## Duke Mathematical Journal

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

### A cup product in the Galois cohomology of number fields

William G. McCallum and Romyar T. Sharifi

#### Abstract

Let *K* be a number field containing the group
*μ*_{n} of *n*th
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
μ_{n}, which
yields a pairing on a subgroup of $K\sp \mathsf{x}$
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 $K=\mathbb {Q}(\mu\sb p)$
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 $\mathbb {Q}(mu\sb p)$
unramified outside *p*, to
relations in the graded ℤ_{p}-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
$\mathbf {P}\sp 1(\overline \mathbb {Q})-\{0,1,\infty\}$, and to
Greenberg's pseudonullity conjecture.

#### Article information

**Source**

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

**Dates**

First available in Project Euclid: 16 April 2004

**Permanent link to this document**

https://projecteuclid.org/euclid.dmj/1082138585

**Digital Object Identifier**

doi:10.1215/S0012-7094-03-12023-2

**Mathematical Reviews number (MathSciNet)**

MR2019977

**Zentralblatt MATH identifier**

1047.11106

**Subjects**

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]

#### Citation

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.