Mathematische Annalen

, Volume 355, Issue 2, pp 481–518 | Cite as

On deformation rings of residually reducible Galois representations and R = T theorems



We introduce a new method of proof for R = T theorems in the residually reducible case. We study the crystalline universal deformation ring R (and its ideal of reducibility I) of a mod p Galois representation ρ 0 of dimension n whose semisimplification is the direct sum of two absolutely irreducible mutually non-isomorphic constituents ρ 1 and ρ 2. Under some assumptions on Selmer groups associated with ρ 1 and ρ 2 we show that R/I is cyclic and often finite. Using ideas and results of (but somewhat different assumptions from) Bellaïche and Chenevier we prove that I is principal for essentially self-dual representations and deduce statements about the structure of R. Using a new commutative algebra criterion we show that given enough information on the Hecke side one gets an R = T-theorem. We then apply the technique to modularity problems for 2-dimensional representations over an imaginary quadratic field and a 4-dimensional representation over Q.

Mathematics Subject Classification (2000)

11F80 11F55 


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  1. 1.
    Agarwal, M., Klosin, K.: Yoshida lifts and the Bloch-Kato conjecture for the convolution L-function. Preprint (2010)Google Scholar
  2. 2.
    Atiyah M.F., Macdonald I.G.: Introduction to Commutative Algebra. Addison-Wesley, Reading (1969)MATHGoogle Scholar
  3. 3.
    Barnet-Lamb, T., Gee, T., Geraghty, D., Taylor, R.: Potential automorphy and change of weight. Preprint (2010).
  4. 4.
    Bass H.: On the ubiquity of Gorenstein rings. Math. Z. 82, 8–28 (1963)MathSciNetMATHCrossRefGoogle Scholar
  5. 5.
    Bellaïche, J., Chenevier, G.: p-adic families of Galois representations and higher rank Selmer groups. Astérisque 324 (2009)Google Scholar
  6. 6.
    Berger T.: On the Eisenstein ideal for imaginary quadratic fields. Compos. Math. 145(3), 603–632 (2009)MathSciNetMATHCrossRefGoogle Scholar
  7. 7.
    Berger T., Klosin K.: A deformation problem for Galois representations over imaginary quadratic fields. Journal de l’Institut de Math. de Jussieu 8(4), 669–692 (2009)MathSciNetMATHCrossRefGoogle Scholar
  8. 8.
    Berger T., Klosin K.: R = T theorem for imaginary quadratic fields. Math. Ann. 349(3), 675–703 (2011)MathSciNetMATHCrossRefGoogle Scholar
  9. 9.
    Bloch, S., Kato, K.: L-functions and Tamagawa numbers of motives. The Grothendieck Festschrift, vol.I. Progr. Math., vol. 86, pp. 333–400. Birkhäuser Boston, Boston (1990)Google Scholar
  10. 10.
    Böcherer, S., Dummigan, N., Schulze-Pillot, R.: Yoshida lifts and Selmer groups. J. Math. Soc. Japan.
  11. 11.
    Brown J.: Saito-Kurokawa lifts and applications to the Bloch-Kato conjecture. Compos. Math. 143(2), 290–322 (2007)MathSciNetMATHGoogle Scholar
  12. 12.
    Calegari F.: Eisenstein deformation rings. Compos. Math. 142(1), 63–83 (2006)MathSciNetMATHCrossRefGoogle Scholar
  13. 13.
    Clozel, L., Harris, M., Taylor, R.: Automorphy for some l-adic lifts of automorphic mod 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 VignérasGoogle Scholar
  14. 14.
    Darmon, H., Diamond, F., Taylor, R.: Fermat’s last theorem. Elliptic Curves, Modular Forms & Fermat’s Last Theorem (Hong Kong, 1993), pp. 2–140. International Press, Cambridge (1997)Google Scholar
  15. 15.
    Dee, J.: Selmer groups of Hecke characters and Chow groups of self products of CM elliptic curves. Preprint (1999). arXiv:math.NT/9901155v1Google Scholar
  16. 16.
    Diamond, F., Flach, M., Guo, L.: The Tamagawa number conjecture of adjoint motives of modular forms. Ann. Sci. École Norm. Sup. 37(4) (2004)Google Scholar
  17. 17.
    Eisenbud, D.: Commutative algebra with a view toward algebraic geometry. Graduate Texts in Mathematics, vol. 150. Springer, New York (1995)Google Scholar
  18. 18.
    Faltings, G., Chai, C.-L.: Degeneration of abelian varieties. Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)], vol. 22. Springer, Berlin (1990). With an appendix by David MumfordGoogle Scholar
