Concept

Modulus (algebraic number theory)

In mathematics, in the field of algebraic number theory, a modulus (plural moduli) (or cycle, or extended ideal) is a formal product of places of a global field (i.e. an algebraic number field or a global function field). It is used to encode ramification data for abelian extensions of a global field. Let K be a global field with ring of integers R. A modulus is a formal product where p runs over all places of K, finite or infinite, the exponents ν(p) are zero except for finitely many p. If K is a number field, ν(p) = 0 or 1 for real places and ν(p) = 0 for complex places. If K is a function field, ν(p) = 0 for all infinite places. In the function field case, a modulus is the same thing as an effective divisor, and in the number field case, a modulus can be considered as special form of Arakelov divisor. The notion of congruence can be extended to the setting of moduli. If a and b are elements of K×, the definition of a ≡∗b (mod pν) depends on what type of prime p is: if it is finite, then where ordp is the normalized valuation associated to p; if it is a real place (of a number field) and ν = 1, then under the real embedding associated to p. if it is any other infinite place, there is no condition. Then, given a modulus m, a ≡∗b (mod m) if a ≡∗b (mod pν(p)) for all p such that ν(p) > 0. Ray class group The ray modulo m is A modulus m can be split into two parts, mf and m∞, the product over the finite and infinite places, respectively. Let Im to be one of the following: if K is a number field, the subgroup of the group of fractional ideals generated by ideals coprime to mf; if K is a function field of an algebraic curve over k, the group of divisors, rational over k, with support away from m. In both case, there is a group homomorphism i : Km,1 → Im obtained by sending a to the principal ideal (resp. divisor) (a). The ray class group modulo m is the quotient Cm = Im / i(Km,1). A coset of i(Km,1) is called a ray class modulo m. Erich Hecke's original definition of Hecke characters may be interpreted in terms of characters of the ray class group with respect to some modulus m.

À propos de ce résultat
Cette page est générée automatiquement et peut contenir des informations qui ne sont pas correctes, complètes, à jour ou pertinentes par rapport à votre recherche. Il en va de même pour toutes les autres pages de ce site. Veillez à vérifier les informations auprès des sources officielles de l'EPFL.

Graph Chatbot

Chattez avec Graph Search

Posez n’importe quelle question sur les cours, conférences, exercices, recherches, actualités, etc. de l’EPFL ou essayez les exemples de questions ci-dessous.

AVERTISSEMENT : Le chatbot Graph n'est pas programmé pour fournir des réponses explicites ou catégoriques à vos questions. Il transforme plutôt vos questions en demandes API qui sont distribuées aux différents services informatiques officiellement administrés par l'EPFL. Son but est uniquement de collecter et de recommander des références pertinentes à des contenus que vous pouvez explorer pour vous aider à répondre à vos questions.