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Concept

Valuation

Résumé

En mathématiques, plus particulièrement en géométrie algébrique et en théorie des nombres, une valuation, ou valuation de Krull, est une mesure de la multiplicité. La notion est une généralisation de la notion de degré ou d'ordre d'annulation d'un polynôme formel en algèbre, du degré de divisibilité par un nombre premier en théorie des nombres, de l'ordre d'un pôle en analyse complexe ou du nombre de points de contact entre deux variétés algébriques en géométrie algébrique. Définition On appelle valuation une application d'un anneau commutatif unitaire (A,+,\times) non nul vers un groupe abélien totalement ordonné (G,+,

Source officielle

https://fr.wikipedia.org/wiki/Valuation

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Concepts associés (26)

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Cours associés (3)

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Séances de cours associées (4)

Séances de cours associées

Chargement

On the Reduction of Points on Abelian Varieties and Tori

Antonella Perucca

Let G be the product of an abelian variety and a torus defined over a number field K. Let R-1, ..., R-n be points in G(K). Let l be a rational prime, and let a(1), ..., a(n) be nonnegative integers. Consider the set of primes p of K satisfying the following condition: the l-adic valuation of the order of (R-i mod p) equals a(i) for every i = 1, ..., n. We show that this set is either finite or has a positive natural density. We characterize the n-tuples a(1), ..., a(n) for which the density is positive. More generally, we study the l-part of the reduction of the points.

2011

Chargement

Explicit construction of self-dual integral normal bases for the square-root of the inverse different

Erik Jarl Pickett

Let K be a finite extension of Q(p), let L/K be a finite abelian Galois extension of odd degree and let D-L be the valuation ring of L. We define A(L/K) to be the unique fractional D-L-ideal with square equal to the inverse different of L/K. For p an odd prime and L/Q(p) contained in certain cyclotomic extensions, Erez has described integral normal bases for A(L)/Q(p) that are self-dual with respect to the trace form. Assuming K/Q(p) to be unramified we generate odd abelian weakly ramified extensions of K using Lubin-Tate formal groups. We then use Dwork's exponential power series to explicitly construct self-dual integral normal bases for the square-root of the inverse different in these extensions. (C) 2009 Elsevier Inc. All rights reserved.

2009

Chargement

Let

F/E

be a finite Galois extension of fields with abelian Galois group

\Gamma

. A self-dual normal basis for

F/E

is a normal basis with the additional property that

Tr_{F/E}(g(x),h(x))=\delta_{g,h}

for

g,h\in\Gamma

. Bayer-Fluckiger and Lenstra have shown that when

char(E)\neq 2

, then

F

admits a self-dual normal basis if and only if

[F:E]

is odd. If

F/E

is an extension of finite fields and

char(E)=2

, then

F

admits a self-dual normal basis if and only if the exponent of

\Gamma

is not divisible by

4

. In this paper we construct self-dual normal basis generators for finite extensions of finite fields whenever they exist. Now let

K

be a finite extension of

\Q_p

, let

L/K

be a finite abelian Galois extension of odd degree and let

\bo_L

be the valuation ring of

L

. We define

A_{L/K}

to be the unique fractional

\bo_L

-ideal with square equal to the inverse different of

L/K

. It is known that a self-dual integral normal basis exists for

A_{L/K}

if and only if

L/K

is weakly ramified. Assuming

p\neq 2

, we construct such bases whenever they exist.

2010

Chargement

Let

F/E

be a finite Galois extension of fields with abelian Galois group

\Gamma

. A self-dual normal basis for

F/E

is a normal basis with the additional property that

Tr_{F/E}(g(x),h(x))=\delta_{g,h}

for

g,h\in\Gamma

. Bayer-Fluckiger and Lenstra have shown that when

char(E)\neq 2

, then

F

admits a self-dual normal basis if and only if

[F:E]

is odd. If

F/E

is an extension of finite fields and

char(E)=2

, then

F

admits a self-dual normal basis if and only if the exponent of

\Gamma

is not divisible by

4

. In this paper we construct self-dual normal basis generators for finite extensions of finite fields whenever they exist. Now let

K

be a finite extension of

\Q_p

, let

L/K

be a finite abelian Galois extension of odd degree and let

\bo_L

be the valuation ring of

L

. We define

A_{L/K}

to be the unique fractional

\bo_L

-ideal with square equal to the inverse different of

L/K

. It is known that a self-dual integral normal basis exists for

A_{L/K}

if and only if

L/K

is weakly ramified. Assuming

p\neq 2

, we construct such bases whenever they exist.

2010