Skip to main content
SpringerLink
Log in

Adjoint selmer groups as Iwasawa modules

  • Published:
Israel Journal of Mathematics Aims and scope Submit manuscript

Abstract

We fix a primep. In this paper, starting from a given Galois representation ϕ having values inp-adic points of a classical groupG, we study the adjoint action of ϕ on thep-adic Lie algebra of the derived group ofG. We call this new Galois representation the adjoint representation Ad(ϕ) of ϕ. Under a suitablep-ordinarity condition (and ramification conditions outsidep), we define, following Greenberg, the Selmer group Sel(Ad(ϕ))/L for each number fieldL. We scrutinize the behavior of Sel(Ad(ϕ))/E ∞ as an Iwasawa module for a fixed ℤ p -extensionE ∞/E of a number fieldE and deduce an exact control theorem. A key ingredient of the proof is the isomorphism between the Pontryagin dual of the Selmer group and the module of Kähler differentials of the universal nearly ordinary deformation ring of ϕ. WhenG=GL(2), ϕ is a modular Galois representation and the base fieldE is totally real, from a recent result of Fujiwara identifying the deformation ring with an appropriatep-adic Hecke algebra, we conclude some fine results on the structure of the Selmer groups, including torsion-property and an exact limit formula ats=0 of the characteristic power series, after removing the trivial zero.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. K. Barré-Sirieix, G. Diaz, F. Gramain and G. Philibert,Une preuve de la conjecture de Mahler-Manin, Inventiones Mathematicae124 (1996), 1–9.

    Article  MathSciNet  MATH  Google Scholar 

  2. D. Blasius and J. D. Rogawski,Motives for Hilbert modular forms, Inventiones Mathematicae114 (1993), 55–87.

    Article  MathSciNet  MATH  Google Scholar 

  3. H. Carayol,Formes modulaires et représentations galoisiennes à valeurs dans un anneau local compact, Contemporary Mathematics165 (1994), 213–237.

    Article  MathSciNet  Google Scholar 

  4. P. Deligne and J. Milne,Tannakian categories, Lecture Notes in Mathematics900, Springer-Verlag, Berlin, 1982, pp. 101–228.

    MATH  Google Scholar 

  5. F. Diamond and R. Taylor,Lifting modular mod l representations, Duke Mathematical Journal74 (1994), 253–269.

    Article  MathSciNet  MATH  Google Scholar 

  6. K. Doi, H. Hida and H. Ishii,Discriminant of Hecke fields and the twisted adjoint L-values for GL(2), Inventiones Mathematicae134 (1998), 547–577.

    Article  MathSciNet  MATH  Google Scholar 

  7. K. Fujiwara,Deformation rings and Hecke algebras in totally real case, preprint, 1996.

  8. R. Greenberg,Iwasawa theory and p-adic deformation of motives, Proceedings of Symposia in Pure Mathematics55, Part 2 (1994), 193–223.

    Article  MATH  Google Scholar 

  9. R. Greenberg,Arithmetic Theory of Elliptic Curves, preprint, to appear in Lecture Notes in Mathematics, Springer.

  10. R. Greenberg and G. Stevens,On the conjecture of Mazur, Tate, and Teitelbaum, Contemporary Mathematics165 (1994), 183–211.

    Article  MathSciNet  MATH  Google Scholar 

  11. M. Harris and R. Taylor,Deformation of automorphic Galois representations, preprint, 1998.

  12. H. Hida,On p-adic Hecke algebras for GL 2 over totally real fields, Annals of Mathematics128 (1988), 295–384.

    Article  MathSciNet  MATH  Google Scholar 

  13. H. Hida,On nearly ordinary Hecke algebras for GL(2)over totally real fields, Advanced Studies in Pure Mathematics17 (1989), 139–169.

    MathSciNet  MATH  Google Scholar 

  14. H. Hida,Nearly ordinary Hecke algebras and Galois representations of several variables, Proceedings of JAMI Inaugural Conference, Supplement to American Journal of Mathematics (1989), 115–134.

  15. H. Hida,p-adic L-functions for base change lifts of GL2 to GL3, in Proceedings of Conference on “Automorphic forms, Shimura varieties, andL-functions”, Perspectives in Mathematics11 (1990), 93–142.

