The Fontaine-Mazur conjecture in the residually reducible case

Pan, Lue

doi:10.1090/jams/991

The Fontaine-Mazur conjecture in the residually reducible case
HTML articles powered by AMS MathViewer

by Lue Pan

J. Amer. Math. Soc. 35 (2022), 1031-1169

DOI: https://doi.org/10.1090/jams/991

Published electronically: November 15, 2021

HTML | PDF | Request permission

Abstract:

We prove new cases of Fontaine-Mazur conjecture on two-dimensional Galois representations over $\mathbb {Q}$ when the residual representation is reducible. Our approach is via a semi-simple local-global compatibility of the completed cohomology and a Taylor-Wiles patching argument for the completed homology in this case. As a key input, we generalize the work of Skinner-Wiles in the ordinary case. In addition, we also treat the residually irreducible case at the end of the paper. Combining with people’s earlier work, we can prove the Fontaine-Mazur conjecture completely in the regular case when $p\geq 5$.

References

E. Artin and J. Tate, Class field theory, W. A. Benjamin, Inc., New York-Amsterdam, 1968. MR 0223335
Gebhard Böckle, On the density of modular points in universal deformation spaces, Amer. J. Math. 123 (2001), no. 5, 985–1007. MR 1854117, DOI 10.1353/ajm.2001.0031
Gebhard Böckle, Deformation rings for some mod 3 Galois representations of the absolute Galois group of $\mathbf Q_3$, Astérisque 330 (2010), 529–542 (English, with English and French summaries). MR 2642411
Laurent Berger and Christophe Breuil, Sur quelques représentations potentiellement cristallines de $\textrm {GL}_2(\mathbf Q_p)$, Astérisque 330 (2010), 155–211 (French, with English and French summaries). MR 2642406
Christophe Breuil and Matthew Emerton, Représentations $p$-adiques ordinaires de $\textrm {GL}_2(\mathbf Q_p)$ et compatibilité local-global, Astérisque 331 (2010), 255–315 (French, with English and French summaries). MR 2667890
Colin J. Bushnell and Guy Henniart, The local Langlands conjecture for $\rm GL(2)$, Grundlehren der mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 335, Springer-Verlag, Berlin, 2006. MR 2234120, DOI 10.1007/3-540-31511-X
Christophe Breuil and Florian Herzig, Ordinary representations of $G(\Bbb {Q}_p)$ and fundamental algebraic representations, Duke Math. J. 164 (2015), no. 7, 1271–1352. MR 3347316, DOI 10.1215/00127094-2916104
Tom Barnet-Lamb, David Geraghty, Michael Harris, and Richard Taylor, A family of Calabi-Yau varieties and potential automorphy II, Publ. Res. Inst. Math. Sci. 47 (2011), no. 1, 29–98. MR 2827723, DOI 10.2977/PRIMS/31
Markus Brodmann and Josef Rung, Local cohomology and the connectedness dimension in algebraic varieties, Comment. Math. Helv. 61 (1986), no. 3, 481–490. MR 860135, DOI 10.1007/BF02621928
Pierre Colmez, Gabriel Dospinescu, and Vytautas Paškūnas, The $p$-adic local Langlands correspondence for $\textrm {GL}_2(\Bbb Q_p)$, Camb. J. Math. 2 (2014), no. 1, 1–47. MR 3272011, DOI 10.4310/CJM.2014.v2.n1.a1
Frank Calegari and Matthew Emerton, On the ramification of Hecke algebras at Eisenstein primes, Invent. Math. 160 (2005), no. 1, 97–144. MR 2129709, DOI 10.1007/s00222-004-0406-z
