Geodesic
THE BREUIL–MÉZARD CONJECTURE FOR POTENTIALLY BARSOTTI–TATE REPRESENTATIONS
Forum of Mathematics, Pi, Tome 2 (2014)

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We prove the Breuil–Mézard conjecture for two-dimensional potentially Barsotti–Tate representations of the absolute Galois group $G_{K}$, $K$ a finite extension of $\mathbb{Q}_{p}$, for any $p>2$ (up to the question of determining precise values for the multiplicities that occur). In the case that $K/\mathbb{Q}_{p}$ is unramified, we also determine most of the multiplicities. We then apply these results to the weight part of Serre’s conjecture, proving a variety of results including the Buzzard–Diamond–Jarvis conjecture. 
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TOBY GEE; MARK KISIN. THE BREUIL–MÉZARD CONJECTURE FOR POTENTIALLY BARSOTTI–TATE REPRESENTATIONS. Forum of Mathematics, Pi, Tome 2 (2014). doi: 10.1017/fmp.2014.1

[BLGG11] Barnet-Lamb, T., Gee, T. and Geraghty, D., ‘The Sato–Tate conjecture for Hilbert modular forms’, J. Amer. Math. Soc. 24(2) (2011), 411–469.Google Scholar

[BLGG12] Barnet-Lamb, T., Gee, T. and Geraghty, D., ‘Congruences between Hilbert modular forms: constructing ordinary lifts’, Duke Math. J. 161(8) (2012), 1521–1580.Google Scholar

[BLGG13a] Barnet-Lamb, T., Gee, T. and Geraghty, D., ‘Congruences between Hilbert modular forms: constructing ordinary lifts, II’, Math. Res. Lett. 20(1) (2013), 67–72.Google Scholar

[BLGG13b] Barnet-Lamb, T., Gee, T. and Geraghty, D., ‘Serre weights for rank two unitary groups’, Math. Ann. 356(4) (2013), 1551–1598.Google Scholar

[BLGHT11] Barnet-Lamb, T., Geraghty, D., Harris, M. and Taylor, R., ‘A family of Calabi–Yau varieties and potential automorphy II’, Publ. Res. Inst. Math. Sci. 47(1) (2011), 29–98.Google Scholar

[BLGGT14a] Barnet-Lamb, T., Gee, T., Geraghty, D. and Taylor, R., ‘Potential automorphy and change of weight’, Ann. of Math. (2) 179(2) (2014), 501–609.Google Scholar

[BLGGT14b] Barnet-Lamb, T., Gee, T., Geraghty, D. and Taylor, R., ‘Local-global compatibility for l = p, II’, Ann. Sci. Éc. Norm. Supér. 47(1) (2014), 165–179.Google Scholar

[BC11] Bellaïche, J. and Chenevier, G., ‘The sign of Galois representations attached to automorphic forms for unitary groups’, Compositio Math. 147(5) (2011), 1337–1352.Google Scholar

[Bla06] Blasius, D., ‘Hilbert modular forms and the Ramanujan conjecture’, inNoncommutative Geometry and Number Theory, Aspects of Mathematics, E37 (Vieweg, Wiesbaden, 2006), 35–56.Google Scholar

[BD13] Breuil, C. and Diamond, F., ‘Formes modulaires de Hilbert modulo et valeurs d’extensions Galoisiennes’, Ann. Sci. Éc. Norm. Supér. (2014), to appear.Google Scholar

[BM02] Breuil, C. and Mézard, A., ‘Multiplicités modulaires et représentations de GL( ) et de Gal( Q ∕ ) en l = p’, Duke Math. J. 115(2) (2002), 205–310; with an appendix by G. Henniart.Google Scholar

[BDJ10] Buzzard, K., Diamond, F. and Jarvis, F., ‘On Serre’s conjecture for mod l Galois representations over totally real fields’, Duke Math. J. 155(1) (2010), 105–161.Google Scholar

[Cal12] Calegari, F., ‘Even Galois representations and the Fontaine–Mazur conjecture II’, J. Amer. Math. Soc. 25(2) (2012), 533–554.Google Scholar

[Car86] Carayol, H., ‘Sur les représentations l-adiques associées aux formes modulaires de Hilbert’, Ann. Sci. Éc. Norm. Supér. (4) 19(3) (1986), 409–468.Google Scholar

[CHT08] Clozel, L., Harris, M. and Taylor, R., ‘Automorphy for some l-adic lifts of automorphic mod l Galois representations’, Publ. Math. Inst. Hautes Études Sci. 108 (2008), 1–181.Google Scholar

[DDT97] Darmon, H., Diamond, F. and Taylor, R., ‘Fermat’s last theorem’, inElliptic Curves, Modular Forms & Fermat’s Last Theorem (Hong Kong, 1993) (International Press, Cambridge, MA, 1997), 2–140.Google Scholar

[Dia07] Diamond, F., ‘A correspondence between representations of local Galois groups and Lie-type groups’, inL-Functions and Galois Representations, London Mathematical Society Lecture Note Series, 320 (Cambridge University Press, Cambridge, 2007), 187–206.Google Scholar

[EG14] Emerton, M. and Gee, T., ‘A geometric perspective on the Breuil–Mézard conjecture’, J. Inst. Math. Jussieu 13(1) (2014), 183–223.Google Scholar

[GL12] Gao, H. and Liu, T., ‘A note on potential diagonalizability of crystalline representations’, Math. Ann. 360(1–2) (2014), 481–487.Google Scholar

[Gee06] Gee, T., ‘A modularity lifting theorem for weight two Hilbert modular forms’, Math. Res. Lett. 13(5–6) (2006), 805–811.Google Scholar

