Modularity lifting theorems for ordinary Galois representations. (English) Zbl 1539.11080
Summary: We generalize results of L. Clozel et al. [Publ. Math., Inst. Hautes Étud. Sci. 108, 1–181 (2008; Zbl 1169.11020)] by proving modularity lifting theorems for ordinary \(l\)-adic Galois representations of any dimension of an imaginary CM or totally real number field. The main theorems are obtained by establishing an \(R^{{{\mathrm{red}}}}={\mathbb {T}}\) theorem over a Hida family. A key part of the proof is to construct appropriate ordinary lifting rings at the primes dividing \(l\) and to determine their irreducible components.
MSC:
| 11F80 | Galois representations |
| 11F70 | Representation-theoretic methods; automorphic representations over local and global fields |
Citations:
Zbl 1169.11020References:
| [1] | Bellaïche, J., Chenevier, G.: Families of Galois representations and Selmer groups. Astérisque 324, xii+314 (2009) · Zbl 1192.11035 |
| [2] | Borel, A., Jacquet, H.: Automorphic forms and automorphic representations, Automorphic forms, representations and \[L\] L-functions (Proc. Sympos. Pure Math., Oregon State Univ., Corvallis, Ore., 1977), Part 1, Proc. Sympos. Pure Math., vol. XXXIII. Amer. Math. Soc., Providence, R.I. (1979) (With a supplement “On the notion of an automorphic representation” by R. P. Langlands, pp. 189-207) |
| [3] | Barnet-Lamb, T., Geraghty, D., Harris, M., Taylor, R.: A family of Calabi-Yau varieties and potential automorphy II. Publ. Res. Inst. Math. Sci. 47(1), 29-98 (2011) · Zbl 1264.11044 · doi:10.2977/PRIMS/31 |
| [4] | Borel, A.: Some finiteness properties of adele groups over number fields. Inst. Hautes Études Sci. Publ. Math. 16, 5-30 (1963) · Zbl 0135.08902 |
| [5] | Bernstein, I.N., Zelevinsky, A.V.: Induced representations of reductive \[{\mathfrak{p}}\] p-adic groups. I. Ann. Sci. École Norm. Sup. (4) 10(4), 441-472 (1977) · Zbl 0412.22015 · doi:10.24033/asens.1333 |
| [6] | Casselman, W.: Introduction to the theory of admissible representations of \[p\] p-adic reductive groups (1995) (Preprint) |
| [7] | Colmez, P., Fontaine, J.-M.: Construction des représentations \[p\] p-adiques semi-stables. Invent. Math. 140(1), 1-43 (2000) · Zbl 1010.14004 · doi:10.1007/s002220000042 |
| [8] | Conrad, B.: Lifting global representations with local properties (Preprint) |
| [9] | Chenevier, G., Harris, M.: Construction of automorphic Galois representations II. Camb. Math. J. 1(1), 53-73 (2013) · Zbl 1310.11062 · doi:10.4310/CJM.2013.v1.n1.a2 |
| [10] | Clozel, L., Harris, M., Taylor, R.: Automorphy for some \[l\] l-adic lifts of automorphic mod \[l\] l Galois representations. Publ. Math. Inst. Hautes Études Sci. 108, 1-181 (2008) (With Appendix A, summarizing unpublished work of Russ Mann, and Appendix B by Marie-France Vignéras) · Zbl 1169.11020 |
| [11] | Fontaine, J.-M.: Représentations \[p\] p-adiques semi-stables. Astérisque 223, 113-184 (1994) [With an appendix by Pierre Colmez, Périodes p-adiques (Bures-sur-Yvette, 1988)] · Zbl 0865.14009 |
| [12] | Geraghty, D., Gee, T.: Companion forms for unitary groups and symplectic groups. Duke Math. J. 161(2), 247-303 (2012) · Zbl 1295.11043 · doi:10.1215/00127094-1507376 |
| [13] | Grothendieck, A.: Éléments de géométrie algébrique. IV. Étude locale des schémas et des morphismes de schémas IV. Inst. Hautes Études Sci. Publ. Math. 32, 361 (1967) · Zbl 0153.22301 |
| [14] | Gross, B.H.: Algebraic modular forms. Isr. J. Math. 113, 61-93 (1999) · Zbl 0965.11020 · doi:10.1007/BF02780173 |