Frank Calegari and Matthew Emerton, Completed cohomology—a survey, Non-abelian fundamental groups and Iwasawa theory, London Math. Soc. Lecture Note Ser., vol. 393, Cambridge Univ. Press, Cambridge, 2012, pp. 239–257. MR 2905536
Ana Caraiani, Matthew Emerton, Toby Gee, David Geraghty, Vytautas Paškūnas, and Sug Woo Shin, Patching and the $p$-adic local Langlands correspondence, Camb. J. Math. 4 (2016), no. 2, 197–287. MR 3529394, DOI 10.4310/CJM.2016.v4.n2.a2
Laurent Clozel, Michael Harris, and Richard Taylor, Automorphy for some $l$-adic lifts of automorphic mod $l$ Galois representations, Publ. Math. Inst. Hautes Études Sci. 108 (2008), 1–181. With Appendix A, summarizing unpublished work of Russ Mann, and Appendix B by Marie-France Vignéras. MR 2470687, DOI 10.1007/s10240-008-0016-1
Frank Calegari and Joel Specter, Pseudorepresentations of weight one are unramified, Algebra Number Theory 13 (2019), no. 7, 1583–1596. MR 4009671, DOI 10.2140/ant.2019.13.1583
J. D. Dixon, M. P. F. du Sautoy, A. Mann, and D. Segal, Analytic pro-$p$ groups, 2nd ed., Cambridge Studies in Advanced Mathematics, vol. 61, Cambridge University Press, Cambridge, 1999. MR 1720368, DOI 10.1017/CBO9780511470882
Henri Darmon, Fred Diamond, and Richard Taylor, Fermat’s last theorem, Current developments in mathematics, 1995 (Cambridge, MA), Int. Press, Cambridge, MA, 1994, pp. 1–154. MR 1474977
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, 1973, pp. 55–105 (French). MR 0347738
Fred Diamond and Richard Taylor, Nonoptimal levels of mod $l$ modular representations, Invent. Math. 115 (1994), no. 3, 435–462. MR 1262939, DOI 10.1007/BF01231768
Matthew Emerton, A local-global compatibility conjecture in the $p$-adic Langlands programme for $\textrm {GL}_{2/{\Bbb Q}}$, Pure Appl. Math. Q. 2 (2006), no. 2, Special Issue: In honor of John H. Coates., 279–393. MR 2251474, DOI 10.4310/PAMQ.2006.v2.n2.a1
Matthew Emerton, Ordinary parts of admissible representations of $p$-adic reductive groups II. Derived functors, Astérisque 331 (2010), 403–459 (English, with English and French summaries). MR 2667883
Matthew Emerton, Local-global compatibility in the p-adic langlands programme for $GL_2/Q$, 2011, http://www.math.uchicago.edu/emerton/pdffiles/lg.pdf.
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
David Geraghty, Modularity lifting theorems for ordinary Galois representations, Math. Ann. 373 (2019), no. 3-4, 1341–1427. MR 3953131, DOI 10.1007/s00208-018-1742-4
Toby Gee and James Newton, Patching and the completed homology of locally symmetric spaces, J. Inst. Math. Jussieu (2020), 1–64.
A. Grothendieck, Éléments de géométrie algébrique. IV. Étude locale des schémas et des morphismes de schémas. III, Inst. Hautes Études Sci. Publ. Math. 28 (1966), 255. MR 217086
Haruzo Hida, Nearly ordinary Hecke algebras and Galois representations of several variables, Algebraic analysis, geometry, and number theory (Baltimore, MD, 1988) Johns Hopkins Univ. Press, Baltimore, MD, 1989, pp. 115–134. MR 1463699
Haruzo Hida, On nearly ordinary Hecke algebras for $\textrm {GL}(2)$ over totally real fields, Algebraic number theory, Adv. Stud. Pure Math., vol. 17, Academic Press, Boston, MA, 1989, pp. 139–169. MR 1097614, DOI 10.2969/aspm/01710139
