The local Langlands correspondence for ${\mathrm{GL}}_n$ in families

Emerton, Matthew; Helm, David

Annales scientifiques de l'École Normale Supérieure, Série 4, Tome 47 (2014) no. 4, pp. 655-722.

Soit $k$ un corps fini de caractéristique $p$ et soit $E$ un corps $\ell$-adique où $\ell \ne p$. Étant donnée une représentation $\rho :G_E\to {\mathrm{GL}}_n\left(A\right)$, où $A$ est une $W\left(k\right)$-algèbre locale, nous étudions le problème de la recherche de $A\left[{\mathrm{GL}}_n\left(E\right)\right]$-module admissible $\pi \left(\rho \right)$ qui « interpole la correspondance de Langlands locale » pour $\rho$ sur les points de $\mathrm{Spec}A$. Nous formulons une version précise de ce problème et montrons qu'il a au plus une solution, à isomorphisme près. Le premier auteur a montré [6] que lorsque $A$ est une algèbre de Hecke, et $\rho :G_{{\mathbb{Q}}_{\ell }}\to {\mathrm{GL}}_2\left(A\right)$ est la représentation naturelle de $G_{{\mathbb{Q}}_{\ell }}$ sur $A$, alors le $A\left[{\mathrm{GL}}_2\left(E\right)\right]$-module $\pi \left(\rho \right)$ existe et apparaît naturellement dans la cohomologie de la tour des courbes modulaires.

Let $k$ be a finite field of characteristic $p$, and let $E$ be an $\ell$-adic field for $\ell \ne p$. Given a representation $\rho :G_E\to {\mathrm{GL}}_n\left(A\right)$ where $A$ is a local $W\left(k\right)$-algebra, we study the problem of finding an admissible $A\left[{\mathrm{GL}}_n\left(E\right)\right]$-module $\pi \left(\rho \right)$ that “interpolates the local Langlands correspondence” for $\rho$ over the points of $\mathrm{Spec}A$. We formulate a precise version of this problem and show that it has at most one solution, up to isomorphism. The first author has shown [6] that when $A$ is a Hecke algebra, and $\rho :G_{{\mathbb{Q}}_{\ell }}\to {\mathrm{GL}}_2\left(A\right)$ is the natural representation of $G_{{\mathbb{Q}}_{\ell }}$ over $A$, the corresponding $A\left[{\mathrm{GL}}_n\left(E\right)\right]$-module $\pi \left(\rho \right)$ exists and arises naturally in the completed cohomology of the tower of modular curves.

Classification : 11S37, 11F33, 11F70, 22E50
Keywords: Local Langlands correspondence, congruences.
Mot clés : Correspondance de Langlands locale, congruences.

Breuil, C.; Schneider, P. First steps towards $p$-adic Langlands functoriality, J. reine angew. Math., Volume 610 (2007), pp. 149-180 (ISSN: 0075-4102) | DOI | MR | Zbl

Bernstein, I. N.; Zelevinsky, A. V. Induced representations of reductive $𝔭$-adic groups. I, Ann. Sci. École Norm. Sup., Volume 10 (1977), pp. 441-472 (ISSN: 0012-9593) | DOI | Numdam | MR | Zbl

Chenevier, G. Une application des variétés de Hecke des groups unitaires (2009) (preprint http://www.math.polytechnique.fr/~chenevier/articles/famgal.pdf ) | MR

Choi, S. H. Local deformation lifting spaces of mod $\ell$ Galois representations, ISBN: 978-1109-25082-4, ProQuest LLC, Ann Arbor, MI (2009) http://gateway.proquest.com/openurl?url_ver=Z39.88-2004&rft_val_fmt=info:ofi/fmt:kev:mtx:dissertation&res_dat=xri:pqdiss&rft_dat=xri:pqdiss:3365229 | MR

Deligne, P., Modular functions of one variable, II (Proc. Internat. Summer School, Univ. Antwerp, Antwerp, 1972) (Lecture Notes in Math.), Volume 349, Springer, Berlin, 1973, pp. 501-597 | MR | Zbl

