Galois groups of $p$-extensions of higher local fields

Victor Abrashkin

doi:10.5802/jtnb.1293

Galois groups of

p

-extensions of higher local fields

Victor Abrashkin ^1,²

¹ Department of Mathematical Sciences, Durham University, Science Laboratories Lower Mountjoy, Durham DH1 3LE, United Kingdom
² Steklov Institute, Gubkina str. 8, 119991, Moscow, Russia

Journal de théorie des nombres de Bordeaux, Volume 36 (2024) no. 2, pp. 671-724

Abstract (Original language)
Résumé

Suppose $𝒦$ is $N$ -dimensional local field of characteristic $p \neq 0$ , $𝒢_{< p}$ is the maximal quotient of period $p$ and nilpotent class $< p$ of $𝒢 = Gal (𝒦_{s e p} / 𝒦)$ , and $𝒦_{< p} \subset 𝒦_{s e p}$ is such that $Gal (𝒦_{< p} / 𝒦) = 𝒢_{< p}$ . We use nilpotent Artin–Schreier theory to identify $𝒢_{< p}$ with the group $G (ℒ)$ obtained from a profinite Lie $𝔽_{p}$ -algebra $ℒ$ via the Campbell–Hausdorff composition law. The canonical $𝒫$ -topology on $𝒦$ is used to define a dense Lie subalgebra $ℒ^{𝒫}$ in $ℒ$ . The algebra $ℒ^{𝒫}$ can be provided with a system of $𝒫$ -topological generators and we prove that all $N$ -dimensional extensions of $𝒦$ in $𝒦_{< p}$ are in the bijection with all $𝒫$ -open subalgebras of $ℒ^{𝒫}$ by the Galois correspondence. These results are applied to higher local fields $K$ of characteristic 0 containing a nontrivial $p$ -th root of unity. If $Γ = Gal (K_{a l g} / K)$ we introduce similarly the quotient $Γ_{< p}$ and present it in the form $G (L)$ , where $L$ is a suitable profinite Lie $𝔽_{p}$ -algebra. Then we introduce a dense $𝔽_{p}$ -Lie subalgebra $L^{𝒫}$ in $L$ , and describe the structure of $L^{𝒫}$ in terms of generators and relations. The general result is illustrated by explicit presentation of $Γ_{< p}$ modulo subgroup of third commutators.

Soit $𝒦$ un corps local de dimension $N$ et de caractéristique $p \neq 0 .$ On note $𝒢_{< p}$ le quotient maximal de $𝒢 = Gal (𝒦_{s e p} / 𝒦)$ de période $p$ et de classe de nilpotence $< p .$ Soit $𝒦_{< p} \subset 𝒦_{s e p}$ tel que $Gal (𝒦_{< p} / 𝒦) = 𝒢_{< p}$ . On utilise la théorie nilpotente d’Artin–Schreier pour identifier $𝒢_{< p}$ avec le groupe $G (ℒ)$ obtenu à partir d’une $𝔽_{p}$ -algèbre de Lie $ℒ$ via la loi de composition de Campbell–Hausdorff. On utilise la topologie canonique sur $𝒦$ dite $𝒫$ -topologie pour définir une sous-algèbre de Lie $ℒ^{𝒫}$ dense dans $ℒ$ . L’algèbre $ℒ^{𝒫}$ peut être munie d’un système de générateurs topologiques et nous prouvons que la correspondance de Galois établit une bijection entre les extensions $N$ -dimensionnelles de $𝒦$ dans $𝒦_{< p}$ et les $𝒫$ -sous-algèbres ouvertes de $ℒ^{𝒫} .$ Ces résultats sont appliqués aux corps locaux supérieurs $K$ de caractéristique 0 contenant une racine $p$ -ième primitive de l’unité. Si $Γ = Gal (K_{a l g} / K),$ on introduit de la même manière le quotient $Γ_{< p}$ de $Γ$ et on le présente sous la forme $G (L)$ , où $L$ est une $𝔽_{p}$ -algèbre de Lie profinie appropriée. On introduit ensuite une $𝔽_{p}$ -sous-algèbre de Lie $L^{𝒫}$ dense dans $L$ et on décrit la structure de $L^{𝒫}$ en termes de générateurs et relations. Le résultat général est illustré par une présentation explicite de $Γ_{< p}$ modulo le sous-groupe engendré par les $3$ -commutateurs.

