Generalized Fesenko reciprocity map

Abstract

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.