Duke Mathematical Journal

Multivariable (φ,Γ)-modules and locally analytic vectors

Laurent Berger

Full-text: Access denied (no subscription detected)

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

Let K be a finite extension of Qp, and let GK=Gal(Q¯p/K). There is a very useful classification of p-adic representations of GK in terms of cyclotomic (φ,Γ)-modules (cyclotomic means that Γ=Gal(K/K) where K is the cyclotomic extension of K). One particularly convenient feature of the cyclotomic theory is the fact that the (φ,Γ)-module attached to any p-adic representation is overconvergent.

Questions pertaining to the p-adic local Langlands correspondence lead us to ask for a generalization of the theory of (φ,Γ)-modules, with the cyclotomic extension replaced by an infinitely ramified p-adic Lie extension K/K. It is not clear what shape such a generalization should have in general. Even in the case where we have such a generalization, namely, the case of a Lubin–Tate extension, most (φ,Γ)-modules fail to be overconvergent.

In this article, we develop an approach that gives a solution to both problems at the same time, by considering the locally analytic vectors for the action of Γ inside some big modules defined using Fontaine’s rings of periods. We show that, in the cyclotomic case, we recover the usual overconvergent (φ,Γ)-modules. In the Lubin–Tate case, we can prove, as an application of our theory, a folklore conjecture in the field stating that (φ,Γ)-modules attached to F-analytic representations are overconvergent.

Article information

Source
Duke Math. J., Volume 165, Number 18 (2016), 3567-3595.

Dates
Received: 14 January 2015
Revised: 1 March 2016
First available in Project Euclid: 12 September 2016

Permanent link to this document
https://projecteuclid.org/euclid.dmj/1473686055

Digital Object Identifier
doi:10.1215/00127094-3674441

Mathematical Reviews number (MathSciNet)
MR3577371

Zentralblatt MATH identifier
06677445

Subjects
Primary: 11S
Secondary: 11F 12H 13J 22E 46S

Keywords
$(\varphi,\Gamma)$-module locally analytic vector $p$-adic period Lubin–Tate group $p$-adic monodromy

Citation

Berger, Laurent. Multivariable $(\varphi,\Gamma)$ -modules and locally analytic vectors. Duke Math. J. 165 (2016), no. 18, 3567--3595. doi:10.1215/00127094-3674441. https://projecteuclid.org/euclid.dmj/1473686055


