A cup product in the Galois cohomology of number fields

Abstract

Let K be a number field containing the group μ_n 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 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.