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Representation Theory

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ISSN 1088-4165

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Admissible unitary completions of locally $\mathbb {Q}_p$-rational representations of $\mathrm {GL}_2(F)$
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by Vytautas Paškūnas PDF
Represent. Theory 14 (2010), 324-354 Request permission


Let $F$ be a finite extension of $\mathbb {Q}_p$, $p>2$. We construct admissible unitary completions of certain representations of $\mathrm {GL}_2(F)$ on $L$-vector spaces, where $L$ is a finite extension of $F$. When $F=\mathbb {Q}_p$ using the results of Berger, Breuil and Colmez we obtain some results about lifting $2$-dimensional mod $p$ representations of the absolute Galois group of $\mathbb {Q}_p$ to crystabelline representations with given Hodge-Tate weights.
  • L. Barthel and R. Livné, Irreducible modular representations of $\textrm {GL}_2$ of a local field, Duke Math. J. 75 (1994), no. 2, 261–292. MR 1290194, DOI 10.1215/S0012-7094-94-07508-X
  • Laurent Berger, Hanfeng Li, and Hui June Zhu, Construction of some families of 2-dimensional crystalline representations, Math. Ann. 329 (2004), no. 2, 365–377. MR 2060368, DOI 10.1007/s00208-004-0529-y
  • L. Berger and C. Breuil, Sur la réduction des représentations cristallines de dimension $2$ en poids moyens, preprint.
  • L. Berger and C. Breuil, Sur quelques représentations potentiellement cristallines de $\mathrm {GL}_2(\mathbb {Q}_p)$, to appear in Astérisque.
  • L. Berger, Représentations modulaires de $\mathrm {GL}_2(\mathbb {Q}_p)$ et représentations galoisiennes de dimension 2, to appear in Astérisque.
  • J. N. Bernstein, Le “centre” de Bernstein, Representations of reductive groups over a local field, Travaux en Cours, Hermann, Paris, 1984, pp. 1–32 (French). Edited by P. Deligne. MR 771671
  • Nicolas Bourbaki, Elements of mathematics. Commutative algebra, Hermann, Paris; Addison-Wesley Publishing Co., Reading, Mass., 1972. Translated from the French. MR 0360549
  • Christophe Breuil and Ariane Mézard, Multiplicités modulaires et représentations de $\textrm {GL}_2(\textbf {Z}_p)$ et de $\textrm {Gal}(\overline \textbf {Q}_p/\textbf {Q}_p)$ en $l=p$, Duke Math. J. 115 (2002), no. 2, 205–310 (French, with English and French summaries). With an appendix by Guy Henniart. MR 1944572, DOI 10.1215/S0012-7094-02-11522-1
  • Christophe Breuil, Sur quelques représentations modulaires et $p$-adiques de $\textrm {GL}_2(\mathbf Q_p)$. I, Compositio Math. 138 (2003), no. 2, 165–188 (French, with English summary). MR 2018825, DOI 10.1023/A:1026191928449
  • Christophe Breuil, Sur quelques représentations modulaires et $p$-adiques de $\textrm {GL}_2(\mathbf Q_p)$. II, J. Inst. Math. Jussieu 2 (2003), no. 1, 23–58 (French, with French summary). MR 1955206, DOI 10.1017/S1474748003000021
  • Christophe Breuil and Peter Schneider, First steps towards $p$-adic Langlands functoriality, J. Reine Angew. Math. 610 (2007), 149–180. MR 2359853, DOI 10.1515/CRELLE.2007.070
  • C. Breuil and V. Paškūnas, Towards a modulo $p$ Langlands correspondence for $\mathrm {GL}_2$, to appear in Mem. Amer. Math. Soc..
  • Armand Brumer, Pseudocompact algebras, profinite groups and class formations, J. Algebra 4 (1966), 442–470. MR 202790, DOI 10.1016/0021-8693(66)90034-2
  • Daniel Bump, Automorphic forms and representations, Cambridge Studies in Advanced Mathematics, vol. 55, Cambridge University Press, Cambridge, 1997. MR 1431508, DOI 10.1017/CBO9780511609572
  • Colin J. Bushnell and Philip C. Kutzko, Smooth representations of reductive $p$-adic groups: structure theory via types, Proc. London Math. Soc. (3) 77 (1998), no. 3, 582–634. MR 1643417, DOI 10.1112/S0024611598000574
  • P.Colmez, Série principale unitaire pour $\mathrm {GL}_2(\mathbb {Q}_p)$ et répresentations triangulines de dimension $2$, preprint 2004.
  • P. Colmez, Représentations de $\mathrm {GL}_2(\mathbb {Q}_p)$ et $(\varphi ,\Gamma )$-modules, to appear in Astérisque.
  • Charles W. Curtis and Irving Reiner, Methods of representation theory. Vol. I, Pure and Applied Mathematics, John Wiley & Sons, Inc., New York, 1981. With applications to finite groups and orders. MR 632548
  • Michel Demazure and Pierre Gabriel, Groupes algébriques. Tome I: Géométrie algébrique, généralités, groupes commutatifs, Masson & Cie, Éditeurs, Paris; North-Holland Publishing Co., Amsterdam, 1970 (French). Avec un appendice Corps de classes local par Michiel Hazewinkel. MR 0302656
