Erol Serbest | Searcharxiv

arXiv2008

Fesenko reciprocity map

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.

math.NT↗

arXiv2008

Generalized Fesenko reciprocity map

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 $κ_K$ of $q=p^f$ elements, satisfying $\pmbμ_p(K^{sep})\subset K$ and $K\subset L\subset K_{ϕ^d}$ where the residue-class degree $[κ_L:κ_K]=d$. More precisely, for such extensions $L/K$, fixing a Lubin-Tate splitting $ϕ$ over $K$, we construct a 1-cocycle, \pmbΦ_{L/K}^{(ϕ)}:\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.

math.NT↗