Skip to main content
×
×
Home

THE BREUIL–MÉZARD CONJECTURE FOR POTENTIALLY BARSOTTI–TATE REPRESENTATIONS

  • TOBY GEE (a1) and MARK KISIN (a2)
Abstract

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.

    • Send article to Kindle

      To send this article to your Kindle, first ensure no-reply@cambridge.org is added to your Approved Personal Document E-mail List under your Personal Document Settings on the Manage Your Content and Devices page of your Amazon account. Then enter the ‘name’ part of your Kindle email address below. Find out more about sending to your Kindle. Find out more about sending to your Kindle.

      Note you can select to send to either the @free.kindle.com or @kindle.com variations. ‘@free.kindle.com’ emails are free but can only be sent to your device when it is connected to wi-fi. ‘@kindle.com’ emails can be delivered even when you are not connected to wi-fi, but note that service fees apply.

      Find out more about the Kindle Personal Document Service.

      THE BREUIL–MÉZARD CONJECTURE FOR POTENTIALLY BARSOTTI–TATE REPRESENTATIONS
      Available formats
      ×
      Send article to Dropbox

      To send this article to your Dropbox account, please select one or more formats and confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your <service> account. Find out more about sending content to Dropbox.

      THE BREUIL–MÉZARD CONJECTURE FOR POTENTIALLY BARSOTTI–TATE REPRESENTATIONS
      Available formats
      ×
      Send article to Google Drive

      To send this article to your Google Drive account, please select one or more formats and confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your <service> account. Find out more about sending content to Google Drive.

