Artin reciprocity

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In mathematics, Artin reciprocity refers to various results connecting Artin L-functions associated to abelian extensions of a number field with Hecke L-functions associated to Hecke's grossencharacters of that number field. The reciprocity law itself was introduced by Emil Artin in the middle of the 1920s.^[1]

The underlying idea is to map a prime ideal (in some ideal class group) to its Frobenius element (in a Galois group).^[2]

In particular cases, it yields all other abelian reciprocity laws such as quadratic reciprocity, cubic reciprocity, and biquadratic reciprocity as special cases. From the 1960s onwards, it has motivated investigations into possible non-abelian reciprocity laws, and the Langlands program.^[2]

[edit] One version of the theorem

A Hecke character (sometimes grossencharacter) of a number field K is defined to be a quasicharacter of the idèle class group of K.

Let K be a number field, E/K an abelian Galois extension of K with Galois group G. Then for any group character σ: G → C^×, there exists a grossencharacter χ of K such that

$L_{E/K}^{\mathrm{Artin}}(\sigma, s) = L_{K}^{\mathrm{Hecke}}(\chi, s)$

where the left hand side is the Artin L-function associated to the extension with character σ and the right hand side is the Hecke L-function associated to the grossencharacter χ of K (see for example section 7.D of ^[3])

[edit] References

^ Hasse, Helmut, History of Class Field Theory in Algebraic Number Theory, Proceedings, Academic Press, 1967, pp. 266-279.
^ ^a ^b Milne, James, Class Field Theory
^ Gelbart, Stephen, Automorphic Forms on Adele Groups, Annals of Mathematics Studies, No. 83, Princeton University Press, 1975. ISBN 0-691-08156-5