Ramification filtration via deformations, II
Victor Abrashkin
TL;DR
The paper develops a geometric, deformation-theoretic framework for describing ramification subgroups of the Galois group of the maximal $p$-extension of a local field in characteristic $p$, extended to modulo $p^M$. It constructs central filtrations and deformations of carefully chosen Lie-algebra data, yielding an explicit description of ramification ideals $ rak L^{(v_0)}$ as preimages under a projection of a deformed operator $B^{†}$ acting on a lifted algebra, and provides explicit generators for $ar{ rak L}^{[v_0]}$ in terms of Lie monomials and coefficients from a structured coefficient set. This work generalizes prior mod $p$ results to arbitrary exponent $p^M$, connects ramification to deformation data via a nilpotent Artin–Schreier framework, and supplies concrete computational tools for constructing ramification subgroups, thereby advancing non-abelian local class field theory and its geometric foundations. The approach highlights a deep link between ramification theory, deformation theory, and $p$-adic differential structures, with potential implications for explicit non-abelian reciprocity laws and higher local fields.
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 $\{\mathcal G_{<p,M}^{(v)}\}_{v\geqslant 0}$ -- 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 $v_0\geqslant 1$, we construct a decreasing central filtration $\mathcal L(w)$, $1\leqslant w\leqslant p$, 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 $Ω$ of $\mathbb{Z} /p^M$ on $\bar{\mathcal L}^{†}$, which induces the identity action on $\bar{\mathcal L}$. Suppose $dΩ=B^{†}$, where $B^{†}\in\operatorname{Diff}\bar{\mathcal L}^{†}$, 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 $Γ_K$ and can be used for further developing of non-abelian local class field theory.