Received: 2023-01-26
Accepted: 2023-03-03
Published online: 2024-11-13

MR Zbl

DOI: 10.5802/jtnb.1293

Classification: 11S15, 11S20
Keywords: Local field, Galois group

Author's affiliations:

Victor Abrashkin ^{1, 2}

¹ Department of Mathematical Sciences, Durham University, Science Laboratories Lower Mountjoy, Durham DH1 3LE, United Kingdom
² Steklov Institute, Gubkina str. 8, 119991, Moscow, Russia

License:

CC-BY-ND 4.0

Copyrights: The authors retain unrestricted copyrights and publishing rights

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Victor Abrashkin. Galois groups of $p$-extensions of higher local fields. Journal de théorie des nombres de Bordeaux, Volume 36 (2024) no. 2, pp. 671-724. doi: 10.5802/jtnb.1293

References
Cited by

[1] Victor A. Abrashkin Ramification filtration of the Galois group of a local field, Proceedings of the St. Petersburg Mathematical Society. Vol. III (Translations. Series 2. American Mathematical Society), Volume 166, American Mathematical Society, 1995, pp. 35-100 | Zbl

[2] Victor A. Abrashkin Ramification filtration of the Galois group of a local field. II, Number theory, algebra and algebraic geometry (Proceedings of the Steklov Institute of Mathematics), Volume 208, Maik Nauka/Interperiodica Publishing, 1995, pp. 18-69 | Zbl

[3] Victor A. Abrashkin On a local analogue of the Grothendieck Conjecture, Int. J. Math., Volume 11 (2000) no. 2, pp. 133-175 | DOI | Zbl | MR

[4] Victor A. Abrashkin Ramification theory for higher dimensional fields, Algebraic number theory and algebraic geometry (Contemporary Mathematics), Volume 300, American Mathematical Society, 2002, pp. 1-16 | DOI | Zbl

[5] Victor A. Abrashkin An analogue of the Grothendieck conjecture for two-dimensional local fields of finite characteristic, Number theory, algebra, and algebraic geometry (Proceedings of the Steklov Institute of Mathematics), Volume 241, Maik Nauka/Interperiodika, 2003, pp. 2-34 | Zbl

[6] Victor A. Abrashkin An analogue of the field-of-norms functor and of the Grothendieck conjecture, J. Algebr. Geom., Volume 16 (2007) no. 4, pp. 671-730 | Zbl | DOI | MR

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

[8] Victor A. Abrashkin 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 | Zbl

[9] Victor A. Abrashkin Groups of automorphisms of local fields of period $p$ and nilpotent class $< p$ . I, Int. J. Math., Volume 28 (2017) no. 6, 1750043, 34 pages | Zbl

[10] Victor A. Abrashkin Groups of automorphisms of local fields of period $p$ and nilpotent class $< p$ . II, Int. J. Math., Volume 28 (2017) no. 10, 1750066, 32 pages | Zbl | MR

[11] Victor A. Abrashkin Ramification filtration via deformations, Sb. Math., Volume 212 (2021) no. 2, pp. 135-169 | Zbl | DOI

[12] Victor A. Abrashkin; Ruth Jenni 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 | Numdam | Zbl

[13] D. G. Benois; Sergeĭ V. Vostokov On $p$ -Extensions of Multidimensional Local Fields, Math. Nachr., Volume 160 (1993), pp. 59-68 | DOI | Zbl | MR

[14] Pierre Berthelot; William Messing Théorie de Dieudonné cristalline. III: Théorèmes d’équivalence et de pleine fidélité, The Grothendieck Festschrift (Progress in Mathematics), Volume 86, Birkhäuser, 1990, pp. 173-247 | Zbl

[15] Fedor Bogomolov; Yuri Tschinkel Commuting elements in Galois groups of function fields, Motives, polylogarithms and Hodge theory. Part I: Motives and polylogarithms (International Press Lecture Series), Volume 3, International Press, 2002, pp. 75-120 | Zbl

