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A Brückner-Vostokov formula for the Hilbert symbol of a formal group was established by Abrashkin under the assumption that roots of unity belong to the base field. The main motivation of this work is to remove this hypothesis. It is obtained by combining methods of ( $ϕ, Γ$ )-modules and a cohomological interpretation of Abrashkin’s technique. To do this, we build ( $ϕ, Γ$ )-modules adapted to the false Tate curve extension and generalize some related tools like the Herr complex with explicit formulas for the...

Arithmetic of formal groups and applications I: Universal norm subgroups.

P. Schneider (1987)

Inventiones mathematicae

Associated orders of certain extensions arising from Lubin-Tate formal groups

Nigel P. Byott (1997)

Journal de théorie des nombres de Bordeaux

Let $k$ be a finite extension of $ℚ_{p}$ , let $k_{1}$ , respectively $k_{3}$ , be the division fields of level $1$ , respectively $3$ , arising from a Lubin-Tate formal group over $k$ , and let $Γ =$ Gal( $k_{3} / k_{1}$ ). It is known that the valuation ring $k_{3}$ cannot be free over its associated order $𝔄$ in $K Γ$ unless $k = ℚ_{p}$ . We determine explicitly under the hypothesis that the absolute ramification index of $k$ is sufficiently large.

Canonical and filling subgroups of formal groups.

Schmitz, David J. (2006)

The New York Journal of Mathematics [electronic only]

Completeness of the absolute Galois group of the rational number field.

Masatoshi Ideka (1977)

Journal für die reine und angewandte Mathematik

Constructing class fields over local fields

Sebastian Pauli (2006)

Journal de Théorie des Nombres de Bordeaux

Let $K$ be a $𝔭$ -adic field. We give an explicit characterization of the abelian extensions of $K$ of degree $p$ by relating the coefficients of the generating polynomials of extensions $L / K$ of degree $p$ to the exponents of generators of the norm group $N_{L / K} (L^{*})$ . This is applied in an algorithm for the construction of class fields of degree $p^{m}$ , which yields an algorithm for the computation of class fields in general.

Counting periodic points of $p$ -adic power series

Hua-Chieh Li (1996)

Compositio Mathematica

Courbes définies sur les corps de séries formelles et loi de réciprocité (Acta Arithmetica 42 (1982), p. 101-106) (Errata)

Jean Douai, C. Touibi (1986)

Acta Arithmetica

Courbes définies sur les corps de séries formelles et loi de réciprocité

Jean Douai, C. Touibi (1982)

Acta Arithmetica

Degré d’une extension de $𝐐_{p}^{nr}$ sur laquelle $J_{0} (N)$ est semi-stable

Mohamed Krir (1996)

Annales de l'institut Fourier

Soit $N$ un entier $\geq 1$ . Pour un nombre premier $p$ on note $Q_{p}^{nr}$ l’extension maximale non ramifiée de $Q_{p}$ . Supposons que $p^{v}$ divise exactement $N$ . Alors, en utilisant les travaux de Carayol et la théorie du corps de classes local, on détermine une extension $E_{v}$ de $Q_{p}^{nr}$ sur laquelle la jacobienne $J_{0}$ de la courbe modulaire de $X_{0} (N)$ admet une réduction semi-stable, puis on donne une estimation de son degré.

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A generalization of class formation by using hypercohomology.

A generalization of the Hasse-Arf theorem.

Abelian local p-class field theory.

Abstract class field theory (formatting of modules).

Almost $C_{p}$ -representation. (Presque $C_{p}$ -représentations.)

An application of the p-adic class number formula.

An explicit formula for the Hilbert symbol of a formal group