Gee, Toby; Geraghty, David The Breuil-Mézard conjecture for quaternion algebras. (La conjecture de Breuil-Mézard pour les algèbres de quaternions.) (English. French summary) Zbl 1395.11085 Ann. Inst. Fourier 65, No. 4, 1557-1575 (2015). Summary: We formulate a version of the Breuil-Mézard conjecture for quaternion algebras, and show that it follows from the Breuil-Mézard conjecture for \(\mathrm{GL}_2\). In the course of the proof we establish a mod \(p\) analogue of the Jacquet-Langlands correspondence for representations of \(\mathrm{GL}_2(k)\), \(k\) a finite field of characteristic \(p\). Cited in 4 Documents MSC: 11F80 Galois representations 11F33 Congruences for modular and \(p\)-adic modular forms Keywords:Galois representations; Breuil-Mézard conjecture PDFBibTeX XMLCite \textit{T. Gee} and \textit{D. Geraghty}, Ann. Inst. Fourier 65, No. 4, 1557--1575 (2015; Zbl 1395.11085) Full Text: DOI arXiv References: [1] Breuil, Christophe; Diamond, Fred, Formes modulaires de Hilbert modulo \(p\) et valeurs d’extensions entre caractères galoisiens, Ann. Sci. Éc. Norm. Supér. (4), 47, 5, 905-974 (2014) · Zbl 1309.11046 [2] Breuil, Christophe; Mézard, Ariane, Multiplicités modulaires et représentations de \({\rm GL}_2({\bf Z}_p)\) et de \({\rm Gal}(\overline{\bf Q}_p/{\bf Q}_p)\) en \(l=p\), Duke Math. J., 115, 2, 205-310 (2002) · Zbl 1042.11030 [3] Carayol, H., Représentations cuspidales du groupe linéaire, Ann. Sci. École Norm. Sup. (4), 17, 2, 191-225 (1984) · Zbl 0549.22009 [4] Diamond, Fred, \(L\)-functions and Galois representations, 320, 187-206 (2007) · Zbl 1230.11069 [5] Gee, Toby; Kisin, Mark, The Breuil-Mézard conjecture for potentially Barsotti-Tate representations, Forum Math. Pi, 2 (2014) · Zbl 1408.11033 [6] Gee, Toby; Savitt, David, Serre weights for quaternion algebras, Compos. Math., 147, 4, 1059-1086 (2011) · Zbl 1282.11042 [7] Harris, Michael; Taylor, Richard, The geometry and cohomology of some simple Shimura varieties, 151 (2001) · Zbl 1036.11027 [8] Kisin, Mark, Potentially semi-stable deformation rings, J. Amer. Math. Soc., 21, 2, 513-546 (2008) · Zbl 1205.11060 [9] Kisin, Mark, The Fontaine-Mazur conjecture for \({\rm GL}_2\), J. Amer. Math. Soc., 22, 3, 641-690 (2009) · Zbl 1251.11045 [10] Kisin, Mark, The structure of potentially semi-stable deformation rings, Proceedings of the International Congress of Mathematicians. Volume II, 294-311 (2010) · Zbl 1273.11090 [11] Kutzko, Philip, Character formulas for supercuspidal representations of \({\rm GL}_l,\;l\) a prime, Amer. J. Math., 109, 2, 201-221 (1987) · Zbl 0618.22006 [12] Paškūnas, Vytautas, On the Breuil-Mézard conjecture, Duke Math. J., 164, 2, 297-359 (2015) · Zbl 1376.11049 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.