scholarly journals Groups of automorphisms of local fields of period p^M and nilpotent class

2017 ◽  
Vol 67 (2) ◽  
pp. 605-635 ◽  
Author(s):  
Victor Abrashkin
Keyword(s):  
Local Fields ◽  
Groups Of Automorphisms ◽  
Nilpotent Class

scholarly journals Groups of automorphisms of local fields of period p and nilpotent class < p, I

2017 ◽  
Vol 28 (06) ◽  
pp. 1750043 ◽  
Author(s):  
Victor Abrashkin
Keyword(s):  
Finite Field ◽  
Explicit Construction ◽  
Field Extension ◽  
Local Fields ◽  
Lie Theory ◽  
Root Of Unity ◽  
Composition Law ◽  
Groups Of Automorphisms ◽  
Nilpotent Class ◽  
Primitive Formula

Suppose [Formula: see text] is a finite field extension of [Formula: see text] containing a primitive [Formula: see text]th root of unity. Let [Formula: see text] be a maximal [Formula: see text]-extension of [Formula: see text] with the Galois group of period [Formula: see text] and nilpotent class [Formula: see text]. In this paper, we develop formalism which allows us to study the structure of [Formula: see text] via methods of Lie theory. In particular, we introduce an explicit construction of a Lie [Formula: see text]-algebra [Formula: see text] and an identification [Formula: see text], where [Formula: see text] is a [Formula: see text]-group obtained from the elements of [Formula: see text] via the Campbell–Hausdorff composition law. In the next paper, we apply this formalism to describe the ramification filtration [Formula: see text] and an explicit form of the Demushkin relation for [Formula: see text].

scholarly journals Groups of automorphisms of local fields of period p and nilpotent class < p, II

2017 ◽  
Vol 28 (10) ◽  
pp. 1750066
Author(s):  
Victor Abrashkin
Keyword(s):  
Finite Field ◽  
Galois Group ◽  
Lie Algebras ◽  
Explicit Form ◽  
Field Extension ◽  
Local Fields ◽  
Root Of Unity ◽  
Groups Of Automorphisms ◽  
Nilpotent Class ◽  
Primitive Formula

Suppose [Formula: see text] is a finite field extension of [Formula: see text] containing a primitive [Formula: see text]th root of unity. Let [Formula: see text] be the maximal quotient of period [Formula: see text] and nilpotent class [Formula: see text] of the Galois group of a maximal [Formula: see text]-extension of [Formula: see text]. We describe the ramification filtration [Formula: see text] and relate it to an explicit form of the Demushkin relation for [Formula: see text]. The results are given in terms of Lie algebras attached to the appropriate [Formula: see text]-groups by the classical equivalence of the categories of [Formula: see text]-groups and Lie algebras of nilpotent class [Formula: see text].

Local Fields

1986 ◽  
Author(s):  
J. W. S. Cassels
Keyword(s):  
Local Fields

scholarly journals Smooth representations of unit groups of split basic algebras over non-Archimedean local fields

2021 ◽  
Vol 0 (0) ◽  
Author(s):  
Carlos A. M. André ◽  
João Dias
Keyword(s):  
Local Field ◽  
Unit Group ◽  
Basic Algebra ◽  
Local Fields ◽  
Dimensional Representation ◽  
Smooth Representation ◽  
One Dimensional ◽  
Finite Dimensional

Abstract We consider smooth representations of the unit group G = A × G=\mathcal{A}^{\times} of a finite-dimensional split basic algebra 𝒜 over a non-Archimedean local field. In particular, we prove a version of Gutkin’s conjecture, namely, we prove that every irreducible smooth representation of 𝐺 is compactly induced by a one-dimensional representation of the unit group of some subalgebra of 𝒜. We also discuss admissibility and unitarisability of smooth representations of 𝐺.

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