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The Fontaine-Mazur conjecture for $ {GL}_2$


Author: Mark Kisin
Journal: J. Amer. Math. Soc. 22 (2009), 641-690
MSC (2000): Primary 11F80
DOI: https://doi.org/10.1090/S0894-0347-09-00628-6
Published electronically: January 21, 2009
MathSciNet review: 2505297
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Abstract: We prove new cases of the Fontaine-Mazur conjecture, that a $ 2$-dimensional $ p$-adic representation $ \rho$ of $ G_{\mathbb{Q}, S}$ which is potentially semi-stable at $ p$ with distinct Hodge-Tate weights arises from a twist of a modular eigenform of weight $ k\geq 2$. Our approach is via the Breuil-Mézard conjecture, which we prove (many cases of) by combining a global argument with recent results of Colmez and Berger-Breuil on the $ p$-adic local Langlands correspondence.


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  • [BB 1] L. Berger, C. Breuil, Sur quelques représentations potentiellement cristallines de $ \operatorname{GL}_{2}(\mathbb{Q}_{p})$, Astérisque, to appear.
  • [BB 2] L. Berger, C. Breuil, Towards a $ p$-adic Langlands program (Course at C.M.S, Hangzhou), 2004.
  • [BCDT] C. Breuil, B. Conrad, F. Diamond, R. Taylor, On the modularity of elliptic curves over $ \mathbb{Q}:$ wild $ 3$-adic exercises, J. Amer. Math. Soc. 14 (2001), 843-939. MR 1839918 (2002d:11058)
  • [BE] C. Breuil, M. Emerton, Représentations $ p$-adiques ordinaires de $ \operatorname{GL}_{2}(\mathbb{Q}_{p})$ et compatibilité local-global, Astérisque, to appear.
  • [BL] L. Barthel, R. Livne, Irreducible modular representations of $ \operatorname{GL}_{2}$ of a local field, Duke Math. J. 75 (1994), 261-292. MR 1290194 (95g:22030)
  • [BLZ] L. Berger, H. Li, H.J. Zhu, Construction of some families of $ 2$-dimensional crystalline representations, Math. Ann. 329 (2004), 365-377. MR 2060368 (2005k:11104)
  • [BM] C. Breuil, A. Mézard, Multiplicités modulaires et représentations de $ \operatorname{GL}_{2}(\mathbb{Z}_{p})$ et de $ \operatorname{Gal} (\bar{\mathbb{Q}}_{p}/\mathbb{Q}_{p})$ en $ l = p$, Duke Math. J. 115 (2002), 205-310, with an appendix by G. Henniart. MR 1944572 (2004i:11052)
  • [Bö] G. Böckle, On the density of modular points in universal deformation spaces, Amer. J. Math. 123 (2001), 985-1007. MR 1854117 (2002g:11070)
  • [Br 1] C. Breuil, Sur quelques représentations modulaires et $ p$-adiques de $ {\operatorname{GL}}_{2}(\mathbb{Q}_{p})$ I, Compositio Math. 138 (2003), 165-188. MR 2018825 (2004k:11062)
  • [Br 2] C. Breuil, Sur quelques représentations modulaires et $ p$-adiques de $ \operatorname{GL}_{2}(\mathbb{Q}_{p})$ II, J. Inst. Math. Jussieu 2 (2003), 1-36. MR 1955206 (2005d:11079)
  • [Br 3] C. Breuil, Invariant $ L$ et série spéciale $ p$-adique, Ann. Scient. de ENS 37 (2004), 559-610. MR 2097893 (2005j:11039)
  • [Ca] H. Carayol, Formes modulaires et représentations galoisiennes à valeurs dans un anneau local complet, $ p$-adic monodromy and the Birch and Swinnerton-Dyer conjecture (Boston, MA, 1991), Contemp. Math., 165, Amer. Math. Soc., 1994, pp. 213-237. MR 1279611 (95i:11059)
  • [CDT] B. Conrad, F. Diamond, R. Taylor, Modularity of certain potentially Barsotti-Tate Galois representations, J. Amer. Math. Soc. 12(2) (1999), 521-567. MR 1639612 (99i:11037)
  • [Co 1] P. Colmez, La série principale unitaire de $ \operatorname{GL}_{2}(\mathbb{Q}_{p})$, preprint (2007).
  • [Co 2] P. Colmez, Représentations de $ \operatorname{GL}_{2}(\mathbb{Q}_{p})$ et $ (\varphi ,\Gamma )$-modules, preprint (2007).
  • [Co 3] P. Colmez, Série principale unitaire pout $ \operatorname{GL}_{2}(\mathbb{Q}_{p})$ et représentations triangulines de dimension $ 2$, preprint (2004).
  • [DDT] H. Darmon, F. Diamond, R. Taylor, Fermat's Last Theorem, Current developments in mathematics, 1995 (Cambridge, MA), pp. 1-154, 1996. MR 1474977 (99d:11067a)
  • [DFG] F. Diamond, M. Flach, L. Guo, The Tamagawa number conjecture for adjoint motives of modular forms, Ann. Sci. Ec. Norm. Sup 37 (2004), 663-727. MR 2103471 (2006e:11089)
  • [Di] F. Diamond, The Taylor-Wiles construction and multiplicity one, Invent. Math. 128 (1997), 379-391. MR 1440309 (98c:11047)
  • [Di 2] F. Diamond, On deformation rings and Hecke rings, Ann. Math 144 (1996), 137-166. MR 1405946 (97d:11172)
