Hostname: page-component-7479d7b7d-t6hkb Total loading time: 0 Render date: 2024-07-13T16:33:12.575Z Has data issue: false hasContentIssue false
  • English
  • Français

THE BREUIL–MÉZARD CONJECTURE FOR POTENTIALLY BARSOTTI–TATE REPRESENTATIONS

Part of: Discontinuous groups and automorphic forms

Published online by Cambridge University Press:  22 December 2014

TOBY GEE  and
MARK KISIN
TOBY GEE
Affiliation:
Department of Mathematics, Imperial College London SW7 2RH, UK; toby.gee@imperial.ac.uk
MARK KISIN
Affiliation:
Department of Mathematics, Harvard University, Cambridge, MA 02138, USA; kisin@math.harvard.edu

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

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.

MSC classification

Primary: 11F33: Congruences for modular and $p$-adic modular forms
Type
Research Article
Information
Forum of Mathematics, Pi , Volume 2 , 2014 , e1
Check for updates
Creative Commons
Creative Common License - CCCreative Common License - BY
This is an Open Access article, distributed under the terms of the Creative Commons Attribution licence (http://creativecommons.org/licenses/by/3.0/), which permits unrestricted re-use, distribution, and reproduction in any medium, provided the original work is properly cited.
Copyright
© The Author(s) 2014

References

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
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
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
Barnet-Lamb, T., Gee, T. and Geraghty, D., ‘Serre weights for rank two unitary groups’, Math. Ann. 356(4) (2013), 1551–1598.Google Scholar
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
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
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
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
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
Breuil, C. and Diamond, F., ‘Formes modulaires de Hilbert modulo $p$ et valeurs d’extensions Galoisiennes’, Ann. Sci. Éc. Norm. Supér. (2014), to appear.Google Scholar
Breuil, C. and Mézard, A., ‘Multiplicités modulaires et représentations de GL2(Z p) et de Gal( Q p∕Q p) en l = p’, Duke Math. J. 115(2) (2002), 205–310; with an appendix by G. Henniart.Google Scholar
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
Calegari, F., ‘Even Galois representations and the Fontaine–Mazur conjecture II’, J. Amer. Math. Soc. 25(2) (2012), 533–554.Google Scholar
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
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
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
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
Emerton, M. and Gee, T., ‘A geometric perspective on the Breuil–Mézard conjecture’, J. Inst. Math. Jussieu 13(1) (2014), 183–223.Google Scholar
Gao, H. and Liu, T., ‘A note on potential diagonalizability of crystalline representations’, Math. Ann. 360(1–2) (2014), 481–487.Google Scholar
Gee, T., ‘A modularity lifting theorem for weight two Hilbert modular forms’, Math. Res. Lett. 13(5–6) (2006), 805–811.Google Scholar
Gee, T., ‘Automorphic lifts of prescribed types’, Math. Ann. 350(1) (2011), 107–144.Google Scholar
Gee, T., ‘On the weights of mod p Hilbert modular forms’, Invent. Math. 184(1) (2011), 1–46.Google Scholar
Gee, T. and Geraghty, D., ‘Companion forms for unitary and symplectic groups’, Duke Math. J. 161(2) (2012), 247–303.Google Scholar
Gee, T. and Geraghty, D., ‘The Breuil–Mézard conjecture for quaternion algebras’, (2013).Google Scholar
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
Gee, T., Liu, T. and Savitt, D., ‘The weight part of Serre’s conjecture for GL(2)’, (2013).Google Scholar
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
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
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
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
Kisin, M., ‘Potentially semi-stable deformation rings’, J. Amer. Math. Soc. 21(2) (2008), 513–546.Google Scholar
Kisin, M., ‘The Fontaine–Mazur conjecture for GL2’, J. Amer. Math. Soc. 22(3) (2009), 641–690.Google Scholar
Kisin, M., ‘Moduli of finite flat group schemes, and modularity’, Ann. of Math. (2) 170(3) (2009), 1085–1180.Google Scholar
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
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
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
Newton, J., ‘Serre weights and Shimura curves’, Proc. Lond. Math. Soc. (3) 108(6) (2014), 1471–1500.Google Scholar
Pilloni, V., ‘The study of 2-dimensional $p$-adic Galois deformations in the $l\neq p$ case’, (2008).Google Scholar
Saito, T., ‘Hilbert modular forms and p-adic Hodge theory’, Compositio Math. 145(5) (2009), 1081–1113.Google Scholar
Sander, F., ‘Hilbert–Samuel multiplicities of certain deformation rings’, (2012).Google Scholar
Schein, M. M., ‘Weights in Serre’s conjecture for Hilbert modular forms: the ramified case’, Israel J. Math. 166 (2008), 369–391.Google Scholar
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
Shotton, J., ‘Local deformation rings and a Breuil–Mézard conjecture when $l\neq p$’, (2013).Google Scholar
Snowden, A., ‘On two dimensional weight two odd representations of totally real fields’, (2009).Google Scholar
Taylor, R., ‘On the meromorphic continuation of degree two L-functions’, Doc. Math. (2006), 729–779; no. extra vol. (electronic).Google Scholar
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
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
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
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