## Duke Mathematical Journal

### Multivariable $(\varphi,\Gamma)$-modules and locally analytic vectors

Laurent Berger

#### Abstract

Let $K$ be a finite extension of $\mathbf{Q}_{p}$, and let $G_{K}=\mathrm{Gal}(\overline{\mathbf{Q}}_{p}/K)$. There is a very useful classification of $p$-adic representations of $G_{K}$ in terms of cyclotomic $(\varphi,\Gamma)$-modules (cyclotomic means that $\Gamma=\mathrm{Gal}(K_{\infty}/K)$ where $K_{\infty}$ is the cyclotomic extension of $K$). One particularly convenient feature of the cyclotomic theory is the fact that the $(\varphi,\Gamma)$-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 $(\varphi,\Gamma)$-modules, with the cyclotomic extension replaced by an infinitely ramified $p$-adic Lie extension $K_{\infty}/K$. 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 $(\varphi,\Gamma)$-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 $\Gamma$ inside some big modules defined using Fontaine’s rings of periods. We show that, in the cyclotomic case, we recover the usual overconvergent $(\varphi,\Gamma)$-modules. In the Lubin–Tate case, we can prove, as an application of our theory, a folklore conjecture in the field stating that $(\varphi,\Gamma)$-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

#### 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

#### 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.