Algebra & Number Theory

Induction parabolique et $(\varphi,\Gamma)$-modules

Christophe Breuil

Full-text: Access denied (no subscription detected)

However, an active subscription may be available with MSP at msp.org/ant.

We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text

Abstract

Soit L une extension finie de p et B un sous-groupe de Borel d’un groupe réductif déployé connexe G sur L de centre connexe. On définit un foncteur contravariant et exact à droite de la catégorie des représentations lisses de B(L) sur pm vers la catégorie des limites projectives de (φ,Γ)-modules étales (pour Gal(¯pp)) sur pm. On montre que ce foncteur est insensible à l’induction parabolique et que, restreint aux représentations de longueur finie dont les constituants sont des sous-quotients de séries principales, il est exact et donne de “vrais” (φ,Γ)-modules. Par passage à la limite projective, on en déduit que, convenablement normalisé, il envoie la G(p)-représentation Π(ρ)ord de Breuil et Herzig (Duke Math. J. 164:7 (2015), 1271–1352) vers le (φ,Γ)-module de la représentation (L|B̂Cρ)ord ρ de Gal(¯pp), reliant ainsi les deux constructions de loc. cit.

Let L be a finite extension of p and B a Borel subgroup of a split reductive connected algebraic group G over L with a connected center. We define a right exact contravariant functor from the category of smooth representations of B(L) over pm to the category of projective limits of étale (φ,Γ)-modules (for Gal(¯pp)) over pm. We show that this functor is insensitive to parabolic induction and that, when restricted to finite length representations with all constituents being subquotients of principal series, it is exact and yields “genuine” (φ,Γ)-modules. By a projective limit process, we deduce that, conveniently normalized, it sends the G(p)-representation Π(ρ)ord of Breuil and Herzig (Duke Math. J. 164:7 (2015), 1271–1352) to the (φ,Γ)-module of the representation (L|B̂Cρ)ord ρ of Gal(¯pp), thus connecting the two constructions of loc. cit.

Article information

Source
Algebra Number Theory, Volume 9, Number 10 (2015), 2241-2291.

Dates
Received: 6 September 2014
Revised: 12 June 2015
Accepted: 8 August 2015
First available in Project Euclid: 16 November 2017

Permanent link to this document
https://projecteuclid.org/euclid.ant/1510842445

Digital Object Identifier
doi:10.2140/ant.2015.9.2241

Mathematical Reviews number (MathSciNet)
MR3437761

Zentralblatt MATH identifier
1377.11119

Subjects
Primary: 11S20: Galois theory
Secondary: 20G25: Linear algebraic groups over local fields and their integers 20G05: Representation theory

Keywords
p-adic Langlands parabolic induction (phi, Gamma)-modules

Citation

Breuil, Christophe. Induction parabolique et $(\varphi,\Gamma)$-modules. Algebra Number Theory 9 (2015), no. 10, 2241--2291. doi:10.2140/ant.2015.9.2241. https://projecteuclid.org/euclid.ant/1510842445