| [15] | Guerberoff, L.: Modularity lifting theorems for Galois representations of unitary type. Compos. Math. 147(4), 1022-1058 (2011) · Zbl 1276.11086 · doi:10.1112/S0010437X10005154 |
| [16] | Hida, H.: On \[p\] p-adic Hecke algebras for \[{{\rm GL}}_2\] GL2 over totally real fields. Ann. Math. (2) 128(2), 295-384 (1988) · Zbl 0658.10034 · doi:10.2307/1971444 |
| [17] | Hida, H.: On nearly ordinary Hecke algebras for \[{{\rm GL}}(2)\] GL(2) over totally real fields, Algebraic number theory. Adv. Stud. Pure Math. 17, 139-169 (1989) (Academic Press, Boston, MA) · Zbl 0742.11026 |
| [18] | Hida, H.: Control theorems of \[p\] p-nearly ordinary cohomology groups for \[{\rm SL}(n)\] SL(n). Bull. Soc. Math. Fr. 123(3), 425-475 (1995) · Zbl 0852.11023 · doi:10.24033/bsmf.2266 |
| [19] | Hida, H.: Automorphic induction and Leopoldt type conjectures for \[{\rm GL}(n)\] GL(n). Asian J. Math. 2(4), 667-710 (1998) (Mikio Sato: a great Japanese mathematician of the twentieth century) · Zbl 0963.11027 |
| [20] | Harris, M., Taylor, R.: The geometry and cohomology of some simple Shimura varieties. Annals of Mathematics Studies, vol. 151. Princeton University Press, Princeton (2001) (With an appendix by Vladimir G. Berkovich) · Zbl 1036.11027 |
| [21] | Jantzen, J.C.: Representations of Algebraic Groups, 2nd edn. Mathematical Surveys and Monographs, vol. 107. American Mathematical Society, Providence, RI (2003) · Zbl 1034.20041 |
| [22] | Kisin, M.: Overconvergent modular forms and the Fontaine-Mazur conjecture. Invent. Math. 153(2), 373-454 (2003) · Zbl 1045.11029 · doi:10.1007/s00222-003-0293-8 |
| [23] | Kisin, M.: Potentially semi-stable deformation rings. J. Am. Math. Soc. 21(2), 513-546 (2008) · Zbl 1205.11060 · doi:10.1090/S0894-0347-07-00576-0 |
| [24] | Labesse, J.-P.: Changement de base CM et séries discrètes, On the stabilization of the trace formula. Shimura Var. Arith. Appl. 1, 429-470 (2011) (Int. Press, Somerville, MA) · Zbl 1255.11027 |
| [25] | Mauger, D.: Algèbres de Hecke quasi-ordinaires universelles. Ann. Sci. École Norm. Sup. (4) 37(2), 171-222 (2004) · Zbl 1196.11074 · doi:10.1016/j.ansens.2004.01.001 |
| [26] | Mazur, B.: Deforming Galois representations, Galois groups over \[{\bf Q}\] Q (Berkeley, CA, 1987). Math. Sci. Res. Inst. Publ. 16, 385-437 (1989) (Springer, New York) · Zbl 0714.11076 |
| [27] | Nekovář, J.: On \[p\] p-adic height pairings, Séminaire de Théorie des Nombres, Paris, 1990-1991, Progr. Math., vol. 108, pp. 127-202. Birkhäuser Boston, Boston (1993) · Zbl 0859.11038 |
| [28] | Neukirch, J., Schmidt, A., Wingberg, K.: Cohomology of number fields, 2nd edn. Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 323. Springer, Berlin (2008) · Zbl 1136.11001 |
| [29] | Shin, S.W.: Galois representations arising from some compact Shimura varieties. Ann. Math. (2) 173(3), 1645-1741 (2011) · Zbl 1269.11053 · doi:10.4007/annals.2011.173.3.9 |
| [30] | Taylor, R.: Automorphy for some \[l\] l-adic lifts of automorphic mod \[l\] l Galois representations. II. Publ. Math. Inst. Hautes Études Sci. 108, 183-239 (2008) · Zbl 1169.11021 |
| [31] | Tilouine, J., Urban, E.: Several-variable \[p\] p-adic families of Siegel-Hilbert cusp eigensystems and their Galois representations. Ann. Sci. École Norm. Sup. (4) 32(4), 499-574 (1999) · Zbl 0991.11016 · doi:10.1016/S0012-9593(99)80021-4 |
| [32] | Zelevinsky, A.V.: Induced representations of reductive \[{\mathfrak{p}}\] p-adic groups. II. On irreducible representations of \[{\rm GL}(n)\] GL(n). Ann. Sci. École Norm. Sup. (4) 13(2), 165-210 (1980) · Zbl 0441.22014 · doi:10.24033/asens.1379 |
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