Mathematische Annalen

, Volume 342, Issue 2, pp 297–308

On Galois groups of unramified pro-p extensions

Article

Abstract

Let p be an odd prime satisfying Vandiver’s conjecture. We consider two objects, the Galois group X of the maximal unramified abelian pro-p extension of the compositum of all Zp-extensions of Qp) and the Galois group \({\mathfrak{G}}\) of the maximal unramified pro-p extension of Q\({(\mu_{p^{\infty}})}\). We give a lower bound for the height of the annihilator of X as an Iwasawa module. Under some mild assumptions on Bernoulli numbers, we provide a necessary and sufficient condition for \({\mathfrak{G}}\) to be abelian. The bound and the condition in the two results are given in terms of special values of a cup product pairing on cyclotomic p-units. We obtain in particular that, for p  <  1,000, Greenberg’s conjecture that X is pseudo-null holds and \({\mathfrak{G}}\) is in fact abelian.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Balister P., Howson S.: Notes on Nakayama’s lemma for compact Λ-modules. Asian Math. J. 1, 224–229 (1997)MATHMathSciNetGoogle Scholar
  2. 2.
    Buhler J., Crandall R., Ernvall R., Metsänkylä T., Shokrollahi M.A.: Irregular primes and cyclotomic invariants to 12 million. J. Symbolic Comput. 31, 89–96 (2001)MATHCrossRefMathSciNetGoogle Scholar
  3. 3.
    Coates J., Fukaya T., Kato K., Sujatha R., Venjakob O.: The GL2 main conjecture for elliptic curves without complex multiplication. Publ. Math. Inst. Hautes Études Sci. 101, 163–208 (2005)MATHMathSciNetGoogle Scholar
  4. 4.
    Greenberg, R.: Iwasawa theory—past and present. In: Class Field Theory: Its Centenary and Prospect. Adv. Stud. Pure. Math. 30, 335–385 (2001)Google Scholar
  5. 5.
    Hachimori Y., Sharifi R.: On the failure of pseudo-nullity of Iwasawa modules. J. Alg. Geom. 14, 567–591 (2005)MATHMathSciNetGoogle Scholar
  6. 6.
    McCallum W., Sharifi R.: A cup product in the Galois cohomology of number fields. Duke Math. J. 120, 269–310 (2003)MATHCrossRefMathSciNetGoogle Scholar
  7. 7.
    Nguyen Quang Do T.: K 3 et formules de Riemann–Hurwitz p-adiques. K-theory 7, 429–441 (1993)MATHCrossRefMathSciNetGoogle Scholar
  8. 8.
    Sharifi R.: Iwasawa theory and the Eisenstein ideal. Duke Math. J. 120, 269–310 (2003)MATHCrossRefMathSciNetGoogle Scholar
  9. 9.
    Venjakob O.: A non-commutative Weierstrass preparation theorem and applications to Iwasawa theory. J. Reine Angew. Math. 559, 153–191 (2003)MATHMathSciNetGoogle Scholar
  10. 10.
    Wingberg K.: On the maximal unramified p-extension of an algebraic number field. J. Reine Angew. Math. 440, 129–156 (1993)MATHMathSciNetGoogle Scholar

Copyright information

© Springer-Verlag 2008

Authors and Affiliations

  1. 1.Department of Mathematics and StatisticsMcMaster UniversityHamiltonCanada

Personalised recommendations