  19. 19.
    Fontaine, J.-M.: Valeurs spéciales des fonctions L des motifs. Astérisque no. 206, Exp. No. 751, 4, 205–249, S’eminaire Bourbaki, vol. 1991/92 (1992)Google Scholar
  20. 20.
    Fontaine, J.-M., Laffaille, G.: Construction de représentations p-adiques. Ann. Sci. École Norm. Sup. (4) 15(4), 547–608 (1982)Google Scholar
  21. 21.
    Fontaine, J.-M., Perrin-Riou, B.: Autour des conjectures de Bloch et Kato: cohomologie galoisienne et valeurs de fonctions L. Motives (Seattle, WA, 1991), Proc. Sympos. Pure Math., vol. 55, pp. 599–706. Amer. Math. Soc., Providence (1994)Google Scholar
  22. 22.
    Geraghty, D.: Modularity lifting theorems for ordinary Galois representations. Preprint (2010)Google Scholar
  23. 23.
    Hida, H.: Modular forms and Galois cohomology. Cambridge Studies in Advanced Mathematics, vol. 69. Cambridge University Press, Cambridge (2000)Google Scholar
  24. 24.
    Jarvis F.: Level lowering for modular mod l representations over totally real fields. Math. Ann. 313(1), 141–160 (1999)MathSciNetMATHCrossRefGoogle Scholar
  25. 25.
    Jorza, A.: Crystalline representations for GL2 over quadratic imaginary fields. Ph.D. thesis, Princeton University (2010)Google Scholar
  26. 26.
    Klosin K.: Congruences among automorphic forms on U(2,2) and the Bloch-Kato conjecture. Annales de l’institut Fourier 59(1), 81–166 (2009)MathSciNetMATHCrossRefGoogle Scholar
  27. 27.
    Laumon, G.: Fonctions zêtas des variétés de Siegel de dimension trois. Astérisque 302, 1–66 (2005). Formes automorphes. II. Le cas du groupe {GSp(4)Google Scholar
  28. 28.
    Livné R.: On the conductors of mod l Galois representations coming from modular forms. J. Number Theory 31(2), 133–141 (1989)MathSciNetMATHCrossRefGoogle Scholar
  29. 29.
    Matsumura, H.: Commutative Ring Theory, 2nd edn. Cambridge Studies in Advanced Mathematics, vol. 8. Cambridge University Press, Cambridge (1989) (Translated from the Japanese by M. Reid)Google Scholar
  30. 30.
    Mazur, B.: An introduction to the deformation theory of Galois representations. Modular Forms and Fermat’s Last Theorem (Boston, MA, 1995), pp. 243–311. Springer, New York (1997)Google Scholar
  31. 31.
    Rubin K.: The “main conjectures” of Iwasawa theory for imaginary quadratic fields. Invent. Math. 103(1), 25–68 (1991)MathSciNetMATHCrossRefGoogle Scholar
  32. 32.
    Rubin, K.: Euler systems. Annals of Mathematics Studies, vol. 147. Princeton University Press, Princeton (2000). Hermann Weyl Lectures. The Institute for Advanced StudyGoogle Scholar
  33. 33.
    Skinner C.M., Wiles A.J.: Ordinary representations and modular forms. Proc. Natl. Acad. Sci. USA 94(20), 10520–10527 (1997)MathSciNetMATHCrossRefGoogle Scholar
  34. 34.
    Skinner, C.M., Wiles, A.J.: Residually reducible representations and modular forms. Inst. Hautes Études Sci. Publ. Math. 89, 5–126 (1999)Google Scholar
  35. 35.
    Taylor R.: Automorphy for some l-adic lifts of automorphic mod l Galois representations. II. Publ. Math. Inst. Hautes Études Sci 108, 183–239 (2008)MATHCrossRefGoogle Scholar
  36. 36.
    Urban E.: On residually reducible representations on local rings. J. Algebra 212(2), 738–742 (1999)MathSciNetMATHCrossRefGoogle Scholar
  37. 37.
    Urban E.: Selmer groups and the Eisenstein-Klingen ideal. Duke Math. J. 106(3), 485–525 (2001)MathSciNetMATHCrossRefGoogle Scholar
  38. 38.
    Weissauer, R.: Four dimensional Galois representations. Astérisque 302, 67–150 (2005). Formes automorphes. II. Le cas du groupe {GSp(4)Google Scholar
  39. 39.
    Weston, T.: On Selmer groups of geometric Galois representations. Thesis, Harvard University, Cambridge (2000)Google Scholar

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© Springer-Verlag 2012

Authors and Affiliations

  1. 1.School of Mathematics and StatisticsUniversity of SheffieldSheffieldUK
  2. 2.Department of MathematicsQueens College, City University of New YorkFlushingUSA

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