Ramification filtration via deformations, II

Abstract

Let $\mathcal{K}$ be a field of formal Laurent series with coefficients in a finite field of characteristic $p$. For $M\ge 1$, let ${\mathcal{G}}_{<p,M}$ be the maximal quotient of the Galois group of $\mathcal{K}$ of period $p^M$ and nilpotent class $<p$ and $\{{\mathcal{G}}_{<p,M}^{(v)}{\}}_{v⩾0}$ -- the ramification subgroups in upper numbering. Let ${\mathcal{G}}_{<p,M}=G(\mathcal{L})$ be the identification of nilpotent Artin-Schreier theory: here $G(\mathcal{L})$ is the group obtained from a suitable profinite Lie $\mathbb{Z}/p^M$-algebra $\mathcal{L}$ via the Campbell-Hausdorff composition law. We develop new techniques to obtain a ``geometrical'' construction of the ideals ${\mathcal{L}}^{(v)}$ such that $G({\mathcal{L}}^{(v)})={\mathcal{G}}_{<p,M}^{(v)}$. Given $v_0⩾1$, we construct a decreasing central filtration $\mathcal{L}(w)$, $1⩽w⩽p$, on $\mathcal{L}$, an epimorphism of Lie $\mathbb{Z}/p^M$-algebras $\bar{\mathcal{V}}:{\bar{\mathcal{L}}}^{†}\to \bar{\mathcal{L}}:=\mathcal{L}/\mathcal{L}(p)$, and a unipotent action $\Omega$ of $\mathbb{Z}/p^M$ on ${\bar{\mathcal{L}}}^{†}$, which induces the identity action on $\bar{\mathcal{L}}$. Suppose $d\Omega =B^{†}$, where $B^{†}\in \mathrm{Diff}{\bar{\mathcal{L}}}^{†}$, and ${\bar{\mathcal{L}}}^{†[v_0]}$ is the ideal of ${\bar{\mathcal{L}}}^{†}$ generated by the elements of $B^{†}({\bar{\mathcal{L}}}^{†})$. Our main result states that the ramification ideal ${\mathcal{L}}^{(v_0)}$ appears as the preimage of the ideal in $\bar{\mathcal{L}}$ generated by $\bar{\mathcal{V}}{B}^{†}({\bar{\mathcal{L}}}^{†[v_0]})$. In the last section we apply this to the explicit construction of generators of ${\bar{\mathcal{L}}}^{(v_0)}$. The paper justifies a geometrical origin of ramification subgroups of ${\Gamma }_K$ and can be used for further developing of non-abelian local class field theory.

38 pages