Export citation

References

  • N. Abe, “On a classification of irreducible admissible modulo $p$ representations of a $p$-adic split reductive group”, Compos. Math. 149:12 (2013), 2139–2168.
  • J. Bergdall and P. Chojecki, “Ordinary representations and companion points for $U(3)$ in the indecomposable case”, prépublication, 2014, hook \posturlhook.
  • L. Berger and M. Vienney, “Irreducible modular representations of the Borel subgroup of ${\rm GL}_2(\Qp)$”, pp. 32–51 in Automorphic Forms and Galois Representations, vol. 1, edited by F. Diamond et al., London Math. Soc. Lecture Note Series 414, Cambridge University Press, 2014.
  • C. Breuil, “Sur quelques représentations modulaires et $p$-adiques de ${\rm GL}\sb 2(\Qp)$: I”, Compositio Math. 138:2 (2003), 165–188.
  • C. Breuil and F. Herzig, “Ordinary representations of $G(\mathbb{Q}\sb p)$ and fundamental algebraic representations”, Duke Math. J. 164:7 (2015), 1271–1352.
  • C. Breuil and V. Paškūnas, Towards a modulo $p$ Langlands correspondence for ${\rm GL}\sb 2$, vol. 216, Mem. Amer. Math. Soc. 1016, 2012.
  • P. Colmez, “Représentations de ${\rm GL}\sb 2(\Qp)$ et $(\phi,\Gamma)$-modules”, pp. 281–509 in Représentations $p$-adiques de groupes $p$-adiques II: Représentations de ${\rm GL}\sb 2(\Qp)$ et $(\phi,\Gamma)$-modules, edited by L. Berger et al., Astérisque 330, 2010.
  • P. Colmez, G. Dospinescu, and V. Paškūnas, “The $p$-adic local Langlands correspondence for ${\rm GL}_2(\mathbb Q_p)$”, Camb. J. Math. 2:1 (2014), 1–47.
  • F. Digne and J. Michel, Representations of finite groups of Lie type, London Mathematical Society Student Texts 21, Cambridge University Press, 1991.
  • M. Emerton, “On a class of coherent rings, with applications to the smooth representation theory of ${\rm{GL}}_2(\Qp)$ in characteristic $p$”, prépublication, 2008, hook http://wvii-w.claymath.org/sites/default/files/emerton.pdf \posturlhook.
  • M. Emerton, “Ordinary parts of admissible representations of $p$-adic reductive groups I. Definition and first properties”, pp. 355–402 in $p$-adic representations of $p$-adic groups III: Global and geometrical methods, Astérisque 331, 2010.
  • M. Erdélyi and G. Zábrádi, “Links between generalized Montréal-functors”, prépublication, 2014.
  • J.-M. Fontaine, “Représentations $p$-adiques des corps locaux (1$^{\textrm{\`ere}}$ partie)”, pp. 249–309 in The Grothendieck Festschrift, vol. II, edited by P. Cartier et al., Progr. Math. 87, Birkhäuser, Boston, 1990.
  • P. Gabriel, “Des catégories abéliennes”, Bull. Soc. Math. France 90 (1962), 323–448.
  • J. Hauseux, “Extensions entre séries principales $p$-adiques et modulo $p$ de $G(F)$”, J. Inst. Math. Jussieu (publié en ligne août 2014).
  • F. Herzig, “The classification of irreducible admissible mod $p$ representations of a $p$-adic ${\rm GL}\sb n$”, Invent. Math. 186:2 (2011), 373–434.
  • J. C. Jantzen, Representations of algebraic groups, $2$-nd ed., Mathematical Surveys and Monographs 107, Amer. Math. Soc., Providence, RI, 2003.
  • S. Morra and B. Schraen, “Structure partielle de certaines représentations supersingulières de ${\mathrm{GL}}_2$”, en préparation.
  • R. Ollivier, “Resolutions for principal series representations of $p$-adic ${\rm GL}\sb n$”, Münster J. Math. 7 (2014), 225–240.
  • P. Schneider and M.-F. Vigneras, “A functor from smooth $o$-torsion representations to $(\phi,\Gamma)$-modules”, pp. 525–601 in On certain $L$-functions, edited by J. Arthur et al., Clay Math. Proc. 13, Amer. Math. Soc., Providence, RI, 2011.
  • B. Schraen, communications personnelles, novembre 2012 et mars 2014.
  • B. Schraen, “Sur la présentation des représentations supersingulières de ${\rm GL}\sb 2(F)$”, J. Reine Angew. Math. 704 (2015), 187–208.
  • J.-P. Serre, Cohomologie galoisienne, $5$-th ed., Lecture Notes in Mathematics 5, Springer, Berlin, 1994.
  • M.-F. Vignéras, “Série principale modulo $p$ de groupes réductifs $p$-adiques”, Geom. Funct. Anal. 17:6 (2008), 2090–2112.
  • G. Zábrádi, “Exactness of the reduction on étale modules”, J. Algebra 331 (2011), 400–415.