    MathSciNet  Google Scholar 

  16. H. Hida,Elementary Theory of L-functions and Eisenstein Series, London Mathematical Society Student Texts26, Cambridge University Press, 1993.

  17. H. Hida,On Selmer groups of adjoint modular Galois representations, inNumber Theory, Paris, London Mathematical Society Lecture Notes Series235 (1996), 89–132.

    MathSciNet  MATH  Google Scholar 

  18. H. Hida,On the search of genuine p-adic modular L-functions for GL(n), Mémoires de la Société Mathématique de France67 (1996).

  19. H. Hida,Hecke algebras, Selmer groups and base change, preprint, 1997.

  20. H. Hida,Modular adjoint Selmer groups of several variables over totally real fields, preprint, 1997.

  21. H. Hida,Non-critical values of adjoint L-functions for SL(2), Proceedings of Symposia in Pure Mathematics66 (1999), Part 1, 123–175.

    Article  MathSciNet  MATH  Google Scholar 

  22. H. Hida,Modular Forms and Galois Cohomology, Cambridge University Press, 1999.

  23. H. Hida,Control theorems for coherent sheaves on Shimura vrieties of PEL-type, preprint, 1999.

  24. H. Hida and Y. Maeda,Non-abelian base change for totally real fields, Special issue of Pacific Journal of Mathematics in memory of Olga Taussky Todd (1997), 189–217.

  25. H. Hida and J. Tilouine,On the anticyclotomic main conjecture for CM fields, Inventiones Mathematicae117 (1994), 89–147.

    Article  MathSciNet  MATH  Google Scholar 

  26. H. Hida, J. Tilouine and E. Urban,Adjoint modular Galois representations and their Selmer groups, Proceedings of the National Academy of Sciences of the United States of America94 (1997), 11121–11124.

    Article  MathSciNet  MATH  Google Scholar 

  27. R. P. Langlands,Base change for GL(2), Annals of Mathematics Studies96, Princeton University Press, 1980.

  28. H. Matsumura,Commutative Ring Theory, Cambridge Studies in Advanced Mathematics8, Cambridge University Press, 1989.

  29. B. Mazur and J. Tilouine,Représentations Galoisiennes différentielles de Kähler et “conjectures principales”, Publications Mathématiques de l’Institut des Hautes Études Scientifiques71 (1990), 65–103.

    Article  MathSciNet  MATH  Google Scholar 

  30. B. Mazur and A. Wiles,On p-adic analytic families of Galois representations, Compositio Mathematica59 (1986), 231–264.

    MathSciNet  MATH  Google Scholar 

  31. T. Ochiai,Control theorem for Selmer groups of p-adic representations, Master Thesis, University of Tokyo, 1998.

  32. R. Taylor,On Galois representations associated to Hilbert modular forms, Inventiones Mathematicae98 (1989), 265–280.

    Article  MathSciNet  MATH  Google Scholar 

  33. R. Taylor and A. Wiles,Ring theoretic properties of certain Hecke modules, Annals of Mathematics142 (1995), 553–572.

    Article  MathSciNet  MATH  Google Scholar 

  34. J. Tilouine,Un sous-groupe p-divisible de la jacobienne de X 1 (N p r)comme module sur l’algèbre de Hecke, Bulletin de la Société Mathématique de France115 (1987), 329–360.

    MathSciNet  MATH  Google Scholar 

  35. J. Tilouine,Deformation of Galois representations and Hecke algebras, Publications of the Mehta Research Institute, Narosa Publications, Delhi, 1996.

  36. E. Urban,Selmer groups and the Eisenstein-Klingen ideal, preprint, 1997.

  37. A. Wiles,Modular elliptic curves and Fermat’s last theorem, Annals of Mathematics142 (1995), 443–551.

    Article  MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, UCLA, 90095-1555, Los Angeles, CA, USA

    Haruzo Hida

Authors
  1. Haruzo Hida
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Haruzo Hida.

Additional information

To the memory of Professor Kenkichi Iwasawa

Research of the author is partially supported by the NSF grant: DMS-9701017.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Hida, H. Adjoint selmer groups as Iwasawa modules. Isr. J. Math. 120, 361–427 (2000). https://doi.org/10.1007/BF02834845

Download citation

  • Received:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF02834845

Keywords

Search

Navigation