Skip to Main Content

Representation Theory

Published by the American Mathematical Society, the Representation Theory (ERT) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-4165

The 2020 MCQ for Representation Theory is 0.7.

What is MCQ? The Mathematical Citation Quotient (MCQ) measures journal impact by looking at citations over a five-year period. Subscribers to MathSciNet may click through for more detailed information.

 

Admissible unitary completions of locally $\mathbb {Q}_p$-rational representations of $\mathrm {GL}_2(F)$
HTML articles powered by AMS MathViewer

by Vytautas Paškūnas PDF
Represent. Theory 14 (2010), 324-354 Request permission

Abstract:

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.
References
  • 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
Similar Articles
  • Retrieve articles in Representation Theory of the American Mathematical Society with MSC (2010): 22-XX, 11-XX
  • Retrieve articles in all journals with MSC (2010): 22-XX, 11-XX
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: https://doi.org/10.1090/S1088-4165-10-00373-0
  • MathSciNet review: 2608966