Groups of automorphisms of local fields of period p M and nilpotent class <p
Annales de l'Institut Fourier, Volume 67 (2017) no. 2, pp. 605-635.

Suppose K is a finite field extension of p containing a p M -th primitive root of unity. For 1s<p denote by K[s,M] the maximal p-extension of K with the Galois group of period p M and nilpotent class s. We apply the nilpotent Artin–Schreier theory together with the theory of the field-of-norms functor to give an explicit description of the Galois groups of K[s,M]/K. As application we prove that the ramification subgroup of the absolute Galois group of K with the upper index v acts trivially on K[s,M] iff v>e K (M+s/(p-1))-(1-δ 1s )/p, where e K is the ramification index of K and δ 1s is the Kronecker symbol.

Soit K une extension finie de p contenant une racine p M -ième primitive de l’unité. Pour 1s<p on note K[s,M] la p-extension maximale de K dont le groupe de Galois est de période p M et de classe de nilpotence s. En utilisant la théorie d’Artin–Schreier nilpotente et la théorie du corps des normes on donne une description explicite du groupe de Galois de K[s,M]/K. Comme application de ce résultat on montre que le sous-groupe de ramification du groupe de Galois absolu de K de ramification supérieure v agit trivialement sur K[s,M] si et seulement si v>e K (M+s/(p-1))-(1-δ 1s )/p, où e K est l’indice de ramification de K et δ 1s est le symbole de Kronecker.

Received:
Revised:
Accepted:
Published online:
DOI: 10.5802/aif.3093
Classification: 11S15, 11S20
Keywords: local fields, upper ramification numbers
Mot clés : Corps locaux, nombres de ramification supérieure.

Abrashkin, Victor 1, 2

1 Department of Mathematical Sciences Durham University Lower Mountjoy, Stockton Rd DH1 3LE (UK)
2 Steklov Institute Gubkina str. 8 119991 Moscow (Russia)
License: CC-BY-ND 4.0
Copyrights: The authors retain unrestricted copyrights and publishing rights
@article{AIF_2017__67_2_605_0,
     author = {Abrashkin, Victor},
     title = {Groups of automorphisms of local fields of period $p^M$ and nilpotent class $<p$},
     journal = {Annales de l'Institut Fourier},
     pages = {605--635},
     publisher = {Association des Annales de l{\textquoteright}institut Fourier},
     volume = {67},
     number = {2},
     year = {2017},
     doi = {10.5802/aif.3093},
     language = {en},
     url = {https://aif.centre-mersenne.org/articles/10.5802/aif.3093/}
}
TY  - JOUR
AU  - Abrashkin, Victor
TI  - Groups of automorphisms of local fields of period $p^M$ and nilpotent class $
JO  - Annales de l'Institut Fourier
PY  - 2017
SP  - 605
EP  - 635
VL  - 67
IS  - 2
PB  - Association des Annales de l’institut Fourier
UR  - https://aif.centre-mersenne.org/articles/10.5802/aif.3093/
DO  - 10.5802/aif.3093
LA  - en
ID  - AIF_2017__67_2_605_0
ER  - 
%0 Journal Article
%A Abrashkin, Victor
%T Groups of automorphisms of local fields of period $p^M$ and nilpotent class $
%J Annales de l'Institut Fourier
%D 2017
%P 605-635
%V 67
%N 2
%I Association des Annales de l’institut Fourier
%U https://aif.centre-mersenne.org/articles/10.5802/aif.3093/
%R 10.5802/aif.3093
%G en
%F AIF_2017__67_2_605_0
Abrashkin, Victor. Groups of automorphisms of local fields of period $p^M$ and nilpotent class $
                  
                

[1] Abrashkin, Victor Automorphisms of local fields of period p and nilpotent class <p (http://arxiv.org/abs/1403.4121)

[2] Abrashkin, Victor Ramification filtration of the Galois group of a local field, Proceedings of the St. Petersburg Mathematical Society III (Amer. Math. Soc. Transl. Ser. 2), Volume 166 (1995), pp. 35-100

[3] Abrashkin, Victor Ramification filtration of the Galois group of a local field. II, Proceedings of Steklov Math. Inst., Volume 208 (1995), pp. 15-62

[4] Abrashkin, Victor Ramification filtration of the Galois group of a local field. III, Izv. Ross. Akad. Nauk, Ser. Mat., Volume 62 (1998) no. 5, pp. 3-48 (English transl. in Izv. Math. 62, no. 5, p. 857-900)

[5] Abrashkin, Victor On a local analogue of the Grothendieck Conjecture, Int. J. Math., Volume 11 (2000) no. 1, pp. 3-43

[6] Abrashkin, Victor Modified proof of a local analogue of the Grothendieck Conjecture, J. Théor. Nombres Bordeaux, Volume 22 (2010) no. 1, pp. 1-50 | DOI

[7] Abrashkin, Victor Galois groups of local fields, Lie algebras and ramification, Arithmetic and Geometry (London Mathematical Society Lecture Note Series), Volume 420, Cambridge University Press, 2015, pp. 1-23

[8] Abrashkin, Victor; Jenni, Ruth The field-of-norms functor and the Hilbert symbol for higher local fields, J. Théor. Nombres Bordeaux, Volume 24 (2012) no. 1, pp. 1-39 | DOI

[9] Fontaine, Jean-Marc Représentations p-adiques des corps locaux. I., The Grothendieck Festschrift, A Collection of Articles in Honor of the 60th Birthday of Alexander Grothendieck, vol. II (Prog. Math.), Volume 87, Birkhäuser, 1990, pp. 249-309

[10] Hall, Marshall jun. The theory of groups, The Macmillan Company, 1959, xiii+434 pages

[11] Khukhro, Evgenii I. p-automorphisms of finite p-groups, London Mathematical Society Lecture Note Series, 246, Cambridge University Press, 1998, xviii+204 pages

[12] Lazard, Michel Sur les groupes nilpotents et les anneaux de Lie, Ann. Sci. Éc. Norm. Supér., Volume 71 (1954), pp. 101-190 | DOI

[13] Mochizuki, Shinichi A version of the Grothendieck conjecture for p-adic local fields, Int. J. Math., Volume 8 (1997) no. 4, pp. 499-506 | DOI

[14] Serre, Jean-Pierre Local fields, Graduate Texts in Mathematics, 67, Springer-Verlag, 1979, vii+241 pages

[15] Winterberger, Jean-Pierre Le corps des normes de certaines extensions infinies des corps locaux; applications, Ann. Sci. Éc. Norm. Supér., Volume 16 (1983), pp. 59-89 | DOI

Cited by Sources: