## Duke Mathematical Journal

### Congruences between Hilbert modular forms: constructing ordinary lifts

#### Abstract

Under mild hypotheses, we prove that if $F$ is a totally real field, and $\overline{\rho}:G_{F}\to\operatorname{GL}_{2}(\overline{\mathbb {F}}_{l})$ is irreducible and modular, then there is a finite solvable totally real extension $F^{\prime}/F$ such that $\overline{\rho}|_{G_{F^{\prime}}}$ has a modular lift which is ordinary at each place dividing $l$. We deduce a similar result for $\overline{\rho}$ itself, under the assumption that at places $v|l$ the representation $\overline {\rho }|_{G_{F_{v}}}$ is reducible. This allows us to deduce improvements to results in the literature on modularity lifting theorems for potentially Barsotti–Tate representations and the Buzzard–Diamond–Jarvis conjecture. The proof makes use of a novel lifting technique, going via rank $4$ unitary groups.

#### Article information

Source
Duke Math. J. Volume 161, Number 8 (2012), 1521-1580.

Dates
First available in Project Euclid: 22 May 2012

http://projecteuclid.org/euclid.dmj/1337690407

Digital Object Identifier
doi:10.1215/00127094-1593326

Mathematical Reviews number (MathSciNet)
MR2931274

Zentralblatt MATH identifier
1297.11028

#### Citation

Barnet-Lamb, Thomas; Gee, Toby; Geraghty, David. Congruences between Hilbert modular forms: constructing ordinary lifts. Duke Math. J. 161 (2012), no. 8, 1521--1580. doi:10.1215/00127094-1593326. http://projecteuclid.org/euclid.dmj/1337690407.

#### References

• [1] J. Arthur and L. Clozel, Simple algebras, base change, and the advanced theory of the trace formula, Ann. of Math. Stud. 120, Princeton Univ. Press, Princeton, 1989.
• [2] T. Barnet-Lamb, T. Gee, and D. Geraghty, The Sato–Tate conjecture for Hilbert modular forms, J. Amer. Math. Soc. 24 (2011), 411–469.
• [3] T. Barnet-Lamb, Serre weights for rank two unitary groups, preprint, arXiv:1106.5586v1 [math.NT]
• [4] T. Barnet-Lamb, T. Gee, D. Geraghty, and R. Taylor, Potential automorphy and change of weight, preprint, arXiv:1010.2561v1 [math.NT]
• [5] T. Barnet-Lamb, D. Geraghty, M. Harris, and R. Taylor, A family of Calabi–Yau varieties and potential automorphy, II, Publ. Res. Inst. Math. Sci. 47 (2011), 29–98.
• [6] J. H. Conway, R. T. Curtis, S. P. Norton, R. A. Parker, and R. A. Wilson, Atlas of finite groups, ten years on, London Math. Soc. Lecture Note Ser. 249, Cambridge Univ. Press, Cambridge, 1998.
• [7] L. Clozel, Changement de base pour les représentations tempérées des groupes réductifs réels, Ann. Sci. École Norm. Sup. (4) 15 (1982), 45–115.
• [8] L. Clozel, M. Harris, and R. Taylor, Automorphy for some l-adic lifts of automorphic mod l Galois representations, Inst. Haute Études Sci. Pub. Math. 108 (2008), 1–181.
• [9] H. Darmon, F. Diamond, and R. Taylor, “Fermat’s last theorem” in Elliptic Curves, Modular Forms and Fermat’s Last Theorem (Hong Kong, 1993), Int. Press, Cambridge, Mass., 1997, 2–140.
• [10] T. Gee, A modularity lifting theorem for weight two Hilbert modular forms, Math. Res. Lett. 13 (2006), 805–811; erratum, Math. Res. Lett. 16 (2009), 57–58.
• [11] T. Gee, Automorphic lifts of prescribed types, Math. Ann. 350 (2011), 107–144.
• [12] T. Gee, On the weights of mod p Hilbert modular forms, Invent. Math. 184 (2011), 1–46.
• [13] T. Gee and D. Geraghty, Companion forms for unitary and symplectic groups, Duke Math. J. 161 (2012), 247–303.
• [14] D. Geraghty, Modularity lifting theorems for ordinary Galois representations, preprint, 2009.
• [15] R. Guralnick, F. Herzig, R. Taylor, and J. Thorne, Adequate subgroups, preprint, arXiv:1107.5993v1 [math.NT]
• [16] M. Harris, “Potential automorphy of odd-dimensional symmetric powers of elliptic curves and applications” in Algebra, Arithmetic, and Geometry: In Honor of Yu. I. Manin, Vol. II, Progr. Math. 270, Birkhäuser, Boston, 2009, 1–21.
• [17] M. Harris, N. Shepherd-Barron, and R. Taylor, A family of Calabi–Yau varieties and potential automorphy, Ann. of Math. (2) 171 (2010), 779–813.
• [18] M. l. Harris and R. Taylor, The Geometry and Cohomology of Some Simple Shimura Varieties, with an appendix by Vladimir G. Berkovich, Ann. Math. Stud. 151, Princeton Univ. Press, Princeton, 2001.
• [19] P. N. Hoffman and J. F. Humphreys, Projective Representations of the Symmetric Groups: Q-Functions and Shifted Tableaux, Oxford Univ. Press, New York, 1992.
• [20] C. Khare and J.-P. Wintenberger, On Serre’s conjecture for 2-dimensional mod p representations of the absolute Galois group of the rationals, Ann. of Math. (2) 169 (2009), 229–253.
• [21] M. Kisin, “Modularity for some geometric Galois representations” in L-Functions and Galois Representations, with an appendix by Ofer Gabber, London Math. Soc. Lecture Note Ser. 320, Cambridge Univ. Press, Cambridge, 2007, 438–470.
• [22] M. Kisin, “Modularity of 2-dimensional Galois representations” in Current Developments in Mathematics, 2005, Int. Press, Somerville, Mass., 2007, 191–230.
• [23] M. Kisin, Potentially semi-stable deformation rings, J. Amer. Math. Soc. 21 (2008), 513–546.
• [24] M. Kisin, The Fontaine–Mazur conjecture for GL2, J. Amer. Math. Soc. 22 (2009), 641–690.
• [25] M. Kisin, Moduli of finite flat group schemes, and modularity, Ann. of Math. (2) 170 (2009), 1085–1180.
• [26] R. Ramakrishna, Deforming Galois representations and the conjectures of Serre and Fontaine–Mazur, Ann. of Math. (2) 156 (2002), 115–154.
• [27] J.-P. Serre, Local Fields, translated from the French by M. J. Greenberg, Grad. Texts in Math. 67, Springer, New York, 1979.
• [28] A. Snowden and A. Wiles, Bigness in compatible systems, preprint, arXiv:0908.1991v3 [math.NT]
• [29] R. Taylor, On the meromorphic continuation of degree two L-functions, Doc. Math. 2006, Extra Vol., 729–779.
• [30] J. Thorne, On the automorphy of l-adic Galois representations with small residual image, to appear J. Inst. Math. Jussieu, preprint, arXiv:1107.5989v1 [math.NT]
• [31] P.-J. White, m-bigness in compatible families, C. R. Math. Acad. Sci. Paris 348 (2010), 1049–1054.
• [32] R. A. Wilson, The finite simple groups: An introduction, Grad. Texts Math. 251, Springer, London, 2009.