Export citation

References

  • [1] Y. Amice, Interpolation $p$-adique, Bull. Soc. Math. France 92 (1964), 117–180.
  • [2] L. Berger, Représentations $p$-adiques et équations différentielles, Invent. Math. 148 (2002), 219–284.
  • [3] L. Berger, Construction de $(\varphi,\Gamma)$-modules: Représentations $p$-adiques et $B$-paires, Algebra Number Theory 2 (2008), 91–120.
  • [4] L. Berger, “Équations différentielles $p$-adiques et $(\varphi,{N})$-modules filtrés” in Représentations $p$-adiques de groupes $p$-adiques, I: Représentations galoisiennes et $(\varphi,\Gamma)$-modules, Astérisque 319, Soc. Math. France, Paris, 2008, 13–38.
  • [5] L. Berger, La correspondance de Langlands locale $p$-adique pour $\mathrm{GL}_{2}(\textbf{Q}_{p})$, Astérisque 339 (2011), 157–180.
  • [6] L. Berger, Multivariable Lubin–Tate $(\varphi,\Gamma)$-modules and filtered $\varphi$-modules, Math. Res. Lett. 20 (2013), 409–428.
  • [7] L. Berger, Lifting the field of norms, J. Éc. polytech. Math. 1 (2014), 29–38.
  • [8] L. Berger and P. Colmez, “Familles de représentations de de Rham et monodromie $p$-adique” in Représentations $p$-adiques de groupes $p$-adiques, I: Représentations galoisiennes et $(\varphi,\Gamma)$-modules, Astérisque 319, Soc. Math. France, Paris, 2008, 303–337.
  • [9] L. Berger and P. Colmez, Théorie de Sen et vecteurs localement analytiques, to appear in Ann. Sci. Éc. Norm. Supér. (4), preprint, arXiv:1405.5430v1 [math.NT].
  • [10] C. Breuil, “The emerging $p$-adic Langlands programme” in Proceedings of the International Congress of Mathematicians, II, Hindustan Book Agency, New Delhi, 2010, 203–230.
  • [11] F. Cherbonnier and P. Colmez, Représentations $p$-adiques surconvergentes, Invent. Math. 133 (1998), 581–611.
  • [12] J. Coates and R. Greenberg, Kummer theory for abelian varieties over local fields, Invent. Math. 124 (1996), 129–174.
  • [13] P. Colmez, Espaces de Banach de dimension finie, J. Inst. Math. Jussieu 1 (2002), 331–439.
  • [14] P. Colmez, Représentations de ${\mathrm{GL}}_{2}(\mathbf{Q}_{p})$ et $(\varphi,\Gamma)$-modules, Astérisque 330 (2010), 281–509.
  • [15] M. De Ieso, Espaces de fonctions de classe $C^{r}$ sur $\mathcal{O}_{F}$, Indag. Math. (N.S.) 24 (2013), 530–556.
  • [16] E. De Shalit, Mahler bases and elementary $p$-adic analysis, to appear in J. Théor. Nombres Bordeaux, preprint, http://jtnb.cedram.org/jtnb-bin/aparaitre/JTNB_0__0_0_A2_0 (accessed 23 August 2016).
  • [17] M. Emerton, Locally analytic vectors in representations of locally $p$-adic analytic groups, to appear in Mem. Amer. Math. Soc., preprint, arXiv:math/0405137v1 [math.RT].
  • [18] J.-M. Fontaine, “Représentations $p$-adiques des corps locaux, I” in The Grothendieck Festschrift, II, Progr. Math. 87, Birkhäuser, Boston, 1990, 249–309.
  • [19] J.-M. Fontaine, “Le corps des périodes $p$-adiques” in Périodes $p$-adiques (Bures-sur-Yvette, 1988), with an appendix by P. Colmez, Astérisque 223, Soc. Math. France, Paris, 1994, 59–111.
  • [20] J.-M. Fontaine and J.-P. Wintenberger, Extensions algébriques et corps des normes des extensions APF des corps locaux, C. R. Math. Acad. Sci. Paris A 288 (1979), A441–A444.
  • [21] J.-M. Fontaine and J.-P. Wintenberger, Le “corps des normes” de certaines extensions algébriques de corps locaux, C. R. Math. Acad. Sci. Paris A 288 (1979), A367–A370.
  • [22] L. Fourquaux, Applications $\textbf{Q}_{p}$-linéaires, continues et Galois-équivariantes de $\textbf{C}_{p}$ dans lui-même, J. Number Theory 129 (2009), 1246–1255.
  • [23] L. Fourquaux and B. Xie, Triangulable $\mathcal{O}_{F}$-analytic $(\varphi_{q},\Gamma)$-modules of rank $2$, Algebra Number Theory 7 (2013), 2545–2592.
  • [24] K. S. Kedlaya, Slope filtrations revisited, Doc. Math. 10 (2005), 447–525.
  • [25] K. S. Kedlaya, Some slope theory for multivariate Robba rings, preprint, arXiv:1311.7468v1 [math.NT].
  • [26] M. Kisin and W. Ren, Galois representations and Lubin-Tate groups, Doc. Math. 14 (2009), 441–461.
  • [27] J. Lubin and J. Tate, Formal complex multiplication in local fields, Ann. of Math. (2) 81 (1965), 380–387.
  • [28] S. Matsuda, Local indices of $p$-adic differential operators corresponding to Artin-Schreier-Witt coverings, Duke Math. J. 77 (1995), 607–625.
  • [29] P. Schneider and J. Teitelbaum, Algebras of $p$-adic distributions and admissible representations, Invent. Math. 153 (2003), 145–196.
  • [30] S. Sen, Ramification in $p$-adic Lie extensions, Invent. Math. 17 (1972), 44–50.
  • [31] J.-P. Serre, Lie Algebras and Lie Groups, 2nd ed., Lecture Notes in Math. 1500, Springer, Berlin, 2006.
  • [32] J. Wengenroth, Derived Functors in Functional Analysis, Lecture Notes in Math. 1810, Springer, Berlin, 2003.
  • [33] J.-P. Wintenberger, Le corps des normes de certaines extensions infinies de corps locaux; applications, Ann. Sci. Éc. Norm. Supér. (4) 16 (1983), 59–89.
  • [34] G. Zábrádi, Generalized Robba rings, with an appendix by P. Schneider, Israel J. Math. 191 (2012), 817–887.