p-adic analytic space of pseudocharacters of a profinite group, and pseudorepresentations over arbitrary rings, . | arXiv

[23.] Colmez, P. Représentations de GL2(Q    p ) et (φ,Γ)-modules, Astérisque, Volume 330 (2010), pp. 281-509 | Numdam | MR | Zbl

[24.] Curtis, C. W.; Reiner, I. Methods of Representation Theory. Volume I, Wiley, New York, 1981 | MR

[25.] Demazure, M.; Gabriel, P. Groupes Algébriques, Tome I, Masson, Paris, 1970 | MR | Zbl

[26.] Demazure, M.; Grothendieck, A. Schémas en Groupes I, Lect. Notes Math, 151, Springer, Berlin, 1970 | MR | Zbl

[27.] G. Dospinescu and B. Schraen, Endomorphism algebras of p-adic representations of p-adic Lie groups, . | arXiv

[28.] Emerton, M. p-adic L-functions and unitary completions of representations of p-adic reductive groups, Duke Math. J., Volume 130 (2005), pp. 353-392 | MR | Zbl

[29.] Emerton, M. A local-global compatibility conjecture in the p-adic Langlands programme for GL2/Q , Pure Appl. Math. Q., Volume 2 (2006), pp. 279-393 | DOI | MR | Zbl

[30.] Emerton, M. Ordinary parts of admissible representations of p-adic reductive groups I. Definition and first properties, Astérisque, Volume 331 (2010), pp. 335-381 | Numdam | MR | Zbl

[31.] Emerton, M. Ordinary parts of admissible representations of p-adic reductive groups II. Derived functors, Astérisque, Volume 331 (2010), pp. 383-438 | Numdam | MR | Zbl

[32.] M. Emerton, Local-global compatibility in the p-adic Langlands programme for GL2/Q. | Zbl

[33.] M. Emerton, Locally analytic vectors in representations of locally p-adic analytic groups, Memoirs of the AMS, to appear. | MR

[34.] Emerton, M.; Paškūnas, V. On effaceability of certain δ-functors, Astérisque, Volume 331 (2010), pp. 439-447 | Numdam | MR | Zbl

[35.] Gabriel, P. Des catégories abéliennes, Bull. Soc. Math. Fr., Volume 90 (1962), pp. 323-448 | Numdam | MR | Zbl

[36.] Ghate, E.; Mézard, A. Filtered modules with coefficients, Trans. Am. Math. Soc., Volume 361 (2009), pp. 2243-2261 | DOI | MR | Zbl

[37.] Hu, Y. Sur quelques représentations supersinguliéres de GL 2 (𝐐 p f ) , J. Algebra, Volume 324 (2010), pp. 1577-1615 | DOI | MR | Zbl

[38.] Kaplansky, I. Commutative Rings, University of Chicago Press, Chicago, 1974 (revised ed.) | MR | Zbl

[39.] Kisin, M. Moduli of finite flat group schemes and modularity, Ann. Math., Volume 170 (2009), pp. 1085-1180 | DOI | MR | Zbl

[40.] Kisin, M. The Fontaine-Mazur conjecture for GL2 , J. Am. Math. Soc., Volume 22 (2009), pp. 641-690 | DOI | MR | Zbl

[41.] Kisin, M. Deformations of G 𝐐 p and GL2(Q    p ) representations, Astérisque, Volume 330 (2010), pp. 511-528 | Numdam | MR | Zbl

[42.] Labute, J. Classification of Demuškin groups, Can. J. Math., Volume 19 (1967), pp. 106-132 | DOI | MR | Zbl

[43.] Lam, T. Y. A First Course in Noncommutative Rings, Springer GTM, 131, 1991 | DOI | MR | Zbl

[44.] Lang, S. Algebra, Springer, Berlin, 2002 (revised) | DOI | Zbl

[45.] M. Lazard, Groupes analytiques p-adiques, Publ. Math. IHES 26 (1965). | Numdam | MR | Zbl

[46.] H. Matsumura, Commutative ring theory, CUP. | MR | Zbl

[47.] Mazur, B. Deforming Galois representations, Galois Groups over Q , Springer, New York (1989), pp. 385-437 | DOI | MR | Zbl

[48.] Mazur, B. An introduction to the deformation theory of Galois representations, Modular Forms and Fermat’s Last Theorem, Springer, New York (1997), pp. 243-311 | DOI | MR | Zbl

[49.] Neukirch, J.; Schmidt, A.; Wingberg, K. Cohomology of Number Fields, Springer, Berlin, 2000 | MR | Zbl

[50.] Nyssen, L. Pseudo-représentations, Math. Ann., Volume 306 (1996), pp. 257-283 | DOI | MR | Zbl

[51.] Ollivier, R. Le foncteur des invariants sous l’action du pro-p-Iwahori de GL(2,F), J. Reine Angew. Math., Volume 635 (2009), pp. 149-185 | MR | Zbl

[52.] Oort, F. Yoneda extensions in abelian categories, Math. Ann., Volume 153 (1964), pp. 227-235 | DOI | MR | Zbl

[53.] D. Prasad, Locally algebraic representations of p-adic groups, appendix to [60].

[54.] Paškūnas, V. Coefficient Systems and Supersingular Representations of GL2(F), Mémoires de la SMF, 99, 2004 | Numdam | MR | Zbl

[55.] Paškūnas, V. On some crystalline representations of GL2(Q    p ), Algebra Number Theory, Volume 3 (2009), pp. 411-421 | DOI | MR | Zbl

[56.] Paškūnas, V. Extensions for supersingular representations of GL2(Q    p ), Astérisque, Volume 331 (2010), pp. 317-353 | Numdam | MR | Zbl

[57.] Paškūnas, V. Admissible unitary completions of locally Q    p -rational representations of GL2(F), Represent. Theory, Volume 14 (2010), pp. 324-354 | DOI | MR | Zbl

[58.] V. Paškūnas, Blocks for mod p representations of GL2(Q    p ), preprint (2011), . | arXiv | MR

[59.] Schneider, P. Nonarchimedean Functional Analysis, Springer, Berlin, 2001 | MR | Zbl

[60.] Schneider, P.; Teitelbaum, J. U(𝔤)-finite locally analytic representations, Represent. Theory, Volume 5 (2001), pp. 111-128 | DOI | MR | Zbl

[61.] Schneider, P.; Teitelbaum, J. Banach space representations and Iwasawa theory, Isr. J. Math., Volume 127 (2002), pp. 359-380 | DOI | MR | Zbl

[62.] Serre, J.-P. Sur la dimension cohomologique des groupes profinis, Topology, Volume 3 (1965), pp. 413-420 | DOI | MR | Zbl

[63.] Serre, J.-P. Linear Representation of Finite Groups, Springer, Berlin, 1977 | DOI | MR

[64.] Serre, J.-P. Cohomologie Galoisienne, Springer, Berlin, 1997 (Cinquième édition, révisée et complétée) | DOI | MR

[65.] Taylor, R. Galois representations associated to Siegel modular forms of low weight, Duke Math. J., Volume 63 (1991), pp. 281-332 | DOI | MR | Zbl

[66.] Bergh, M. Blowing up of non-commutative smooth surfaces, Mem. Amer. Math. Soc., Volume 154 (2001), p. 734 | MR | Zbl

[67.] Vignéras, M.-F. Representations modulo p of the p-adic group GL(2,F), Compos. Math., Volume 140 (2004), pp. 333-358 | DOI | MR | Zbl

Cité par Sources :