Even Galois representations and the Fontaine–Mazur conjecture. II

Calegari, Frank

doi:10.1090/S0894-0347-2011-00721-2

Even Galois representations and the Fontaine–Mazur conjecture. II
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by Frank Calegari;

J. Amer. Math. Soc. 25 (2012), 533-554

DOI: https://doi.org/10.1090/S0894-0347-2011-00721-2

Published electronically: October 3, 2011

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Abstract:

We prove, under mild hypotheses, that there are no irreducible two-dimensional potentially semi-stable even $p$-adic Galois representations of $\mathrm {Gal}(\overline {\mathbf {Q}})$ with distinct Hodge–Tate weights. This removes the ordinary hypotheses required in the author’s previous work. We construct examples of irreducible two-dimensional residual representations that have no characteristic zero geometric deformations.

References

James Arthur and Laurent Clozel, Simple algebras, base change, and the advanced theory of the trace formula, Annals of Mathematics Studies, vol. 120, Princeton University Press, Princeton, NJ, 1989. MR 1007299
Thomas Barnet-Lamb, Toby Gee, and David Geraghty, The Sato-Tate conjecture for Hilbert modular forms, J. Amer. Math. Soc. 24 (2011), no. 2, 411–469. MR 2748398, DOI 10.1090/S0894-0347-2010-00689-3
—, Congruences between Hilbert modular forms: constructing ordinary lifts, preprint, 2010.
Tom Barnet-Lamb, Toby Gee, David Geraghty, and Richard Taylor, Potential automorphy and change of weight, preprint, 2010.
Tom Barnet-Lamb, David Geraghty, Michael Harris, and Richard Taylor, A family of Calabi-Yau varieties and potential automorphy II, to appear P.R.I.M.S., 2009.
Christophe Breuil and Ariane Mézard, Multiplicités modulaires et représentations de $\textrm {GL}_2(\textbf {Z}_p)$ et de $\textrm {Gal}(\overline \textbf {Q}_p/\textbf {Q}_p)$ en $l=p$, Duke Math. J. 115 (2002), no. 2, 205–310 (French, with English and French summaries). With an appendix by Guy Henniart. MR 1944572, DOI 10.1215/S0012-7094-02-11522-1
Gebhard Böckle, On the density of modular points in universal deformation spaces, Amer. J. Math. 123 (2001), no. 5, 985–1007. MR 1854117
Don Blasius and Jonathan D. Rogawski, Tate classes and arithmetic quotients of the two-ball, The zeta functions of Picard modular surfaces, Univ. Montréal, Montreal, QC, 1992, pp. 421–444. MR 1155236
Frank Calegari, Even Galois representations and the Fontaine–Mazur conjecture, Invent. Math. 185 (2011), 1–16.
Ana Caraiani, Local-global compatibility and the action of monodromy on nearby cycles, preprint arXiv:1010.2188, 2010.
Brian Conrad, Ofer Gabber, and Gopal Prasad, Pseudo-reductive groups, New Mathematical Monographs, vol. 17, Cambridge University Press, Cambridge, 2010. MR 2723571, DOI 10.1017/CBO9780511661143
Gaëtan Chenevier, Une application des variétés de Hecke des groupes unitaires, preprint, 2009.
—, On the infinite fern of Galois representations of unitary type, preprint, 2010.
Laurent Clozel, Michael Harris, and J.-P. Labesse, Stabilisation de la formule des traces, variétés de Shimura, et applications arithmétiques, to appear (séminaire G.R.F.A. Paris 7).
Frank Calegari and Barry Mazur, Nearly ordinary Galois deformations over arbitrary number fields, J. Inst. Math. Jussieu 8 (2009), no. 1, 99–177. MR 2461903, DOI 10.1017/S1474748008000327
Pierre Colmez, Représentations de $\textrm {GL}_2(\mathbf Q_p)$ et $(\phi ,\Gamma )$-modules, Astérisque 330 (2010), 281–509 (French, with English and French summaries). MR 2642409
Brian Conrad, Cohomological descent, preprint.
P. Deligne, Formes modulaires et représentations de $\textrm {GL}(2)$, Modular functions of one variable, II (Proc. Internat. Summer School, Univ. Antwerp, Antwerp, 1972) Lecture Notes in Math., Vol. 349, Springer, Berlin-New York, 1973, pp. 55–105 (French). MR 347738
A. J. de Jong, Smoothness, semi-stability and alterations, Inst. Hautes Études Sci. Publ. Math. 83 (1996), 51–93. MR 1423020
Matthew Emerton, Local–global compatibility in the $p$-adic Langlands programme, preprint.