Haruzo Hida, Hilbert modular forms and Iwasawa theory, Oxford Mathematical Monographs, The Clarendon Press, Oxford University Press, Oxford, 2006. MR 2243770, DOI 10.1093/acprof:oso/9780198571025.001.0001
Yongquan Hu and Fucheng Tan, The Breuil-Mézard conjecture for non-scalar split residual representations, Ann. Sci. Éc. Norm. Supér. (4) 48 (2015), no. 6, 1383–1421 (English, with English and French summaries). MR 3429471, DOI 10.24033/asens.2272
Mark Kisin, Overconvergent modular forms and the Fontaine-Mazur conjecture, Invent. Math. 153 (2003), no. 2, 373–454. MR 1992017, DOI 10.1007/s00222-003-0293-8
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
Mark Kisin, Moduli of finite flat group schemes, and modularity, Ann. of Math. (2) 170 (2009), no. 3, 1085–1180. MR 2600871, DOI 10.4007/annals.2009.170.1085
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
Stephen Lichtenbaum, On the values of zeta and $L$-functions. I, Ann. of Math. (2) 96 (1972), 338–360. MR 360527, DOI 10.2307/1970792
Huishi Li and Freddy van Oystaeyen, Zariskian filtrations, $K$-Monographs in Mathematics, vol. 2, Kluwer Academic Publishers, Dordrecht, 1996. MR 1420862, DOI 10.1007/978-94-015-8759-4
Hideyuki Matsumura, Commutative algebra, 2nd ed., Mathematics Lecture Note Series, vol. 56, Benjamin/Cummings Publishing Co., Inc., Reading, Mass., 1980. MR 575344
Hideyuki Matsumura, Commutative ring theory, 2nd ed., Cambridge Studies in Advanced Mathematics, vol. 8, Cambridge University Press, Cambridge, 1989. Translated from the Japanese by M. Reid. MR 1011461
B. Mazur, Deforming Galois representations, Galois groups over $\textbf {Q}$ (Berkeley, CA, 1987) Math. Sci. Res. Inst. Publ., vol. 16, Springer, New York, 1989, pp. 385–437. MR 1012172, DOI 10.1007/978-1-4613-9649-9_{7}
Jürgen Neukirch, Alexander Schmidt, and Kay Wingberg, Cohomology of number fields, 2nd ed., Grundlehren der mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 323, Springer-Verlag, Berlin, 2008. MR 2392026, DOI 10.1007/978-3-540-37889-1
James Newton and Jack A. Thorne, Adjoint Selmer groups of automorphic Galois representations of unitary type, to appear in J. Eur. Math. Soc., 2020.
Vytautas Paškūnas, On some crystalline representations of $\textrm {GL}_2(\Bbb Q_p)$, Algebra Number Theory 3 (2009), no. 4, 411–421. MR 2525557, DOI 10.2140/ant.2009.3.411
Vytautas Paškūnas, Extensions for supersingular representations of $\textrm {GL}_2(\Bbb Q_p)$, Astérisque 331 (2010), 317–353 (English, with English and French summaries). MR 2667891
Vytautas Paškūnas, The image of Colmez’s Montreal functor, Publ. Math. Inst. Hautes Études Sci. 118 (2013), 1–191. MR 3150248, DOI 10.1007/s10240-013-0049-y
Vytautas Paškūnas, Blocks for $\textrm {mod}\,p$ representations of $\textrm {GL}_2(\Bbb Q_p)$, Automorphic forms and Galois representations. Vol. 2, London Math. Soc. Lecture Note Ser., vol. 415, Cambridge Univ. Press, Cambridge, 2014, pp. 231–247. MR 3444235
Vytautas Paškūnas, On 2-dimensional 2-adic Galois representations of local and global fields, Algebra Number Theory 10 (2016), no. 6, 1301–1358. MR 3544298, DOI 10.2140/ant.2016.10.1301
Vytautas Paškūnas, On some consequences of a theorem of J. Ludwig, arXiv e-prints, 2018, https://arxiv.org/abs/1804.07567.
Vytautas Paškūnas and Shen-Ning Tung, Finiteness properties of the category of mod $p$ representations of $\mathrm {GL}_2(\mathbb {Q}_p)$, to appear in Forum Math. Sigma, 2021.
