Generalized Fesenko reciprocity map

Abstract:

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 $\pmb {\mu }_p(K^{\mathrm {sep}})\subset K$ 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 \begin{equation*} \pmb {\Phi }_{L/K}^{(\varphi )}:\mathrm {Gal}(L/K)\rightarrow K^\times /N_{L_0/K}L_0^\times \times U_{\widetilde {\mathbb X}(L/K)}^\diamond /Y_{L/L_0}, \end{equation*} 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.