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 Mat. Sb., 2021, Volume 212, Number 2, Pages 3–37 (Mi msb9322)

Ramification filtration via deformations

V. A. Abrashkinab

a Department of Mathematical Sciences, Durham University, Durham, UK
b Steklov Mathematical Institute of Russian Academy of Sciences, Moscow

Abstract: Let $\mathscr K$ be a field of formal Laurent series with coefficients in a finite field of characteristic $p$, $\mathscr G_{<p}$ the maximal quotient of the Galois group of $\mathscr K$ of period $p$ and nilpotency class $<p$ and {$\{\mathscr G_{<p}^{(v)}\}_{v\geqslant 1}$} the filtration by ramification subgroups in the upper numbering. Let $\mathscr G_{<p}=G(\mathscr L)$ 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 $G(\mathscr L^{(v)})=\mathscr G_{<p}^{(v)}$ 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 $\mathscr G_{<p}$ from the set of its jumps.
Bibliography: 13 titles.

Keywords: local field, ramification subgroups.

DOI: https://doi.org/10.4213/sm9322

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English version:
Sbornik: Mathematics, 2021, 212:2, 135–169

Bibliographic databases:

UDC: 512.625
MSC: 11S15, 11S20

Citation: V. A. Abrashkin, “Ramification filtration via deformations”, Mat. Sb., 212:2 (2021), 3–37; Sb. Math., 212:2 (2021), 135–169

Citation in format AMSBIB
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