Emerton, M. Local-global compatibility for mod $p$ and $p$-adic modular forms (in preparation)

Emerton, M. A local-global compatibility conjecture in the $p$-adic Langlands programme for ${\mathrm{GL}}_{2/\mathbb{Q}}$ , Pure Appl. Math. Q., Volume 2 (2006), pp. 279-393 (ISSN: 1558-8599) | DOI | MR | Zbl

Helm, D. On the modified mod $p$ local Langlands correspondence for ${\mathrm{GL}}_2\left({\mathbb{Q}}_{\ell }\right)$ (preprint; http://www2.imperial.ac.uk/~dhelm/correspondence.pdf ) | MR | Zbl

Helm, D. The Bernstein center of the category of smooth $W\left(k\right)\left[{\mathrm{GL}}_n\left(F\right)\right]$-modules (preprint arXiv:1201.1874 ) | MR

Helm, D. Towards a Galois-theoretic interpretation of the integral Bernstein center (in preparation)

Helm, D. Whittaker models and the integral Bernstein center for ${\mathrm{GL}}_n$ (preprint http://www2.imperial.ac.uk/~dhelm/bernstein3.pdf ) | MR

Harris, M.; Taylor, R., Annals of Math. Studies, 151, Princeton Univ. Press, Princeton, NJ, 2001, 276 pages (ISBN: 0-691-09090-4) | MR | Zbl

Jacquet, H.; Shalika, J. The Whittaker models of induced representations, Pacific J. Math., Volume 109 (1983), pp. 107-120 http://projecteuclid.org/euclid.pjm/1102720206 (ISSN: 0030-8730) | DOI | MR | Zbl

Kudla, S. S., Motives (Seattle, WA, 1991) (Proc. Sympos. Pure Math.), Volume 55, Amer. Math. Soc., Providence, RI, 1994, pp. 365-391 | MR | Zbl

Tadić, M. Induced representations of $\mathrm{GL}(n,A)$ for $p$-adic division algebras $A$ , J. reine angew. Math., Volume 405 (1990), pp. 48-77 (ISSN: 0075-4102) | DOI | MR | Zbl

Tate, J., Automorphic forms, representations and $L$-functions (Proc. Sympos. Pure Math., Oregon State Univ., Corvallis, Ore., 1977), Part 2 (Proc. Sympos. Pure Math., XXXIII), Amer. Math. Soc., Providence, R.I., 1979, pp. 3-26 | MR | Zbl

Vignéras, M.-F. Correspondance de Langlands semi-simple pour $\mathrm{GL}(n,F)$ modulo $l\ne p$ , Invent. Math., Volume 144 (2001), pp. 177-223 (ISSN: 0020-9910) | DOI | MR | Zbl

Vignéras, M.-F., Progress in Math., 137, Birkhäuser, 1996 (ISBN: 0-8176-3929-2) | MR | Zbl

Vignéras, M.-F. Cohomology of sheaves on the building and $R$-representations, Invent. Math., Volume 127 (1997), pp. 349-373 (ISSN: 0020-9910) | DOI | MR | Zbl

Vignéras, M.-F. Induced $R$-representations of $p$-adic reductive groups, Selecta Math. (N.S.), Volume 4 (1998), pp. 549-623 (ISSN: 1022-1824) | DOI | MR | Zbl

Zelevinsky, A. V. Induced representations of reductive $𝔭$-adic groups. II. On irreducible representations of $\mathrm{GL}\left(n\right)$ , Ann. Sci. École Norm. Sup., Volume 13 (1980), pp. 165-210 (ISSN: 0012-9593) | DOI | Numdam | MR | Zbl

Zelevinsky, A. V. The $p$-adic analogue of the Kazhdan-Lusztig conjecture, Funktsional. Anal. i Prilozhen., Volume 15 (1981), pp. 9-21 (ISSN: 0374-1990) | MR | Zbl