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 $p$, $\mathscr G_{<p}$ 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.

Détail

Keywords: local field, ramification subgroups.

Bibliographie

[1] V. A. Abrashkin, “A ramification filtration of the Galois group of a local field”, Proceedings of the St. Petersburg Mathematical Society, v. III, Amer. Math. Soc. Transl. Ser. 2, 166, Amer. Math. Soc., Providence, RI, 1995, 35–100 | DOI | MR | MR | Zbl | Zbl

[2] V. A. Abrashkin, “Ramification filtration of the Galois group of a local field. II”, Proc. Steklov Inst. Math., 208 (1995), 15–62 | MR | Zbl

[3] V. A. Abrashkin, “A ramification filtration of the Galois group of a local field. III”, Izv. Math., 62:5 (1998), 857–900 | DOI | DOI | MR | Zbl

[4] V. A. Abrashkin, “On a local analogue of the Grothendieck conjecture”, Internat. J. Math., 11:2 (2000), 133–175 | DOI | MR | Zbl

[5] V. Abrashkin, “Modified proof of a local analogue of the Grothendieck conjecture”, J. Théor. Nombres Bordeaux, 22:1 (2010), 1–50 | DOI | MR | Zbl

[6] V. Abrashkin, R. Jenni, “The field-of-norms functor and the Hilbert symbol for higher local fields”, J. Théor. Nombres Bordeaux, 24:1 (2012), 1–39 | DOI | MR | Zbl

[7] V. Abrashkin, “Galois groups of local fields, Lie algebras and ramification”, Arithmetic and geometry, London Math. Soc. Lecture Note Ser., 420, Cambridge Univ. Press, Cambridge, 2015, 1–23 | DOI | MR | Zbl

[8] V. Abrashkin, “Groups of automorphisms of local fields of period $p$ and nilpotent class $

$. I”, Internat. J. Math., 28:6 (2017), 1750043, 34 pp. | DOI | MR | Zbl

[9] V. Abrashkin, “Groups of automorphisms of local fields of period $p$ and nilpotent class $

$. II”, Internat. J. Math., 28:10 (2017), 1750066, 32 pp. | DOI | MR | Zbl

[10] V. A. Abrashkin, “An analogue of the Grothendieck conjecture for two-dimensional local fields of finite characteristic”, Proc. Steklov Inst. Math., 241 (2003), 2–34 | MR | Zbl

[11] J.-M. Fontaine, “Représentations $p$-adiques des corps locaux (1-ere partie)”, The Grothendieck Festschrift, A collection of articles in honor of the 60th birthday of A. Grothendieck, v. II, Progr. Math., 87, Birkhäuser Boston, Boston, MA, 1990, 249–309 | DOI | MR | Zbl

[12] E. I. Khukhro, $p$-automorphisms of finite $p$-groups, London Math. Soc. Lecture Note Ser., 246, Cambridge Univ. Press, Cambridge, 1998, xviii+204 pp. | DOI | MR | Zbl

[13] M. Lazard, “Sur les groupes nilpotents et les anneaux de Lie”, Ann. Sci. Ecole Norm. Sup. (3), 71:2 (1954), 101–190 | DOI | MR | Zbl