Gerd Faltings, $p$-adic Hodge theory, J. Amer. Math. Soc. 1 (1988), no. 1, 255–299. MR 924705, DOI 10.1090/S0894-0347-1988-0924705-1
Jean-Marc Fontaine and Barry Mazur, Geometric Galois representations, Elliptic curves, modular forms, & Fermat’s last theorem (Hong Kong, 1993) Ser. Number Theory, I, Int. Press, Cambridge, MA, 1995, pp. 41–78. MR 1363495
Jean-Marc Fontaine, Représentations $l$-adiques potentiellement semi-stables, Astérisque 223 (1994), 321–347 (French). Périodes $p$-adiques (Bures-sur-Yvette, 1988). MR 1293977
David Geraghty, Modularity lifting theorems for ordinary Galois representations, preprint, 2009.
Robert Guralnick, Florian Herzig, Richard Taylor, and Jack Thorne, Adequate subgroups, Appendix to [Tho10], 2010.
Michael Harris, Nick Shepherd-Barron, and Richard Taylor, A family of Calabi-Yau varieties and potential automorphy, Ann. of Math. (2) 171 (2010), no. 2, 779–813. MR 2630056, DOI 10.4007/annals.2010.171.779
Michael Harris and Richard Taylor, The geometry and cohomology of some simple Shimura varieties, Annals of Mathematics Studies, vol. 151, Princeton University Press, Princeton, NJ, 2001. With an appendix by Vladimir G. Berkovich. MR 1876802
H. Jacquet and J. A. Shalika, On Euler products and the classification of automorphic forms. II, Amer. J. Math. 103 (1981), no. 4, 777–815. MR 623137, DOI 10.2307/2374050
Mark Kisin, Modularity of 2-dimensional Galois representations, Current developments in mathematics, 2005, Int. Press, Somerville, MA, 2007, pp. 191–230. MR 2459302
Mark Kisin, Potentially semi-stable deformation rings, J. Amer. Math. Soc. 21 (2008), no. 2, 513–546. MR 2373358, DOI 10.1090/S0894-0347-07-00576-0
Mark Kisin, The Fontaine-Mazur conjecture for $\textrm {GL}_2$, J. Amer. Math. Soc. 22 (2009), no. 3, 641–690. MR 2505297, DOI 10.1090/S0894-0347-09-00628-6
Jürgen Klüners, A polynomial with Galois group $\textrm {SL}_2(11)$, J. Symbolic Comput. 30 (2000), no. 6, 733–737. Algorithmic methods in Galois theory. MR 1800035, DOI 10.1006/jsco.2000.0380
Chandrashekhar Khare and Jean-Pierre Wintenberger, Serre’s modularity conjecture. I, Invent. Math. 178 (2009), no. 3, 485–504. MR 2551763, DOI 10.1007/s00222-009-0205-7
Chandrashekhar Khare and Jean-Pierre Wintenberger, Serre’s modularity conjecture. II, Invent. Math. 178 (2009), no. 3, 505–586. MR 2551764, DOI 10.1007/s00222-009-0206-6
Gunter Malle, Multi-parameter polynomials with given Galois group, J. Symbolic Comput. 30 (2000), no. 6, 717–731. Algorithmic methods in Galois theory. MR 1800034, DOI 10.1006/jsco.2000.0379
B. Mazur, An “infinite fern” in the universal deformation space of Galois representations, Collect. Math. 48 (1997), no. 1-2, 155–193. Journées Arithmétiques (Barcelona, 1995). MR 1464022
Ravi Ramakrishna, Deforming Galois representations and the conjectures of Serre and Fontaine-Mazur, Ann. of Math. (2) 156 (2002), no. 1, 115–154. MR 1935843, DOI 10.2307/3597186
Jean-Pierre Serre, On a theorem of Jordan, Bull. Amer. Math. Soc. (N.S.) 40 (2003), no. 4, 429–440. MR 1997347, DOI 10.1090/S0273-0979-03-00992-3
Sug Woo Shin, Galois representations arising from some compact Shimura varieties, Ann. of Math. (2) 173 (2011), no. 3, 1645–1741. MR 2800722, DOI 10.4007/annals.2011.173.3.9
Andrew Snowden, Weight two representations of totally real fields, preprint.
Richard Taylor, The image of complex conjugation in l-adic representations associated to automorphic forms, to appear in Algebra & Number Theory, 2010.
Jack Thorne, On the automorphy of $l$-adic Galois representations with small residual image, to appear in the Journal of the Institute of Mathematics of Jussieu, 2010.
Takeshi Tsuji, $p$-adic Hodge theory in the semi-stable reduction case, Proceedings of the International Congress of Mathematicians, Vol. II (Berlin, 1998), 1998, pp. 207–216. MR 1648071
Richard Taylor and Teruyoshi Yoshida, Compatibility of local and global Langlands correspondences, J. Amer. Math. Soc. 20 (2007), no. 2, 467–493. MR 2276777, DOI 10.1090/S0894-0347-06-00542-X