  • Schémas en groupes. I: Propriétés générales des schémas en groupes, Lecture Notes in Mathematics, Vol. 151, Springer-Verlag, Berlin-New York, 1970 (French). Séminaire de Géométrie Algébrique du Bois Marie 1962/64 (SGA 3); Dirigé par M. Demazure et A. Grothendieck. MR 0274458
  • F. Diamond, A correspondence between representations of local Galois groups and Lie-type groups, Proceedings of the LMS Durham Symposium on L-functions and Galois Representations, 2004.
  • M. Emerton, Locally analytic vectors in representations of locally $p$-adic analytic groups, to appear in Memoirs of the AMS.
  • Matthew Emerton, Locally analytic representation theory of $p$-adic reductive groups: a summary of some recent developments, $L$-functions and Galois representations, London Math. Soc. Lecture Note Ser., vol. 320, Cambridge Univ. Press, Cambridge, 2007, pp. 407–437. MR 2392361, DOI 10.1017/CBO9780511721267.012
  • Pierre Gabriel, Des catégories abéliennes, Bull. Soc. Math. France 90 (1962), 323–448 (French). MR 232821, DOI 10.24033/bsmf.1583
  • G. Henniart, Sur l’unicité des types pour $\mathrm {GL}_2$, appendix to [8].
  • Serge Lang, Algebra, 3rd ed., Graduate Texts in Mathematics, vol. 211, Springer-Verlag, New York, 2002. MR 1878556, DOI 10.1007/978-1-4613-0041-0
  • Michel Lazard, Groupes analytiques $p$-adiques, Inst. Hautes Études Sci. Publ. Math. 26 (1965), 389–603 (French). MR 209286
  • Saunders Mac Lane, Homology, Classics in Mathematics, Springer-Verlag, Berlin, 1995. Reprint of the 1975 edition. MR 1344215
  • Jürgen Neukirch, Alexander Schmidt, and Kay Wingberg, Cohomology of number fields, Grundlehren der mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 323, Springer-Verlag, Berlin, 2000. MR 1737196
  • Vytautas Paskunas, Coefficient systems and supersingular representations of $\textrm {GL}_2(F)$, Mém. Soc. Math. Fr. (N.S.) 99 (2004), vi+84 (English, with English and French summaries). MR 2128381
  • Vytautas Paskunas, On the restriction of representations of $\textrm {GL}_2(F)$ to a Borel subgroup, Compos. Math. 143 (2007), no. 6, 1533–1544. MR 2371380, DOI 10.1112/S0010437X07002862
  • Vytautas Paškūnas, On some crystalline representations of $\textrm {GL}_2(\Bbb Q_p)$, Algebra Number Theory 3 (2009), no. 4, 411–421. MR 2525557, DOI 10.2140/ant.2009.3.411
  • D. Prasad, Locally algebraic representations of $p$-adic groups, appendix to [35].
  • Peter Schneider, Nonarchimedean functional analysis, Springer Monographs in Mathematics, Springer-Verlag, Berlin, 2002. MR 1869547, DOI 10.1007/978-3-662-04728-6
  • P. Schneider, J. Teitelbaum, and Dipendra Prasad, $U({\mathfrak {g}})$-finite locally analytic representations, Represent. Theory 5 (2001), 111–128. With an appendix by Dipendra Prasad. MR 1835001, DOI 10.1090/S1088-4165-01-00109-1
  • P. Schneider and J. Teitelbaum, Banach space representations and Iwasawa theory, Israel J. Math. 127 (2002), 359–380. MR 1900706, DOI 10.1007/BF02784538
  • Jean-Pierre Serre, Linear representations of finite groups, Graduate Texts in Mathematics, Vol. 42, Springer-Verlag, New York-Heidelberg, 1977. Translated from the second French edition by Leonard L. Scott. MR 0450380, DOI 10.1007/978-1-4684-9458-7
  • Otmar Venjakob, Characteristic elements in noncommutative Iwasawa theory, J. Reine Angew. Math. 583 (2005), 193–236. MR 2146857, DOI 10.1515/crll.2005.2005.583.193
  • Marie-France Vignéras, Représentations $l$-modulaires d’un groupe réductif $p$-adique avec $l\ne p$, Progress in Mathematics, vol. 137, Birkhäuser Boston, Inc., Boston, MA, 1996 (French, with English summary). MR 1395151
  • M.-F. Vignéras, Admissibilite des representations $p$-adiques et lemme de Nakayama, preprint 2007.
  • Marie-France Vignéras, A criterion for integral structures and coefficient systems on the tree of $\textrm {PGL}(2,F)$, Pure Appl. Math. Q. 4 (2008), no. 4, Special Issue: In honor of Jean-Pierre Serre., 1291–1316 (English, with French summary). MR 2441702, DOI 10.4310/PAMQ.2008.v4.n4.a13
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Additional Information
  • Vytautas Paškūnas
  • Affiliation: Fakultät für Mathematik, Universität Bielefeld, Postfach 100131, D-33501 Bielefeld, Germany
  • Received by editor(s): September 15, 2008
  • Published electronically: April 7, 2010
  • © Copyright 2010 American Mathematical Society
    The copyright for this article reverts to public domain 28 years after publication.
  • Journal: Represent. Theory 14 (2010), 324-354
  • MSC (2010): Primary 22-XX; Secondary 11-XX
  • DOI:
  • MathSciNet review: 2608966