Partant de l’hypothèse, inspirée par des travaux récents sur la correspondance de Langlands géométrique, que l’analogue de la correspondance de Langlands locale, pour les représentations modulo p de groupes p-adiques, pourrait prendre la forme d’une équivalence de catégories (supérieures) plutôt qu’une bijection d’ensembles, cet article présente une séries de spéculations et de questions sur les propriétés d’une telle équivalence hypothétique. Du côté galoisien de la correspondance, on trouverait une catégorie de faisceaux sur le champ ind-algébrique construit par Emerton et Gee, ou éventuellement une version dérivée de ce champ ; du côté automorphe de la correspondance, on trouverait une catégorie dérivée de dg-modules sur l’algèbre de Hecke dérivée étudiée par Schneider. Les deux côtés sont assez mystérieux, mais certaines des questions proposées dans cet article pourraient être accessibles.
Starting with the hypothesis, inspired by recent work on the geometric Langlands correspondence, that the analogue for mod p representations of p-adic groups of the local Langlands correspondence might be an equivalence of (higher) categories rather than a bijection of sets, this paper presents a series of speculations and questions about the properties of such a hypothetical equivalence. The Galois side of the correspondence would be a category of sheaves on the ind-algebraic stack constructed by Emerton and Gee, or perhaps a derived variant thereof ; the automorphic side of the correspondence would be a derived category of dg-modules over the derived Hecke algebra studied by Schneider. Both sides are quite mysterious, but some of the questions proposed in this paper may be accessible.
@article{AFST_2016_6_25_2-3_403_0,
author = {Michael Harris},
title = {Speculations on the mod $p$ representation theory of $p$-adic groups},
journal = {Annales de la Facult\'e des sciences de Toulouse : Math\'ematiques},
pages = {403--418},
publisher = {Universit\'e Paul Sabatier, Toulouse},
volume = {Ser. 6, 25},
number = {2-3},
year = {2016},
doi = {10.5802/afst.1499},
zbl = {1362.22022},
mrnumber = {3530163},
language = {en},
url = {https://afst.centre-mersenne.org/articles/10.5802/afst.1499/}
}
TY - JOUR AU - Michael Harris TI - Speculations on the mod $p$ representation theory of $p$-adic groups JO - Annales de la Faculté des sciences de Toulouse : Mathématiques PY - 2016 SP - 403 EP - 418 VL - 25 IS - 2-3 PB - Université Paul Sabatier, Toulouse UR - https://afst.centre-mersenne.org/articles/10.5802/afst.1499/ DO - 10.5802/afst.1499 LA - en ID - AFST_2016_6_25_2-3_403_0 ER -
%0 Journal Article %A Michael Harris %T Speculations on the mod $p$ representation theory of $p$-adic groups %J Annales de la Faculté des sciences de Toulouse : Mathématiques %D 2016 %P 403-418 %V 25 %N 2-3 %I Université Paul Sabatier, Toulouse %U https://afst.centre-mersenne.org/articles/10.5802/afst.1499/ %R 10.5802/afst.1499 %G en %F AFST_2016_6_25_2-3_403_0
Michael Harris. Speculations on the mod $p$ representation theory of $p$-adic groups. Annales de la Faculté des sciences de Toulouse : Mathématiques, Serie 6, Volume 25 (2016) no. 2-3, pp. 403-418. doi : 10.5802/afst.1499. https://afst.centre-mersenne.org/articles/10.5802/afst.1499/
[1] Abe (N.), Henniart (G.), Herzig (F.), Vignéras (M.-F.).— A classification of irreducible modulo representations of a reductive -adic group, manuscript (2014).
[2] Arinkin (D.), Gaitsgory (D.).— Singular support of coherent sheaves and the geometric Langlands conjecture, preprint (2012). | DOI | MR
[3] Breuil (C.), Paškūnas (V.).— Towards a modulo Langlands correspondence for , Memoirs of Amer. Math. Soc., 216 (2012). | DOI | MR
[4] Breuil (C.), Schneider (P.).— First steps towards p-adic Langlands functoriality, J. Reine Angew. Math., 610 p. 149-180 (2007). | DOI | MR | Zbl
[5] Barthel (L.), Livné (R.).— Irreducible modular representations of of a local field, Duke Math. J. 75, no 2, p. 261-292 (1994). | DOI | MR | Zbl
[6] Barthel (L.), Livné (R.).— Modular representations of GL(2) of a local field: the ordinary, unramified case, J. Number Theory 55, p. 1-27 (1995). | DOI
[7] Ben-Zvi (D.), Francis (J.), Nadler (D.).— Integral transforms and Drinfeld centers in derived algebraic geometry, J. Amer. Math. Soc., 23, p. 909-966 (2010). | DOI | MR | Zbl
[8] Ben-Zvi (D.), Nadler (D.).— Nonlinear traces, preprint 2013, arXiv:1305.7175v3 [math.AG].
