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We consider a dynamical system consisting of a pair of commuting power series under composition, one noninvertible and another nontorsion invertible, of height one with coefficients in the $p$ -adic integers. Assuming that each point of the dynamical system generates a Galois extension over the base field, we show that these extensions are in fact abelian, and, using results from the theory of the field of norms, we also show that the dynamical system must include a torsion series. From an earlier...

Galois structure of ideals in wildly ramified abelian $p$ -extensions of a $p$ -adic field, and some applications

Nigel P. Byott (1997)

Journal de théorie des nombres de Bordeaux

Let $K$ be a finite extension of $ℚ_{p}$ with ramification index $e$ , and let $L / K$ be a finite abelian $p$ -extension with Galois group $Γ$ and ramification index $p^{n}$ . We give a criterion in terms of the ramification numbers $t_{i}$ for a fractional ideal $𝔓^{h}$ of the valuation ring $S$ of $L$ not to be free over its associated order $𝔄 (K Γ; 𝔓^{h})$ . In particular, if $t_{n} - [t_{n} / p] < p^{n - 1} e$ then the inverse different can be free over its associated order only when $t_{i} \equiv - 1$ (mod $p^{n}$ ) for all $i$ . We give three consequences of this. Firstly, if $𝔄 (K Γ; S)$ is a Hopf order and $S$ is $𝔄 (K Γ; S)$ -Galois...

Genre et genre résiduel des corps de fonctions valués

Michel Matignon (1985/1986)

Groupe de travail d'analyse ultramétrique

Henselian Discrete Valued Fields Admitting One-Dimensional Local Class Field Theory

Chipchakov, I. (2004)

Serdica Mathematical Journal

2000 Mathematics Subject Classification: 11S31 12E15 12F10 12J20.This paper gives a characterization of Henselian discrete valued fields whose finite abelian extensions are uniquely determined by their norm groups and related essentially in the same way as in the classical local class field theory. It determines the structure of the Brauer groups and character groups of Henselian discrete valued strictly primary quasilocal (or PQL-) fields, and thereby, describes the forms of the local reciprocity...

Jacobi sums and explicit reciprocity laws

David E. Rohrlich (1986)

Compositio Mathematica

K2-analogs of Hasse's norm theorems.

Anthony Bak, Ulf Rehmann (1984)

Commentarii mathematici Helvetici

Les constantes locales de l'équation fonctionelle de la fonction L d'Artin d'une représentation orthogonale.

Pierre Deligne (1976)

Inventiones mathematicae

Lien algébrique entre les deux facteurs de la formule analytique du nombre de classes dans les corps abéliens

Bernard Oriat (1986)

Acta Arithmetica

Lifting the field of norms

Laurent Berger (2014)

Journal de l’École polytechnique — Mathématiques

Let $K$ be a finite extension of $Q_{p}$ . The field of norms of a $p$ -adic Lie extension $K_{\infty} / K$ is a local field of characteristic $p$ which comes equipped with an action of $Gal (K_{\infty} / K)$ . When can we lift this action to characteristic $0$ , along with a compatible Frobenius map? In this note, we formulate precisely this question, explain its relevance to the theory of $(ϕ, Γ)$ -modules, and give a condition for the existence of certain types of lifts.

Local Class Field Theory via Lubin-Tate Theory

Teruyoshi Yoshida (2008)

Annales de la faculté des sciences de Toulouse Mathématiques

We give a self-contained exposition of local class field theory, via Lubin-Tate theory and the Hasse-Arf theorem, refining the arguments of Iwasawa [9].

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Dualité sur un corps local à corps résiduel algébriquement clos

Endomorphism rings of almost full formal groups.

Exercises dyadiques.

Explicit reciprocity laws for Lubin-Tate modules.

Explicit reciprocity laws on relative Lubin-Tate groups

Extensions infinies identiquement ramifiées.

Formal group law homomorphisms over $𝒪_{ℂ_{p}}$ .

Galois extensions of height-one commuting dynamical systems