Ramification filtration via deformations

V. A. Abrashkin

Sbornik. Mathematics, Tome 212 (2021) no. 2, pp. 135-169

Résumé

Let $\mathscr K$ be a field of formal Laurent series with coefficients in a finite field of characteristic $p$, $\mathscr G_{$ the maximal quotient of the Galois group of $\mathscr K$ of period $p$ and nilpotency class $$ and {$\{\mathscr G_{$} the filtration by ramification subgroups in the upper numbering. Let $\mathscr G_{$ be the identification of nilpotent Artin-Schreier theory: here $G(\mathscr L)$ is the group obtained from a suitable profinite Lie $\mathbb{F}_p$-algebra $\mathscr L$ via the Campbell-Hausdorff composition law. We develop a new technique for describing the ideals $\mathscr L^{(v)}$ such that $G(\mathscr L^{(v)})=\mathscr G_{$ and constructing their generators explicitly. Given $v_0\geqslant 1$, we construct an epimorphism of Lie algebras $\overline\eta^{\dagger}\colon \mathscr L\to \overline{\mathscr L}^{\dagger}$ and an action $\Omega_U$ of the formal group of order $p$, $\alpha_p=\operatorname{Spec}\mathbb{F}_p[U]$, $U^p=0$, on $\overline{\mathscr L}^{\dagger}$. Suppose $d\Omega_U=B^{\dagger}U$, where $B^{\dagger}\in\operatorname{Diff}\overline{\mathscr L}^{\dagger}$, and $\overline{\mathscr L}^{\dagger}[v_0]$ is the ideal of $\overline{\mathscr L}^{\dagger}$ generated by the elements of $B^{\dagger}(\overline{\mathscr L}^{\dagger})$. The main result in the paper states that $\mathscr L^{(v_0)}=(\overline\eta^{\dagger})^{-1}\overline{\mathscr L}^{\dagger}[v_0]$. In the last sections we relate this result to the explicit construction of generators of $\mathscr L^{(v_0)}$ obtained previously by the author, develop a more efficient version of it and apply it to recover the whole ramification filtration of $\mathscr G_{$ from the set of its jumps. Bibliography: 13 titles.

Détail

Keywords: local field, ramification subgroups.