Mathematische Annalen

, Volume 342, Issue 2, pp 297–308 | Cite as

On Galois groups of unramified pro-p extensions

  • Romyar T. SharifiEmail author


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 Z p -extensions of Q p ) 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.


Galois Group Algebraic Extension Decomposition Group Inertia Subgroup Iwasawa Theory 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Balister P., Howson S.: Notes on Nakayama’s lemma for compact Λ-modules. Asian Math. J. 1, 224–229 (1997)zbMATHMathSciNetGoogle 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)zbMATHCrossRefMathSciNetGoogle 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)zbMATHMathSciNetGoogle 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)zbMATHMathSciNetGoogle Scholar
  6. 6.
    McCallum W., Sharifi R.: A cup product in the Galois cohomology of number fields. Duke Math. J. 120, 269–310 (2003)zbMATHCrossRefMathSciNetGoogle Scholar
  7. 7.
    Nguyen Quang Do T.: K 3 et formules de Riemann–Hurwitz p-adiques. K-theory 7, 429–441 (1993)zbMATHCrossRefMathSciNetGoogle Scholar
  8. 8.
    Sharifi R.: Iwasawa theory and the Eisenstein ideal. Duke Math. J. 120, 269–310 (2003)zbMATHCrossRefMathSciNetGoogle Scholar
  9. 9.
    Venjakob O.: A non-commutative Weierstrass preparation theorem and applications to Iwasawa theory. J. Reine Angew. Math. 559, 153–191 (2003)zbMATHMathSciNetGoogle Scholar
  10. 10.
    Wingberg K.: On the maximal unramified p-extension of an algebraic number field. J. Reine Angew. Math. 440, 129–156 (1993)zbMATHMathSciNetGoogle Scholar

Copyright information

© Springer-Verlag 2008

Authors and Affiliations

  1. 1.Department of Mathematics and StatisticsMcMaster UniversityHamiltonCanada

Personalised recommendations