A cup product in the Galois cohomology of number fields
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 the ``cup product conjecture'' above is equivalent to the nontriviality of all pairings \(\langle \ldotp, \ldotp \rangle_{r}\) obtained by projecting \((\ldotp, \ldotp )_{S}\) on the eigenspaces. The authors use their Theorem 7.2 to derive a criterion which can be checked numerically for \(p = 37\) (the first irregular prime). The cup product conjecture yields information on the form of relations in the maximal pro-\(p\)-quotient \({\mathcal G}\) of \(G_{K, S}.\) In the Vandiver setting, these relations \(\rho\) can be indexed by the irregular pairs \((p, r),\) and it is a matter of bookkeeping to write down an expression for \(\rho\) in \(\text{gr}^{2} ({\mathcal G})\) in terms of values of \(L_{p}\)-functions (for the ``linear'' terms) and of the pairing \(\langle \ldotp, \ldotp \rangle_{r}\) (for the ``quadratic'' terms). For \(p = 37,\) one gets in this way an explicit nontrivial relation in \(\text{gr}^{2} ({\mathcal G}).\) A related subject is the theory of the pro-\(p\)-fundamental group of \({\mathbb P}^{1} - \{ 0, 1, \infty\}.\) Deligne conjectures that \(\text{Lie}_{{\mathbb Z}_{p}} ({\mathcal G}) \otimes {\mathbb Q}_{p}\) is free on certain distinguished elements, but the se\-cond named author has shown that Greenberg's ``multiple conjecture'' (see below) implies that \(\text{Lie}_{{\mathbb Z}_{p}} ({\mathcal G})\) itself is not free. The cup product conjecture gives nontrivial relations in \(\text{Lie}_{{\mathbb Z}_{p}} ({\mathcal G})\) in terms of the pairing \((\ldotp, \ldotp)_{S}.\) An Iwasawa-theoretic consequence of the cup product conjecture concerns Greenberg's multiple conjecture, which asserts that over the compositum \(\widetilde K\) of all \({\mathbb Z}_{p}\)-extensions of a number field \(K,\) the Galois group of the maximal \((p)\)-ramified Abelian pro-\(p\)-extension of \(\widetilde K\) has no \({\mathbb Z}_{p} G(\widetilde K/K)\)-torsion. By way of example, if \(K = {\mathbb Q}(\mu_{p})\) and the \(p\)-class group of \(K\) is cyclic, then the nontriviality of \((\ldotp, \ldotp)_{S}\) restricted to the cyclotomic units implies Greenberg's conjecture.
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