TY - JOUR

T1 - Calculations in the Generalized Lubin–Tate Theory

AU - Vostokov, S. V.

AU - Leonova, E. O.

N1 - Funding Information: This work was supported by the Russian Science Foundation, grant no. 16-11-10200. Publisher Copyright: © 2020, Pleiades Publishing, Ltd. Copyright: Copyright 2020 Elsevier B.V., All rights reserved.

PY - 2020/4/1

Y1 - 2020/4/1

N2 - Abstract: In this paper, various extensions of local fields are considered. For arbitrary finite extension K of the field of p-adic numbers, the maximum Abelian extension KAb/K and the corresponding Galois group can be described using the well-known Lubin–Tate theory. It is represented as a direct product of groups obtained using the maximum unramified extension of K and a fully ramified extension obtained using the roots of some endomorphisms of Lubin–Tate formal groups. We consider the so-called “generalized Lubin–Tate formal groups” and extensions obtained by adding the roots of their endomorphisms to the field under consideration. Using the fact that a correctly chosen generalized formal group coincides with the classical one over unramified finite extension Tm of degree m of field K, it was possible to obtain the Galois group of the extension (Tm)Ab/K. The main result of the work, is an explicit description of the Galois group of the extension (Kur)Ab/K, where Kur is the maximum unramified extension of K. Similar methods are also used to study ramified extensions of the field K.

AB - Abstract: In this paper, various extensions of local fields are considered. For arbitrary finite extension K of the field of p-adic numbers, the maximum Abelian extension KAb/K and the corresponding Galois group can be described using the well-known Lubin–Tate theory. It is represented as a direct product of groups obtained using the maximum unramified extension of K and a fully ramified extension obtained using the roots of some endomorphisms of Lubin–Tate formal groups. We consider the so-called “generalized Lubin–Tate formal groups” and extensions obtained by adding the roots of their endomorphisms to the field under consideration. Using the fact that a correctly chosen generalized formal group coincides with the classical one over unramified finite extension Tm of degree m of field K, it was possible to obtain the Galois group of the extension (Tm)Ab/K. The main result of the work, is an explicit description of the Galois group of the extension (Kur)Ab/K, where Kur is the maximum unramified extension of K. Similar methods are also used to study ramified extensions of the field K.

KW - formal group laws

KW - maximum unramified extension

UR - http://www.scopus.com/inward/record.url?scp=85085876014&partnerID=8YFLogxK

U2 - 10.1134/S1063454120020168

DO - 10.1134/S1063454120020168

M3 - Article

AN - SCOPUS:85085876014

VL - 53

SP - 131

EP - 135

JO - Vestnik St. Petersburg University: Mathematics

JF - Vestnik St. Petersburg University: Mathematics

SN - 1063-4541

IS - 2

ER -