A cup product in the Galois cohomology of number fields
type: Item
instance of: scholarly article
title: A cup product in the Galois cohomology of number fields
zbMATH Open document ID: 1047.11106
DOI: 10.1215/S0012-7094-03-12023-2
author: William G. McCallum
author: Romyar T. Sharifi
published in: Duke Mathematical Journal
publication date: 2004-03-07T00:00:00Z
full work available at URL: https://arxiv.org/abs/math/0202161
review text: Let \(K\) be a number field containing the group \(\mu_{n}\) of \(n\)th roots of unity and \(S\) a set of primes of \(K\) containing those above \(n\) and all real archimedean primes. Let \(G_{K,S}\) denote the Galois group of the maximal extension of \(K\) unramified outside \(S.\) This paper deals with the cup product map \((\ldotp , \ldotp)_{S}: H^{1} (G_{K, S}, \mu_{n} )^{\otimes 2} \rightarrow H^{2} ( G_{K, S}, \mu_{n}^{\otimes 2})\) in the special case where its image is contained in the kernel of the localization map \(H^{2} (G_{S, K}, \mu_{n}^{\otimes 2}) \to \bigoplus_{v \in S} H^{2} ( G_{v}, \mu_{n}^{\otimes 2} ),\) which is, by Poitou-Tate duality, isomorphic to \(C_{K, S} \otimes \mu_{n},\) \(C_{K, S}\) denoting the \(S\)-class group of \(K.\) The archetypal example is given by \(n =\) a prime \(p\), \(K = {\mathbb Q}(\mu_{p}),\) \(S = \{(1-\xi_{p})\},\) the localization map being then the zero map. The authors develop several formulae for the cup product \((\ldotp, \ldotp)_{S}\) in terms of ideals in a Kummer extension of \(K\) (Theorem 2.4), in terms of genus theory (Proposition 6.2) or (for \(n= p)\) in relation with the Hilbert symbol (Theorem 7.2). The situation being inherently nonlocal, interesting arithmetical consequences are expected. In particular, the induced pairing \({\mathcal O}^{\otimes 2}_{K, S} \to H^{2} ( G_{K, S}, \mu_{n}^{\otimes 2} )\) gives rise to a canonical map \(K^{M}_{2}( {\mathcal O}_{K, S})/n \to K_{2}({\mathcal O}_{K, S})/n.\) Restricting to the case \(K = {\mathbb Q}(\mu_{p}),\) \(p\) an odd prime satisfying Vandiver's conjecture, the authors conjecture that the natural map \(K^{M}_{2} ({\mathcal O}_{K, S}) \otimes {\mathbb Z}_{p} \to K_{2}({\mathcal O}_{K, S}) \otimes {\mathbb Z}_{p}\) is surjective. Actually, for all irregular pairs \((p, r)\) (where \(r\) is even and \(p\) divides the Bernoulli number \(B_{r}),\) Vandiver's conjecture implies that the eigenspaces \(A_{K}(\omega^{p-r}) \otimes \mu_{p}\) are cyclic, hence…
Mathematics Subject Classification ID: 11R34
Mathematics Subject Classification ID: 11R23
Mathematics Subject Classification ID: 11R29
Mathematics Subject Classification ID: 11R70
Mathematics Subject Classification ID: 20F34
zbMATH DE Number: 2051859
reviewed by: Thong Nguyen Quang Do
MaRDI profile type: Publication
arXiv ID: math/0202161
OpenAlex ID: W2099415038
description: scientific article