      THE BREUIL–MÉZARD CONJECTURE FOR POTENTIALLY BARSOTTI–TATE REPRESENTATIONS
      Available formats
      ×
Copyright
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.
References
Hide All
[BLGG11]Barnet-Lamb, T., Gee, T. and Geraghty, D., ‘The Sato–Tate conjecture for Hilbert modular forms’, J. Amer. Math. Soc. 24(2) (2011), 411469.
[BLGG12]Barnet-Lamb, T., Gee, T. and Geraghty, D., ‘Congruences between Hilbert modular forms: constructing ordinary lifts’, Duke Math. J. 161(8) (2012), 15211580.
[BLGG13a]Barnet-Lamb, T., Gee, T. and Geraghty, D., ‘Congruences between Hilbert modular forms: constructing ordinary lifts, II’, Math. Res. Lett. 20(1) (2013), 6772.
[BLGG13b]Barnet-Lamb, T., Gee, T. and Geraghty, D., ‘Serre weights for rank two unitary groups’, Math. Ann. 356(4) (2013), 15511598.
[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), 2998.
[BLGGT14a]Barnet-Lamb, T., Gee, T., Geraghty, D. and Taylor, R., ‘Potential automorphy and change of weight’, Ann. of Math. (2) 179(2) (2014), 501609.
[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), 165179.
[BC11]Bellaïche, J. and Chenevier, G., ‘The sign of Galois representations attached to automorphic forms for unitary groups’, Compositio Math. 147(5) (2011), 13371352.
[Bla06]Blasius, D., ‘Hilbert modular forms and the Ramanujan conjecture’, inNoncommutative Geometry and Number Theory, Aspects of Mathematics, E37 (Vieweg, Wiesbaden, 2006), 3556.
[BD13]Breuil, C. and Diamond, F., ‘Formes modulaires de Hilbert modulo et valeurs d’extensions Galoisiennes’, Ann. Sci. Éc. Norm. Supér. (2014), to appear.
[BM02]Breuil, C. and Mézard, A., ‘Multiplicités modulaires et représentations de GL2(Zp) et de Gal(QpQp) en l = p’, Duke Math. J. 115(2) (2002), 205310; with an appendix by G. Henniart.
[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), 105161.
[Cal12]Calegari, F., ‘Even Galois representations and the Fontaine–Mazur conjecture II’, J. Amer. Math. Soc. 25(2) (2012), 533554.
[Car86]Carayol, H., ‘Sur les représentations l-adiques associées aux formes modulaires de Hilbert’, Ann. Sci. Éc. Norm. Supér. (4) 19(3) (1986), 409468.
[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), 1181.
[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), 2140.
[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), 187206.
[EG14]Emerton, M. and Gee, T., ‘A geometric perspective on the Breuil–Mézard conjecture’, J. Inst. Math. Jussieu 13(1) (2014), 183223.
[GL12]Gao, H. and Liu, T., ‘A note on potential diagonalizability of crystalline representations’, Math. Ann. 360(1–2) (2014), 481487.
[Gee06]Gee, T., ‘A modularity lifting theorem for weight two Hilbert modular forms’, Math. Res. Lett. 13(5–6) (2006), 805811.
[Gee11a]Gee, T., ‘Automorphic lifts of prescribed types’, Math. Ann. 350(1) (2011), 107144.
[Gee11b]Gee, T., ‘On the weights of mod p Hilbert modular forms’, Invent. Math. 184(1) (2011), 146.
[GG12]Gee, T. and Geraghty, D., ‘Companion forms for unitary and symplectic groups’, Duke Math. J. 161(2) (2012), 247303.
[GG13]Gee, T. and Geraghty, D., ‘The Breuil–Mézard conjecture for quaternion algebras’, (2013).
[GLS12]Gee, T., Liu, T. and Savitt, D., ‘Crystalline extensions and the weight part of Serre’s conjecture’, Algebra Number Theory 6(7) (2012), 15371559.
[GLS13]Gee, T., Liu, T. and Savitt, D., ‘The weight part of Serre’s conjecture for GL(2)’, (2013).
[GLS14]Gee, T., Liu, T. and Savitt, D., ‘The Buzzard–Diamond–Jarvis conjecture for unitary groups’, J. Amer. Math. Soc. 27(2) (2014), 389435.
[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.
[KM74]Katz, N. M. and Messing, W., ‘Some consequences of the Riemann hypothesis for varieties over finite fields’, Invent. Math. 23 (1974), 7377.
[KW09]Khare, C. and Wintenberger, J.-P., ‘On Serre’s conjecture for 2-dimensional mod p representations of Gal(∕ℚ)’, Ann. of Math. (2) 169(1) (2009), 229253.
[Kis08]Kisin, M., ‘Potentially semi-stable deformation rings’, J. Amer. Math. Soc. 21(2) (2008), 513546.
[Kis09a]Kisin, M., ‘The Fontaine–Mazur conjecture for GL2’, J. Amer. Math. Soc. 22(3) (2009), 641690.
[Kis09b]Kisin, M., ‘Moduli of finite flat group schemes, and modularity’, Ann. of Math. (2) 170(3) (2009), 10851180.
[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), 294311.
[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), 429470.
[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.
[New13]Newton, J., ‘Serre weights and Shimura curves’, Proc. Lond. Math. Soc. (3) 108(6) (2014), 14711500.
[Pil08]Pilloni, V., ‘The study of 2-dimensional -adic Galois deformations in the case’, (2008).
[Sai09]Saito, T., ‘Hilbert modular forms and p-adic Hodge theory’, Compositio Math. 145(5) (2009), 10811113.
[San12]Sander, F., ‘Hilbert–Samuel multiplicities of certain deformation rings’, (2012).
[Sch08]Schein, M. M., ‘Weights in Serre’s conjecture for Hilbert modular forms: the ramified case’, Israel J. Math. 166 (2008), 369391.
[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.
[Sho13]Shotton, J., ‘Local deformation rings and a Breuil–Mézard conjecture when ’, (2013).
[Sno09]Snowden, A., ‘On two dimensional weight two odd representations of totally real fields’, (2009).
[Tay06]Taylor, R., ‘On the meromorphic continuation of degree two L-functions’, Doc. Math. (2006), 729779; no. extra vol. (electronic).
[Tho12]Thorne, J., ‘On the automorphy of l-adic Galois representations with small residual image’, J. Inst. Math. Jussieu 11(4) (2012), 855920; with an appendix by R. Guralnick, F. Herzig, R. Taylor and Thorne.
[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), 3362.
[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), 254266.
[Vig89b]Vignéras, M.-F., ‘Représentations modulaires de GL(2, F) en caractéristique l, F corps p-adique, pl’, Compositio Math. 72(1) (1989), 3366.
Recommend this journal

Email your librarian or administrator to recommend adding this journal to your organisation's collection.

Forum of Mathematics, Pi
  • ISSN: -
  • EISSN: 2050-5086
  • URL: /core/journals/forum-of-mathematics-pi
Please enter your name
Please enter a valid email address
Who would you like to send this to? *
×
MathJax

MSC classification

Metrics

Full text views

Total number of HTML views: 0
Total number of PDF views: 0 *
Loading metrics...

Abstract views

Total abstract views: 0 *
Loading metrics...

* Views captured on Cambridge Core between <date>. This data will be updated every 24 hours.

Usage data cannot currently be displayed