TY - JOUR

T1 - A cup product in the Galois cohomology of number fields

AU - McCallum, William G.

AU - Sharifi, Romyar T.

PY - 2003/11/1

Y1 - 2003/11/1

N2 - 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× 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 = ℚ(μp) 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 K, to relations in the Galois group of the maximal pro-p extension of ℚ(μp) unramified outside p, to relations in the graded ℤp-Lie algebra associated to the representation of the absolute Galois group of ℚ in the outer automorphism group of the pro-p fundamental group of P1 (ℚ̄) - {0,1, ∞}, and to Greenberg's pseudonullity conjecture.

AB - 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× 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 = ℚ(μp) 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 K, to relations in the Galois group of the maximal pro-p extension of ℚ(μp) unramified outside p, to relations in the graded ℤp-Lie algebra associated to the representation of the absolute Galois group of ℚ in the outer automorphism group of the pro-p fundamental group of P1 (ℚ̄) - {0,1, ∞}, and to Greenberg's pseudonullity conjecture.

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U2 - 10.1215/S0012-7094-03-12023-2

DO - 10.1215/S0012-7094-03-12023-2

M3 - Article

AN - SCOPUS:0347020857

VL - 120

SP - 269

EP - 310

JO - Duke Mathematical Journal

JF - Duke Mathematical Journal

SN - 0012-7094

IS - 2

ER -