The Breuil–Mézard Conjecture for quaternion algebras

Gee, Toby; Geraghty, David

doi:10.5802/aif.2967

The Breuil–Mézard Conjecture for quaternion algebras
[La conjecture de Breuil–Mézard pour les algèbres de quaternions]

Gee, Toby ¹ ; Geraghty, David ²

¹ Department of Mathematics Imperial College London 180 Queen’s Gate London SW7 2RH (UK)
² Department of Mathematics Boston College Carney Hall 301 Chestnut Hill MA 02467 (USA)

Annales de l'Institut Fourier, Tome 65 (2015) no. 4, pp. 1557-1575.

Résumé
Abstract

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 : 10.5802/aif.2967

Classification : 11F80, 11F33
Keywords: Galois representations, Breuil–Mézard Conjecture
Mot clés : Représentations galoisiennes, Conjecture de Breuil–Mézard

Affiliations des auteurs :

Gee, Toby ¹ ; Geraghty, David ²

¹ Department of Mathematics Imperial College London 180 Queen’s Gate London SW7 2RH (UK)
² Department of Mathematics Boston College Carney Hall 301 Chestnut Hill MA 02467 (USA)

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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://www.numdam.org/articles/10.5802/aif.2967/

Bibliographie
Cité par

[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

[2] Breuil, Christophe; Mézard, Ariane Multiplicités modulaires et représentations de ${GL}_{2} (Z_{p})$ et de $Gal ({\bar{Q}}_{p} / Q_{p})$ en $l = p$ , Duke Math. J., Volume 115 (2002) no. 2, pp. 205-310 (With an appendix by Guy Henniart) | DOI | MR | Zbl

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

[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 | DOI | MR | Zbl

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

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

[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 | Zbl

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

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

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

[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 | DOI | MR | Zbl

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

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