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Nilpotent local class field theory. (English) Zbl 1027.11089

From the introduction: Let \(G\) be any profinite group and \(A\) an Abelian profinite group. Let \(L(G)= \bigoplus_{n=1}^\infty G^{(n)}/G^{(n+1)}\) be the graded Lie algebra associated with \(G\) by means of the lower central series \((G^{(n)})_{n\geq 1}\) and let \({\mathcal L}(A)= \bigoplus_{n=1}^\infty L_n(A)\) be the universal graded Lie algebra associated with \(A\). Any homomorphism \(\varphi\) of \(A\) into \(G/G^{(2)}\) gives rise to a homomorphism \(\varphi_*\) of \({\mathcal L}(A)\) into \(L(G)\).
In this paper we study the special situation where \(A\) is the profinite completion \(\widehat{K}^\times\) of the multiplicative group \(K^\times\) of a local field \(K\), i.e. a field which is completed with respect to a discrete valuation with finite residue class field. The group \(G\) is the absolute Galois group \(G_K\) of \(K\) and \(\varphi\) is the Artin isomorphism of \(\widehat{K}^\times\) onto \(G_K/ G_K^{(2)}\).
The surjectivity of \(\varphi\) implies the same for \(\varphi_*\). The goal of this paper is the determination of the kernel of \(\varphi_*\). This is equivalent to the determination of the kernel of the component homomorphisms \[ \varphi_*(l): {\mathcal L}(A(l))\to L(G_K(l)), \] where \(l\) is any prime and \(B(l)\) is the maximal pro-\(l\) quotient of a profinite group \(B\). The difficult case occurs when \(l=p\), the residual characteristic of \(K\). If \(K\) is of characteristic \(p\), or if \(K\) is of characteristic zero and does not contain a primitive \(p\)th root of the unity, this kernel is zero. So we assume that \(K\) is of characteristic zero and contains a primitive \(p^\kappa\)th root of unity \(\zeta\) with \(\kappa\) chosen largest possible. In this case \(G_K(p)\) is a Demushkin group so that the cup-product \[ H^1(G_K(p), \mathbb{Z}/p^\kappa\mathbb{Z})\times H^1 (G_K(p), \mathbb{Z}/p^\kappa\mathbb{Z})\to H^2(G_K(p), \mathbb{Z}/p^\kappa\mathbb{Z})= \mathbb{Z}/p^\kappa\mathbb{Z} \] is nondegenerate. We now assume that \(p\) is odd. In this case, the form is alternating and so we obtain by duality an element in \[ G_K^{\text{ab}}/ (G_K^{\text{ab}})^{p^\kappa}\wedge G_K^{\text{ab}}/ (G_K^{\text{ab}})^{p^\kappa}. \] Using the Artin isomorphism, this determines an element \(\tau\in{\mathcal L}_2(A(p)) \otimes \mathbb{Z}/p^\kappa\mathbb{Z}\) which is determined by \(G_K\) up to a unit of \(\mathbb{Z}/p^\kappa\mathbb{Z}\). The main result is the following theorem:
The kernel of \(\varphi_*(p)\) is the ideal of \({\mathcal L}(A(p))\) that is generated by the elements having the form \([\text{ad} (\lambda) (\zeta), \text{ad} (\lambda) (\tau)]\), where \(\lambda\) is an element of the enveloping algebra of \({\mathcal L}(A(p))\).
The present paper originated from the thesis of the second author [Graduierte nilpotente Klassenkörpertheorie, Dissertation, Berlin (1995)] directed by the first, and assisted by important suggestions of the third author. Section 5 on ‘The module structure of \({\mathcal L}(G_K(p))\) and \(\operatorname {ker}\varphi_K(p)\)’ was added by the third author.

MSC:

11S31 Class field theory; \(p\)-adic formal groups
11S20 Galois theory
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