[9] Buzzard (K.), Diamond (F.), Jarvis (F.).— On Serre’s conjecture for mod Galois representations over totally real fields, Duke Math. J., 155, p. 105-161 (2010). | DOI | MR
[10] Caraiani (A.), Emerton (M.), Gee (T.), Geraghty (D.), Paškūnas (V.), and Shin (S. W.).— Patching and the -adic local Langlands correspondence, preprint (2013). | DOI | MR
[11] Chenevier (G.).— The p-adic analytic space of pseudocharacters of a profinite group and pseudorepresentations over arbitrary rings, Automorphic forms and Galois representations, vol 1, London Mathematical Society Lecture Note Series 414, p. 221-285 (2014). | DOI | Zbl
[12] Emerton (M.), Gee (T.).— “Scheme-theoretic images” of morphisms of stacks, preprint (2015).
[13] Faltings (G.).— The trace formula and DrinfeldÕs upper halfplane, Duke Math. J., 76 p. 467-481 (1994). | DOI | MR | Zbl
[14] Fargues (L.).— -torseurs en théorie de Hodge -adique, preprint, 2015, at http://webusers.imj-prg.fr/ laurent.fargues/Prepublications.html.
[15] Harris (M.).— Mathematics without Apologies: Portrait of a Problematic Vocation, Princeton: Princeton University Press (2015). | DOI | Zbl
[16] Hartl (U.), Hellmann (E.).— The universal family of semi-stable -adic Galois representations, preprint (2013). | DOI | MR
[17] Helm (D.).— Whittaker models and the integral Bernstein center for , preprint, arxiv:1210.1789. | DOI | MR | Zbl
[18] Kohlhaase (J.).— Smooth duality in natural characteristic, preprint, on line at http://www.esaga.uni-due.de/jan.kohlhaase/publications/. | DOI | MR | Zbl
[19] Lafforgue (V.).— Chtoucas pour les groupes réductifs et paramétrisation de Langlands globale, preprint 2013 (version 4). | DOI | Zbl
[20] Loday (J.-L.).— Cyclic Homology, second edition, Berlin, Heidelberg, New York: Springer (1998). | DOI | Zbl
[21] Ollivier (R.).— Le foncteur des invariants sous l’action du pro--Iwahori de , J. reine angew. Math., 635 p. 149-185 (2009). | DOI | Zbl
[22] Ollivier (R.).— Compatibility between Satake and Bernstein-type isomorphisms in characteristic , preprint (2012). | DOI | Zbl
[23] Paškūnas (V.).— The image of Colmez’s Montreal functor, Publ. Math. IHES, 118 p. 1-191 (2013). | DOI | MR | Zbl
[24] Schneider (P.).— Smooth representations and Hecke modules in characteristic , preprint 2007. | DOI
[25] Scholze (P.).— Remarks [on] the Cohomology of the Lubin-Tate Tower, notes taken by D. Collins, available at http://www.msri.org/workshops/731/schedules/17615. | MR | Zbl
[26] Scholze (P.).— Lectures on -adic geometry, Fall 2014, notes by J. Weinstein, available at https://math.berkeley.edu/ jared/Math274/ScholzeLectures.pdf | Zbl
[27] Strauch (M.).— Deformation spaces of one-dimensional formal groups and their cohomology Advances in Mathematics, 217 p. 889-951 (2008). | DOI | MR | Zbl
[28] Toën (B.), Vezzosi (G.).— Infinies-categories monoïdales rigides, traces et caractères de Chern, e-print arXiv:0903.3292
[29] Vignéras (M.-F.).— Pro-p-Iwahori Hecke ring and supersingular Fp-representations. Mathematische Annalen 331, 523-556. Erratum volume 333, no. 3, p. 699-701 (2005). | DOI | MR
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