  • [FL] J. M Fontaine, G. Laffaille, Construction de représentations $ p$-adiques, Ann. Sci. École Norm. Sup. 15 (1983), 547-683. MR 707328 (85c:14028)
  • [FM] J.M. Fontaine, B. Mazur, Geometric Galois Representations, Elliptic curves, modular forms, and Fermat's last theorem (Hong Kong, 1993), Internat. Press, Cambridge, MA, pp. 41-78, 1995. MR 1363495 (96h:11049)
  • [Fo] J.M. Fontaine, Représentations $ p$-adiques semi-stables, Périodes $ p$-adiques, Astérisque 223, Société Mathématique de France, pp. 113-184, 1994. MR 1293972 (95g:14024)
  • [GD] A. Grothendieck, J. Dieudonné, Elèments de géometrie algèbrique I, II, III, IV, Inst. des Hautes Études Sci. Publ. Math. 4, 8, 11, 17, 20, 24, 28, 32 (1961-67). MR 0217083 (36:177a); MR 0217084 (36:177b); MR 0217085 (36:177c); MR 0163911 (29:1210); MR 0173675 (30:3885); MR 0199181 (33:7330); MR 0217086 (36:178); MR 0238860 (39:220)
  • [Ge] T. Gee, Automorphic lifts of prescribed types, preprint (2006).
  • [GM] F. Gouvêa, B. Mazur, On the density of modular representations, Computational perspectives on number theory (Chicago, IL, 1995), AMS/IP Stud. Adv. Math., 7, 1998, pp. 127-142. MR 1486834 (99a:11056)
  • [GS] R. Greenberg, G. Stevens, On the conjecture of Mazur, Tate, and Teitelbaum, $ p$-adic monodromy and the Birch and Swinnerton-Dyer conjecture (Boston, MA, 1991), Contemp. Math. 165, pp. 183-211, 1994. MR 1279610 (95j:11057)
  • [Ki 1] M. Kisin, Potentially semi-stable deformation rings, J. AMS 21 (2008), 513-546. MR 2373358
  • [Ki 2] M. Kisin, Moduli of finite flat group schemes and modularity, Ann. of Math., to appear.
  • [Ki 3] M. Kisin, Geometric deformations of modular Galois representations, Invent. Math. 157 (2004), 275-328. MR 2076924 (2006a:11067)
  • [Ki 4] M. Kisin, Overconvergent modular forms and the Fontaine-Mazur conjecture, Invent. Math. 153, 373-454. MR 1992017 (2004f:11053)
  • [Ki 5] M. Kisin, Modularity of $ 2$-adic Barsotti-Tate representations, preprint (2007).
  • [Ki 6] M. Kisin, Deformations of $ G_{\mathbb{Q}_{p}}$ and $ \operatorname{GL}_{2}(\mathbb{Q}_{p})$ representations (Appendix to [Co 2]), preprint (2008).
  • [KW 1] C. Khare, J-P. Wintenberger, Serre's modularity conjecture: The case of odd conductor (I), preprint (2006).
  • [KW 2] C. Khare, J-P. Wintenberger, Serre's modularity conjecture: The case of odd conductor (II), preprint (2006).
  • [Ma] H. Matsumura, Commutative Algebra, Mathematics Lecture Note Series, The Benjamin Cummings Publishing Company, 1980. MR 575344 (82i:13003)
  • [Maz] B. Mazur, An introduction to the deformation theory of Galois representations, Modular forms and Fermat's last theorem (Boston, MA, 1995), Springer, New York, pp. 243-311, 1997. MR 1638481
  • [Ny] L. Nyssen, Pseudo-représentations, Math. Ann. 306 (1996), 257-283. MR 1411348 (98a:20013)
  • [SW 1] C. Skinner, A. Wiles, Residually reducible representations and modular forms, Inst. Hautes Études Sci. Publ. Math. IHES 89 (1999), 5-126. MR 1793414 (2002b:11072)
  • [SW 2] C. Skinner, A. Wiles, Nearly ordinary deformations of irreducible residual representations, Ann. Fac. Sci. Toulouse Math (6) 10 (2001), 185-215. MR 1928993 (2004b:11073)
  • [Ta 1] R. Taylor, Galois representations associated to Siegel modular forms of low weight, Duke Math. J. 63 (1991), 281-332. MR 1115109 (92j:11044)
  • [Ta 2] R. Taylor, On the meromorphic continuation of degree $ 2$ $ L$-functions, Documenta John Coates' Sixtieth Birthday (2006), 729-779. MR 2290604 (2008c:11154)
  • [TW] R. Taylor and A. Wiles, Ring-theoretic properties of certain Hecke algebras, Ann. of Math. (2) 141 (1995), 553-572. MR 1333036 (96d:11072)
  • [Wi] A. Wiles, Modular elliptic curves and Fermat's last theorem, Ann. of Math. (2) 141 (1995), 443-551. MR 1333035 (96d:11071)

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Additional Information

Mark Kisin
Affiliation: Department of Mathematics, University of Chicago, 5734 S. University Avenue, Chicago, Illinois 60637
Email: kisin@math.uchicago.edu

DOI: https://doi.org/10.1090/S0894-0347-09-00628-6
Received by editor(s): June 25, 2007
Published electronically: January 21, 2009
Additional Notes: The author was partially supported by NSF grant DMS-0400666 and a Sloan Research Fellowship.
Article copyright: © Copyright 2009 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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