[Gee11a] Gee, T., ‘Automorphic lifts of prescribed types’, Math. Ann. 350(1) (2011), 107–144.Google Scholar

[Gee11b] Gee, T., ‘On the weights of mod p Hilbert modular forms’, Invent. Math. 184(1) (2011), 1–46.Google Scholar

[GG12] Gee, T. and Geraghty, D., ‘Companion forms for unitary and symplectic groups’, Duke Math. J. 161(2) (2012), 247–303.Google Scholar

[GG13] Gee, T. and Geraghty, D., ‘The Breuil–Mézard conjecture for quaternion algebras’, (2013).Google Scholar

[GLS12] Gee, T., Liu, T. and Savitt, D., ‘Crystalline extensions and the weight part of Serre’s conjecture’, Algebra Number Theory 6(7) (2012), 1537–1559.Google Scholar

[GLS13] Gee, T., Liu, T. and Savitt, D., ‘The weight part of Serre’s conjecture for GL(2)’, (2013).Google Scholar

[GLS14] Gee, T., Liu, T. and Savitt, D., ‘The Buzzard–Diamond–Jarvis conjecture for unitary groups’, J. Amer. Math. Soc. 27(2) (2014), 389–435.Google Scholar

[HT01] Harris, M. and Taylor, R., ‘The geometry and cohomology of some simple Shimura varieties’, inAnnals of Mathematics Studies, Vol. 151 (Princeton University Press, Princeton, NJ, 2001); with an appendix by V. G. Berkovich.Google Scholar

[KM74] Katz, N. M. and Messing, W., ‘Some consequences of the Riemann hypothesis for varieties over finite fields’, Invent. Math. 23 (1974), 73–77.Google Scholar

[KW09] Khare, C. and Wintenberger, J.-P., ‘On Serre’s conjecture for 2-dimensional mod p representations of Gal(ℚ∕ℚ)’, Ann. of Math. (2) 169(1) (2009), 229–253.Google Scholar

[Kis08] Kisin, M., ‘Potentially semi-stable deformation rings’, J. Amer. Math. Soc. 21(2) (2008), 513–546.Google Scholar

[Kis09a] Kisin, M., ‘The Fontaine–Mazur conjecture for GL’, J. Amer. Math. Soc. 22(3) (2009), 641–690.Google Scholar

[Kis09b] Kisin, M., ‘Moduli of finite flat group schemes, and modularity’, Ann. of Math. (2) 170(3) (2009), 1085–1180.Google Scholar

[Kis10] Kisin, M., ‘The structure of potentially semi-stable deformation rings’, inProceedings of the International Congress of Mathematicians, Vol. II (Hindustan Book Agency, New Delhi, 2010), 294–311.Google Scholar

[Lab11] Labesse, J.-P., ‘Changement de base C et séries discrètes’, inOn the Stabilization of the Trace formula, Stab. Trace Formula Shimura Var. Arith. Appl., 1 (Int. Press, Somerville, MA, 2011), 429–470.Google Scholar

[Mat89] Matsumura, H., ‘Commutative ring theory’, inCambridge Studies in Advanced Mathematics, 2nd edn, Vol. 8 (Cambridge University Press, Cambridge, 1989), ; translated from the Japanese by M. Reid.Google Scholar

[New13] Newton, J., ‘Serre weights and Shimura curves’, Proc. Lond. Math. Soc. (3) 108(6) (2014), 1471–1500.Google Scholar

[Pil08] Pilloni, V., ‘The study of 2-dimensional -adic Galois deformations in the case’, (2008).Google Scholar

[Sai09] Saito, T., ‘Hilbert modular forms and p-adic Hodge theory’, Compositio Math. 145(5) (2009), 1081–1113.Google Scholar

[San12] Sander, F., ‘Hilbert–Samuel multiplicities of certain deformation rings’, (2012).Google Scholar

[Sch08] Schein, M. M., ‘Weights in Serre’s conjecture for Hilbert modular forms: the ramified case’, Israel J. Math. 166 (2008), 369–391.Google Scholar

[Ser77] Serre, J.-P., ‘Linear representations of finite groups’, inGraduate Texts in Mathematics, Vol. 42 (Springer-Verlag, New York, 1977); translated from the second French edition by L. L. Scott.Google Scholar

[Sho13] Shotton, J., ‘Local deformation rings and a Breuil–Mézard conjecture when ’, (2013).Google Scholar

[Sno09] Snowden, A., ‘On two dimensional weight two odd representations of totally real fields’, (2009).Google Scholar

[Tay06] Taylor, R., ‘On the meromorphic continuation of degree two L-functions’, Doc. Math. (2006), 729–779; no. extra vol. (electronic).Google Scholar

[Tho12] Thorne, J., ‘On the automorphy of l-adic Galois representations with small residual image’, J. Inst. Math. Jussieu 11(4) (2012), 855–920; with an appendix by R. Guralnick, F. Herzig, R. Taylor and Thorne.Google Scholar

[Tit66] Tits, J., ‘Classification of algebraic semisimple groups’, inAlgebraic Groups and Discontinuous Subgroups (Proceedings of Symposia in Pure Mathematics, Boulder, CO, 1965) (American Mathematical Society, Providence, RI, 1966), 33–62.Google Scholar

[Vig89a] Vignéras, M.-F., ‘Correspondance modulaire galois-quaternions pour un corps p-adique’, inNumber Theory (Ulm, 1987), Lecture Notes in Mathematics, 1380 (Springer, New York, 1989), 254–266.Google Scholar

[Vig89b] Vignéras, M.-F., ‘Représentations modulaires de GL(2, F) en caractéristique l, F corps p-adique, p≠l’, Compositio Math. 72(1) (1989), 33–66.Google Scholar

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