Generalized Fesenko reciprocity map

K. I. Ikeda ; E. Serbest

Algebra i analiz, Tome 20 (2008) no. 4, pp. 118-159.

Résumé

The paper is a natural continuation and generalization of the works of Fesenko and of the authors. Fesenko's theory is carried over to infinite $APF$-Galois extensions $L$ over a local field $K$ with a finite residue-class field $\kappa_K$ of $q=p^f$ elements, satisfying $\mathbf{\mu}_p(K^\mathrm{sep})\subset K$ and $K\subset L\subset K_{\varphi^d}$, where the residue-class degree $[\kappa_L:\kappa_K]$ is equal to $d$. More precisely, for such extensions $L/K$ and a fixed Lubin–Tate splitting $\varphi$ over $K$, a 1-cocycle $$ \mathbf{\Phi}_{L/K}^{(\varphi)}\colon\mathrm{Gal}(L/K)\to K^\times/N_{L_0/K}L_0^\times\times U_{\widetilde{\mathbb X}(L/K)}^\diamond/Y_{L/L_0} $$ where $L_0=L\cap K^{nr}$, is constructed, and its functorial and ramification-theoretic properties are studied. The case of $d=1$ recovers the theory of Fesenko.

Détail

Keywords: local fields, higher-ramification theory, $APF$-extensions Fontaine–Wintenberger field of norms, Fesenko reciprocity map, generalized Fesenko reciprocity map, non-abelian local class field theory.

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