[16] Colas Bordavid Profinite completion and double-dual: isomorphisms and counter-examples (2080) (https://hal.science/hal-00208000v1/, version 1)

[17] Ivan B. Fesenko Abelian local $p$ -class field theory, Math. Ann., Volume 301 (1995) no. 3, pp. 561-586 | Zbl | DOI | MR

[18] Uwe Jannsen; Kay Wingberg Die Struktur der absoluten Galoisgruppe $𝔭$ -adischer Zahlkörper, Invent. Math., Volume 70 (1982), pp. 71-98 | Zbl | DOI | MR

[19] Kazuya Kato A generalization of local class field theory by using $K$ -groups I, J. Fac. Sci., Univ. Tokyo, Sect. I A, Volume 26 (1979), pp. 303-376 | Zbl | MR

[20] Kazuya Kato A generalization of local class field theory by using $K$ -groups. II, J. Fac. Sci., Univ. Tokyo, Sect. I A, Volume 27 (1980) no. 3, pp. 603-683 | Zbl

[21] Kazuya Kato Existence theorem for higher local fields, Invitation to higher local fields (Münster, 1999) (Geometry and Topology Monographs), Volume 3, Geometry and Topology Publications, 2000, pp. 165-195 | Zbl | MR

[22] Helmut Koch Galois theory of $p$ -extensions, Springer Monographs in Mathematics, Springer, 2002, xiii+190 pages | DOI | MR

[23] Aleksandra I. Madunts; Igor B. Zhukov Multidimensional complete fields: topology and other basic constructions, Proceedings of the St. Petersburg Mathematical Society. Vol. III (Translations. Series 2. American Mathematical Society), Volume 166, American Mathematical Society, 1995, pp. 1-34 | Zbl

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

[25] Alekseĭ N. Parshin Class fields and algebraic $K$ -theory (Russian), Usp. Mat. Nauk, Volume 30 (1975) no. 1, pp. 253-254 | Zbl | MR

[26] Alekseĭ N. Parshin Local class field theory. (Russian), Tr. Mat. Inst. Steklova, Volume 165 (1985), pp. 143-170 | Zbl | MR

[27] Alekseĭ N. Parshin Galois cohomology and Brauer group of local fields. (Russian), Proc. Steklov Inst. Math., Volume 183 (1990), pp. 159-169 | Zbl

[28] Anthony J. Scholl Higher fields of norms and ( $ϕ, Γ$ )-modules, Doc. Math. (2006), pp. 685-709 | Zbl | MR

[29] Jean-Pierre Serre Cohomologie Galoisienne, Lecture Notes in Mathematics, 5, Springer, 1964 | Zbl | MR

[30] Jean-Pierre Serre Structure de certains pro- $p$ -groupes (d’aprés Demushkin), Bourbaki Seminar. Vol. 8. Years 1962/63-1963/64. Expos. 241–276, Société Mathématique de France, 1995, pp. 145-155

[31] Sergei V. Vostokov Explicit construction of class field theory for a multidimensional local field, Izv. Akad. Nauk SSSR, Ser. Mat., Volume 49 (1985) no. 2, pp. 283-308 | Zbl

[32] Kay Wingberg Galois groups of Poincare type, Galois groups over $ℚ$ (Mathematical Sciences Research Institute Publications), Volume 16, Springer, 1989, pp. 439-449 | Zbl | DOI

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

[34] Igor Zhukov Milnor and topological K-groups of multidimensional complete fields, Algebra Anal., Volume 9 (1997) no. 1, pp. 98-147 | Zbl

[35] Igor Zhukov Higher dimensional local fields, Invitation to higher local fields (Münster, 1999) (Geometry and Topology Monographs), Volume 3, Geometry and Topology Publications, 2000, pp. 5-18 | Zbl | MR

[36] Igor Zhukov On ramification theory in the case of an imperfect residue field, Sb. Math., Volume 194 (2003) no. 12, pp. 1747-1774 | Zbl | DOI

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