Kenneth A. Ribet, A modular construction of unramified $p$-extensions of $Q(\mu _{p})$, Invent. Math. 34 (1976), no. 3, 151–162. MR 419403, DOI 10.1007/BF01403065
Peter Scholze, On the $p$-adic cohomology of the Lubin-Tate tower, Ann. Sci. Éc. Norm. Supér. (4) 51 (2018), no. 4, 811–863 (English, with English and French summaries). With an appendix by Michael Rapoport. MR 3861564, DOI 10.24033/asens.2367
Goro Shimura, The special values of the zeta functions associated with Hilbert modular forms, Duke Math. J. 45 (1978), no. 3, 637–679. MR 507462
Jack Shotton, Local deformation rings for $\textrm {GL}_2$ and a Breuil-Mézard conjecture when $\ell \ne p$, Algebra Number Theory 10 (2016), no. 7, 1437–1475. MR 3554238, DOI 10.2140/ant.2016.10.1437
Christopher Skinner, A note on the $p$-adic Galois representations attached to Hilbert modular forms, Doc. Math. 14 (2009), 241–258. MR 2538615
Tobias Schmidt and Matthias Strauch, Dimensions of some locally analytic representations, Represent. Theory 20 (2016), 14–38. MR 3455080, DOI 10.1090/ert/475
The Stacks Project Authors, Stacks project, https://stacks.math.columbia.edu, 2018.
C. M. Skinner and A. J. Wiles, Residually reducible representations and modular forms, Inst. Hautes Études Sci. Publ. Math. 89 (1999), 5–126 (2000). MR 1793414
C. M. Skinner and Andrew J. Wiles, Nearly ordinary deformations of irreducible residual representations, Ann. Fac. Sci. Toulouse Math. (6) 10 (2001), no. 1, 185–215 (English, with English and French summaries). MR 1928993, DOI 10.5802/afst.988
Richard Taylor, Galois representations associated to Siegel modular forms of low weight, Duke Math. J. 63 (1991), no. 2, 281–332. MR 1115109, DOI 10.1215/S0012-7094-91-06312-X
Richard Taylor, On icosahedral Artin representations. II, Amer. J. Math. 125 (2003), no. 3, 549–566. MR 1981033, DOI 10.1353/ajm.2003.0021
Richard Taylor, On the meromorphic continuation of degree two $L$-functions, Doc. Math. Extra Vol. (2006), 729–779. MR 2290604
Richard Taylor, Automorphy for some $l$-adic lifts of automorphic mod $l$ Galois representations. II, Publ. Math. Inst. Hautes Études Sci. 108 (2008), 183–239. MR 2470688, DOI 10.1007/s10240-008-0015-2
Jack A. Thorne, Automorphy lifting for residually reducible $l$-adic Galois representations, J. Amer. Math. Soc. 28 (2015), no. 3, 785–870. MR 3327536, DOI 10.1090/S0894-0347-2014-00812-2
Shen-Ning Tung, On the automorphy of 2-dimensional potentially semi-stable deformation rings of $g_{\mathbb {q}_p}$, to appear in Algebra Number Theory, 2018.
Shen-Ning Tung, On the modularity of 2-adic potentially semi-stable deformation rings, Math. Z. 298 (2021), no. 1-2, 107–159. MR 4257080, DOI 10.1007/s00209-020-02588-4
Lawrence C. Washington, The non-$p$-part of the class number in a cyclotomic $\textbf {Z}_{p}$-extension, Invent. Math. 49 (1978), no. 1, 87–97. MR 511097, DOI 10.1007/BF01399512
A. Wiles, On $p$-adic representations for totally real fields, Ann. of Math. (2) 123 (1986), no. 3, 407–456. MR 840720, DOI 10.2307/1971332
A. Wiles, On ordinary $\lambda$-adic representations associated to modular forms, Invent. Math. 94 (1988), no. 3, 529–573. MR 969243, DOI 10.1007/BF01394275