Duke Mathematical Journal

A cup product in the Galois cohomology of number fields

William G. McCallum and Romyar T. Sharifi

Full-text: Access denied (no subscription detected)

We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text


Let K be a number field containing the group μn of nth roots of unity, and let S be a set of primes of K including all those dividing n and all real archimedean places. We consider the cup product on the first Galois cohomology group of the maximal S-ramified extension of K with coefficients in μn, which yields a pairing on a subgroup of K x containing the S-units. In this general situation, we determine a formula for the cup product of two elements that pair trivially at all local places.

Our primary focus is the case in which K=( μ p ) for n=p, an odd prime, and S consists of the unique prime above p in K. We describe a formula for this cup product in the case that one element is a pth root of unity. We explain a conjectural calculation of the restriction of the cup product to p-units for all p≤10,000$ and conjecture its surjectivity for all p satisfying Vandiver's conjecture. We prove this for the smallest irregular prime p=37 via a computation related to the Galois module structure of p-units in the unramified extension of K of degree p.

We describe a number of applications: to a product map in K-theory, to the structure of S-class groups in Kummer extensions of S, to relations in the Galois group of the maximal pro-p extension of ( μ p ) unramified outside p, to relations in the graded ℤp-Lie algebra associated to the representation of the absolute Galois group of ȑA in the outer automorphism group of the pro-p fundamental group of P 1 ( ¯ ){0,1,} , and to Greenberg's pseudonullity conjecture.

Article information

Duke Math. J., Volume 120, Number 2 (2003), 269-310.

First available in Project Euclid: 16 April 2004

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 11R34: Galois cohomology [See also 12Gxx, 19A31]
Secondary: 11R23: Iwasawa theory 11R29: Class numbers, class groups, discriminants 11R70: $K$-theory of global fields [See also 19Fxx] 20F34: Fundamental groups and their automorphisms [See also 57M05, 57Sxx]


McCallum, William G.; Sharifi, Romyar T. A cup product in the Galois cohomology of number fields. Duke Math. J. 120 (2003), no. 2, 269--310. doi:10.1215/S0012-7094-03-12023-2. https://projecteuclid.org/euclid.dmj/1082138585

Export citation