# Ramification filtration via deformations

## TL;DR

This paper introduces a new technique using deformations and Lie algebra actions to describe and generate ramification filtration ideals in Galois groups over local fields of characteristic p.

## Contribution

It develops a novel method to explicitly describe ramification ideals via deformations and Lie algebra actions, improving understanding of ramification filtrations in Galois groups.

## Key findings

- Established a new description of ramification ideals using deformations.
- Constructed explicit generators for ramification ideals.
- Connected deformation techniques with the explicit structure of ramification filtrations.

## Abstract

Let $\mathcal K$ be a field of formal Laurent series with coefficients in a finite field of characteristic $p$, $\mathcal G_{<p}$ -- the maximal quotient of $\operatorname{Gal} (\mathcal K_{sep}/\mathcal K)$ of period $p$ and nilpotent class $<p$ and $\{\mathcal G_{<p}^{(v)}\}_{v\geqslant 0}$ -- its filtration by ramification subgroups in the upper numbering. Let $\mathcal G_{<p}=G(\mathcal L)$ be the identification of nilpotent Artin-Schreier theory: here $G(\mathcal L)$ is the group obtained from a suitable profinite Lie $\mathbb{F}_p$-algebra $\mathcal L$ via the Campbell-Hausdorff composition law. We develop a new technique to describe the ideals $\mathcal L^{(v)}$ such that $G(\mathcal L^{(v)})=\mathcal G_{<p}^{(v)}$ and to find their generators. Given $v_0\geqslant 1$ we construct epimorphism of Lie algebras $\bar\eta ^{\dag }:\mathcal L\longrightarrow \bar{\mathcal L}^{\dag }$ and an action $\Omega_U$ of the formal group of order $p$, $\alpha =_p=\operatorname{Spec}\,\mathbb{F}_p[U]$, $U^p=0$, on $\bar{\mathcal L}^{\dag }$. Suppose $d\Omega_U=B^{\dag }U$, where $B^{\dag }\in\operatorname{Diff}\bar{\mathcal L}^{\dag }$, and $\bar{\mathcal L}^{\dag }[v_0]$ is the ideal of $\bar{\mathcal L}^{\dag }$ generated by the elements of $B^{\dag }(\bar{\mathcal L}^{\dag })$. The main result of the paper states that $\mathcal L^{(v_0)}=(\bar\eta ^{\dag })^{-1}\bar{\mathcal L}^{\dag }[v_0]$. In the last sections we relate this result to the explicit construction of generators of $\mathcal L^{(v_0)}$ obtained earlier by the author, develop its more efficient version and apply it to the recovering of the whole ramification filtration of $\mathcal G_{<p}$ from the set of its jumps.