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Let $$K$$ be a global field. The aim of this speculative paper is to discuss the possibility of constructing the non-abelian version of global class field theory of $$K$$ by “glueing” the non-abelian local class field theories of $$K_\nu $$ in the sense of Koch, for each $$\nu \in \mathbb{h }_K$$ , following Chevalley’s philosophy of ideles, and further discuss the relationship of this theory with the global reciprocity principle of Langlands.

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In recent papers, Fesenko has defined the non-abelian local reciprocity map for every totally-ramified arithmetically profinite ($APF$) Galois extension of a given local field $K$ by extending the works of Hazewinkel and Neukirch-Iwasawa. The theory of Fesenko extends the previous non-abelian generalizations of local class field theory given by Koch-de Shalit and by A. Gurevich. In this paper, which is research-expository in nature, we give a detailed account of Fesenko's work including all the skipped proofs.

This paper which is a continuation of [2], is essentially expository in nature, although some new results are presented. LetK be a local field with finite residue class fieldKk. We first define (cf. Definition 2.4) the conductorf(E/K) of an arbitrary finite Galois extensionE/K in the sense of non-abelian local class field theory as wherenG is the break in the upper ramification filtration ofG = Gal(E/K) defined by\(G^{n_G } \ne G^{n_{G + \delta } } = 1,\forall \delta \in \mathbb{R}_{_ \ne ^ > 0} \). Next, we study the basic properties of the idealf(E/K) inOk in caseE/K is a metabelian extension utilizing Koch-de Shalit metabelian local class field theory (cf. [8]).

Let K be a global field. The aim of this paper is to study the basic properties of the global non-Abelian norm-residue symbol \(\mathsf{NR}_{K}^{\underline{\varphi }^{\mathrm{Weil}} }\) of K, which is defined following the Chevalley-Miyake philosophy of ideles by “glueing” the local non-Abelian norm-residue symbol \(\{\bullet,K_{\nu }\}_{\varphi _{\nu }}\) of K ν in the sense of Koch, for each \(\nu \in\mathbb{h}_{K}\).

In this paper, which is the natural continuation and generalization of Fesenko's non-abelian reciprocity map, we extend the theory of Fesenko to infinite $APF$-Galois extensions $L$ over a local field $K$, with finite residue-class field $\kappa_K$ of $q=p^f$ elements, satisfying $\pmb{\mu}_p(K^{sep})\subset K$ and $K\subset L\subset K_{\phi^d}$ where the residue-class degree $[\kappa_L:\kappa_K]=d$. More precisely, for such extensions $L/K$, fixing a Lubin-Tate splitting $\phi$ over $K$, we construct a 1-cocycle, \pmb{\Phi}_{L/K}^{(\phi)}:\text{Gal}(L/K)\to K^\times/N_{L_0/K}L_0^\times\times U_{\widetilde{\mathbb X}(L/K)}^\diamond /Y_{L/L_0}, where $L_0=L\cap K^{nr}$, and study its functorial and ramification-theoretic properties. The case $d=1$ recovers the theory of Fesenko., Comment: 36 pages. To appear in Algebra i Analiz

In this article, we investigate (up to (Q) over bar) the values of the standard L-function L-Mm(K)(s, Psi) of the algebra M-m(K) over a CM-field K attached to a certain Hecke character Psi of K (defined through a Hecke character psi of the maximal totally real subfield F of K) at s = (k + 2n)/eta(sic); where m less than or equal to k is an element of Z is fixed and depends on the character psi, For All n is an element of Z(+). (C) 2000 Editions scientifiques et medicales Elsevier SAS.