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Towards a modulo $p$ Langlands correspondence for ${\mathrm {GL}}_2$
About this Title
Christophe Breuil, C.N.R.S. & I.H.É.S., Le Bois-Marie, 35 route de Chartres, 91440 Bures-sur-Yvette, France and Vytautas Paškūnas, Fakultät für Mathematik, Universität Bielefeld, Postfach 100131, D-33501 Bielefeld, Germany
Publication: Memoirs of the American Mathematical Society
Publication Year:
2012; Volume 216, Number 1016
ISBNs: 978-0-8218-5227-9 (print); 978-0-8218-8525-3 (online)
DOI: https://doi.org/10.1090/S0065-9266-2011-00623-4
Published electronically: May 23, 2011
Keywords: Supersingular,
$\mathrm {mod}\, p$ Langlands correspondence,
Serre weights
MSC: Primary 22E50, 11F80, 11F70
Table of Contents
Chapters
- 1. Introduction
- 2. Representation theory of $\Gamma$ over $\bar {\mathbb F}_p$ I
- 3. Representation theory of $\Gamma$ over $\bar {\mathbb F}_p$ II
- 4. Representation theory of $\Gamma$ over $\bar {\mathbb F}_p$ III
- 5. Results on $K$-extensions
- 6. Hecke algebra
- 7. Computation of $\mathbb {R}^1\mathcal {I}$ for principal series
- 8. Extensions of principal series
- 9. General theory of diagrams and representations of ${\mathrm {GL}}_2$
- 10. Examples of diagrams
- 11. Generic Diamond weights
- 12. The unicity Lemma
- 13. Generic Diamond diagrams
- 14. The representations $D_{0}(\rho )$ and $D_1(\rho )$
- 15. Decomposition of generic Diamond diagrams
- 16. Generic Diamond diagrams for $f\in \{1,2\}$
- 17. The representation $R(\sigma )$
- 18. The extension Lemma
- 19. Generic Diamond diagrams and representations of ${\mathrm {GL}}_2$
- 20. The case $F=\mathbb Q_{p}$
Abstract
We construct new families of smooth admissible $\bar {\mathbb {F}}_p$-representations of $\mathrm {GL}_2(F)$, where $F$ is a finite extension of $\mathbb {Q}_p$. When $F$ is unramified, these representations have the $\mathrm {GL}_2({\mathcal O}_F)$-socle predicted by the recent generalizations of Serre’s modularity conjecture. Our motivation is a hypothetical mod $p$ Langlands correspondence.- J. L. Alperin, Local representation theory, Cambridge Studies in Advanced Mathematics, vol. 11, Cambridge University Press, Cambridge, 1986. Modular representations as an introduction to the local representation theory of finite groups. MR 860771
- Henning Haahr Andersen, Jens Jørgensen, and Peter Landrock, The projective indecomposable modules of $\textrm {SL}(2,\,p^{n})$, Proc. London Math. Soc. (3) 46 (1983), no. 1, 38–52. MR 684821, DOI 10.1112/plms/s3-46.1.38
- Matthew Bardoe and Peter Sin, The permutation modules for $\textrm {GL}(n+1,\textbf {F}_q)$ acting on $\textbf {P}^n(\textbf {F}_q)$ and $\textbf {F}^{n-1}_q$, J. London Math. Soc. (2) 61 (2000), no. 1, 58–80. MR 1745400, DOI 10.1112/S002461079900839X
- L. Barthel and R. Livné, Irreducible modular representations of $\textrm {GL}_2$ of a local field, Duke Math. J. 75 (1994), no. 2, 261–292. MR 1290194, DOI 10.1215/S0012-7094-94-07508-X
- Laurent Berger, Représentations modulaires de $\textrm {GL}_2(\mathbf Q_p)$ et représentations galoisiennes de dimension 2, Astérisque 330 (2010), 263–279 (French, with English and French summaries). MR 2642408
- Christophe Breuil, Sur quelques représentations modulaires et $p$-adiques de $\textrm {GL}_2(\mathbf Q_p)$. I, Compositio Math. 138 (2003), no. 2, 165–188 (French, with English summary). MR 2018825, DOI 10.1023/A:1026191928449
- Breuil C., Representations of Galois and of $\operatorname {GL}_2$ in characteristic $p$, graduate course at Columbia university, Fall 2007, available at http://www.ihes.fr/$\sim$breuil/PUBLICATIONS/New-York.pdf
- Breuil C., Diagrammes de Diamond et $(\varphi ,\Gamma )$-modules, Israel J. Math. 182, 2011, 349–382.
- Breuil C., Sur un problème de compatibilité local-global modulo $p$ pour $\operatorname {GL}_2$ (with an appendix by L. Dembélé), preprint 2009, available at http://www.ihes.fr/$\sim$breuil/PUBLICATIONS/compamodp.pdf
- Buzzard K., draft, March 17, 2006.
