The Breuil–Mézard Conjecture for quaternion algebras
[La conjecture de Breuil–Mézard pour les algèbres de quaternions]
Annales de l'Institut Fourier, Tome 65 (2015) no. 4, pp. 1557-1575.

Nous formulons une version de la conjecture de Breuil–Mézard pour les algèbres de quaternions. Nous montrons que cette version est une consequence de la version originale pour GL 2 . Une partie de la démonstration est la construction d’un analogue modulo p de la correspondance de Jacquet–Langlands pour les représentations de GL 2 (k) ou k est un corps fini de caractéristique p.

We formulate a version of the Breuil–Mézard conjecture for quaternion algebras, and show that it follows from the Breuil–Mézard conjecture for GL 2 . In the course of the proof we establish a mod p analogue of the Jacquet–Langlands correspondence for representations of GL 2 (k), k a finite field of characteristic p.

DOI : https://doi.org/10.5802/aif.2967
Classification : 11F80,  11F33
Mots clés : Représentations galoisiennes, Conjecture de Breuil–Mézard
@article{AIF_2015__65_4_1557_0,
     author = {Gee, Toby and Geraghty, David},
     title = {The {Breuil{\textendash}M\'ezard} {Conjecture} for quaternion algebras},
     journal = {Annales de l'Institut Fourier},
     pages = {1557--1575},
     publisher = {Association des Annales de l{\textquoteright}institut Fourier},
     volume = {65},
     number = {4},
     year = {2015},
     doi = {10.5802/aif.2967},
     language = {en},
     url = {http://archive.numdam.org/articles/10.5802/aif.2967/}
}
TY  - JOUR
AU  - Gee, Toby
AU  - Geraghty, David
TI  - The Breuil–Mézard Conjecture for quaternion algebras
JO  - Annales de l'Institut Fourier
PY  - 2015
DA  - 2015///
SP  - 1557
EP  - 1575
VL  - 65
IS  - 4
PB  - Association des Annales de l’institut Fourier
UR  - http://archive.numdam.org/articles/10.5802/aif.2967/
UR  - https://doi.org/10.5802/aif.2967
DO  - 10.5802/aif.2967
LA  - en
ID  - AIF_2015__65_4_1557_0
ER  - 
Gee, Toby; Geraghty, David. The Breuil–Mézard Conjecture for quaternion algebras. Annales de l'Institut Fourier, Tome 65 (2015) no. 4, pp. 1557-1575. doi : 10.5802/aif.2967. http://archive.numdam.org/articles/10.5802/aif.2967/

[1] Breuil, Christophe; Diamond, Fred Formes modulaires de Hilbert modulo p et valeurs d’extensions entre caractères galoisiens, Ann. Sci. Éc. Norm. Supér. (4), Volume 47 (2014) no. 5, pp. 905-974 | MR 3294620

[2] Breuil, Christophe; Mézard, Ariane Multiplicités modulaires et représentations de GL 2 (Z p ) et de Gal (Q ¯ p /Q p ) en l=p, Duke Math. J., Volume 115 (2002) no. 2, pp. 205-310 (With an appendix by Guy Henniart) | Article | MR 1944572 | Zbl 1042.11030

[3] Carayol, H. Représentations cuspidales du groupe linéaire, Ann. Sci. École Norm. Sup. (4), Volume 17 (1984) no. 2, pp. 191-225 | Numdam | MR 760676 | Zbl 0549.22009

[4] Diamond, Fred A correspondence between representations of local Galois groups and Lie-type groups, L-functions and Galois representations (London Math. Soc. Lecture Note Ser.), Volume 320, Cambridge Univ. Press, Cambridge, 2007, pp. 187-206 | Article | MR 2392355 | Zbl 1230.11069

[5] Gee, Toby; Kisin, Mark The Breuil-Mézard conjecture for potentially Barsotti-Tate representations, Forum Math. Pi, Volume 2 (2014), pp. e1, 56 | Article | MR 3292675

[6] Gee, Toby; Savitt, David Serre weights for quaternion algebras, Compos. Math., Volume 147 (2011) no. 4, pp. 1059-1086 | Article | MR 2822861 | Zbl 1282.11042

[7] Harris, Michael; Taylor, Richard The geometry and cohomology of some simple Shimura varieties, Annals of Mathematics Studies, 151, Princeton University Press, Princeton, NJ, 2001, pp. viii+276 (With an appendix by Vladimir G. Berkovich) | MR 1876802 | Zbl 1036.11027

[8] Kisin, Mark Potentially semi-stable deformation rings, J. Amer. Math. Soc., Volume 21 (2008) no. 2, pp. 513-546 | Article | MR 2373358 | Zbl 1205.11060

[9] Kisin, Mark The Fontaine-Mazur conjecture for GL 2 , J. Amer. Math. Soc., Volume 22 (2009) no. 3, pp. 641-690 | Article | MR 2505297 | Zbl 1251.11045

[10] Kisin, Mark The structure of potentially semi-stable deformation rings, Proceedings of the International Congress of Mathematicians. Volume II (2010), pp. 294-311 | MR 2827797 | Zbl 1273.11090

[11] Kutzko, Philip Character formulas for supercuspidal representations of GL l ,l a prime, Amer. J. Math., Volume 109 (1987) no. 2, pp. 201-221 | Article | MR 882420 | Zbl 0618.22006

[12] Paškūnas, Vytautas On the Breuil-Mézard conjecture, Duke Math. J., Volume 164 (2015) no. 2, pp. 297-359 | Article | MR 3306557

Cité par Sources :