A cup product in the Galois cohomology of number fields

Let (kappla) be a number field containing the group (mu)n of nth roots of unity, and let S be a set of primes of (kappla) including all those dividing n and all real archimedean places. We consider the cup product on the first Galois cohomology group of the maximal Sramified extension of (kappla) with coefficients in (mu)n, which yields a pairing on a subgroup of (kapppla)^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 (kappla)=Q((mu)p) for n=(rho), an odd prime, and S consists of the unique prime above p in (kappla). 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(more than)10,000$ and conjecture its surjectivity for all (rho) satisfying Vandiverʹs conjecture. We prove this for the smallest irregular prime (rho)=37 via a computation related to the Galois module structure of p-units in the unramified extension of (kappla) of degree (rho).We describe a number of applications: to a product map in (kappla)-theory, to the structure of S-class groups in Kummer extensions of S, to relations in the Galois group of the maximal pro-(rho) extension of Q unramified outside p, to relations in the graded Z(rho)-Lie algebra associated to the representation of the absolute Galois group of rA in the outer automorphism group of the pro-(rho) fundamental group of P^1,and to Greenbergʹs pseudonullity conjecture.