- Kevin Buzzard, Fred Diamond, and Frazer Jarvis, On Serre’s conjecture for mod $\ell$ Galois representations over totally real fields, Duke Math. J. 155 (2010), no. 1, 105–161. MR 2730374, DOI 10.1215/00127094-2010-052
- Chang S., Extensions of rank one $(\varphi ,\Gamma )$-modules, Ph.D. univ. Brandeis, 2006.
- Chang S., Diamond F., Extensions of $(\varphi ,\Gamma )$-modules and crystalline representations, Compositio Math. 147, 2011, 375–427.
- Pierre Colmez, Représentations de $\textrm {GL}_2(\mathbf Q_p)$ et $(\phi ,\Gamma )$-modules, Astérisque 330 (2010), 281–509 (French, with English and French summaries). MR 2642409
- Fred Diamond, A correspondence between representations of local Galois groups and Lie-type groups, $L$-functions and Galois representations, London Math. Soc. Lecture Note Ser., vol. 320, Cambridge Univ. Press, Cambridge, 2007, pp. 187–206. MR 2392355, DOI 10.1017/CBO9780511721267.006
- Emerton M., draft (email), March 4, 2006.
- Matthew Emerton, Ordinary parts of admissible representations of $p$-adic reductive groups II. Derived functors, Astérisque 331 (2010), 403–459 (English, with English and French summaries). MR 2667883
- Gee T., On the weight of mod $p$ Hilbert modular forms, Inventiones Math. 184, 2011, 1–46.
- Gee T., Savitt D., Serre weights for mod $p$ Hilbert modular forms: the totally ramified case, to appear in J. Reine Angew. Math.
- Herzig F., The classification of irreducible admissible mod $p$ representations of a $p$-adic $\operatorname {GL}_n$, to appear in Inventiones Math.
- Hu Y., Diagrammes canoniques et représentations modulo $p$ de $\operatorname {GL}_2(F)$, to appear in J. Inst. Math. Jussieu.
- Hu Y., Sur quelques représentations supersingulières de $\operatorname {GL}_2({\mathbb Q}_{p^f})$, J. Algebra 324, 2010, 1577–1615.
- A. Vincent Jeyakumar, Principal indecomposable representations for the group $\textrm {SL}(2,\,q)$, J. Algebra 30 (1974), 444–458. MR 342601, DOI 10.1016/0021-8693(74)90216-6
- Rachel Ollivier, Le foncteur des invariants sous l’action du pro-$p$-Iwahori de $\textrm {GL}_2(F)$, J. Reine Angew. Math. 635 (2009), 149–185 (French, with English summary). MR 2572257, DOI 10.1515/CRELLE.2009.078
- Paškūnas V., Coefficient systems and supersingular representations of $\operatorname {GL}_2(F)$, Mém. Soc. Math. de France 99, 2004.
- Paškūnas V., Admissible unitary completions of locally ${\mathbb Q}_{p}$-rational representations of $\operatorname {GL}_2(F)$, Representation Theory 14, 2010, 324–354.
- Vytautas Paškūnas, Extensions for supersingular representations of $\textrm {GL}_2(\Bbb Q_p)$, Astérisque 331 (2010), 317–353 (English, with English and French summaries). MR 2667891
- Michael M. Schein, Weights in Serre’s conjecture for Hilbert modular forms: the ramified case, Israel J. Math. 166 (2008), 369–391. MR 2430440, DOI 10.1007/s11856-008-1035-9
- Schein M., An irreducibility criterion for supersingular mod $p$ representations of $\operatorname {GL}_2(F)$, for $F$ a totally ramified extension of ${\mathbb Q}_{p}$, to appear in Transactions Amer. Math. Soc.
- Serre J.-P., Cohomologie galoisienne, Lecture Notes in Maths 5, Springer, 1997.
- Jean-Pierre Serre, Linear representations of finite groups, Springer-Verlag, New York-Heidelberg, 1977. Translated from the second French edition by Leonard L. Scott; Graduate Texts in Mathematics, Vol. 42. MR 0450380
- B. Shreekantha Upadhyaya, Composition factors of the principal indecomposable modules for the special linear group $\textrm {SL}(2,\,q)$, J. London Math. Soc. (2) 17 (1978), no. 3, 437–445. MR 500628, DOI 10.1112/jlms/s2-17.3.437
- Marie-France Vignéras, Representations modulo $p$ of the $p$-adic group $\textrm {GL}(2,F)$, Compos. Math. 140 (2004), no. 2, 333–358. MR 2027193, DOI 10.1112/S0010437X03000071
- Vignéras M.-F., The Colmez functor for $\operatorname {GL}(2,F)$, preprint 2009.