Ramification in local Galois groups -- the second central step
Let \(F\) be a local field with finite residue field and let \(F^{\text{sep}}\) be a separable closure of \(F\). Let \(F^{\text{ab}}/F\) be the maximum abelian subextension of \(F^{\text{sep}}/F\) and let \(\tilde{F}/F\) be the maximal subextension of \(F^{\text{sep}}/F\) whose Galois group is nilpotent of class \(\leq 2\). This survey paper is a study of the Galois group \(\text{Gal}(\tilde{F}/F)\). The group \(\text{Gal}(F^{\text{ab}}/F)\) is well-understood thanks to class field theory, so the author concentrates his attention on the abelian group \(\text{Gal}(\tilde{F}/F^{\text{ab}})\). A basic result is that \(\text{Gal}(\tilde{F}/F^{\text{ab}})\cong FF^{\times}\), where \(FF^{\times}\) denotes the inverse limit of the system of exterior products \((F^{\times}/N)\wedge_{\mathbb Z}(F^{\times}/N)\), for \(N\) ranging over the closed subgroups of \(F^{\times}\) with finite index. The main focus of this paper is on summarizing what is known about the upper ramification filtration of \(\text{Gal}(\tilde{F}/F^{\text{ab}})\). A complete description of the filtration is not known, but the results that are presented here show that it is certainly far more complicated than the ramification filtration of \(\text{Gal}(F^{\text{ab}}/F)\). As an application, information about the ramification filtration of \(\text{Gal}(\tilde{F}/F^{\text{ab}})\) is used to get information about the Swan conductors of certain finite-dimensional complex representations of \(\text{Gal}(F^{\text{sep}}/F)\).
scientific article; zbMATH DE number 5543151