A Cup Product in the Galois Cohomology of Number Fields
| dc.creator | McCallum, William G. | |
| dc.creator | Sharifi, Romyar T. | |
| dc.date | 2002-02-17 | |
| dc.date | 2003-12-12 | |
| dc.date.accessioned | 2026-07-07T07:52:12Z | |
| dc.date.available | 2026-07-07T07:52:12Z | |
| dc.description | Let K be a number field containing the group of n-th roots of unity and S 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 n-th roots of unity, which yields a pairing on a subgroup of the multiplicative group of K containing the S-units. In this general situation, we determine a formula for the cup product of two elements which pair trivially at all local places. Our primary focus is the case that K is the cyclotomic field of p-th roots of unity 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 p-th 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 K unramified outside p, to relations in the graded Z_p-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 P^1 minus three points, and to Greenberg's pseudo-nullity conjecture. | |
| dc.description | final version | |
| dc.identifier | https://arxiv.org/abs/math/0202161 | |
| dc.identifier | http://arxiv.org/abs/math/0202161 | |
| dc.identifier | Duke Math. J. 120 (2003) 269-310 | |
| dc.identifier.uri | http://salesiana.dossiersoluciones.com/handle/123456789/125794 | |
| dc.subject | Number Theory | |
| dc.subject | 11R34; 11R29; 11R70 | |
| dc.title | A Cup Product in the Galois Cohomology of Number Fields | |
| dc.type | text |