Artin reciprocity

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In mathematics, Artin reciprocity (or the global reciprocity law) realizes the abelianization of a Galois group of an extension $L / K$ of global fields in terms of the arithmetic of $K$ . As this is (besides the existence theorem) necessary to establish a classification of all abelian extensions of $K$ and to understand the behavior of the prime places, Artin reciprocity can be understood as the main theorem of (global) class field theory.

As the name suggests, Artin reciprocity yields all other abelian reciprocity laws such as quadratic reciprocity, cubic reciprocity, and biquadratic reciprocity as special cases (albeit not trivially). The theorem was introduced by Emil Artin in the middle of the 1920s.^[1]

[edit] Statement in contemporary language

Let $L / K$ be a finite Galois extension of global fields, then one has the identity

$\hat{H}^{0}( \text{Gal}(L/K), C_L) \simeq\hat{H}^{-2}( \text{Gal}(L/K), \mathbb{Z})$

with $\hat{H}$ - the Tate cohomology groups and $C L$ - the idèle class group. If one works out the cohomology groups this reads

$C_K/{\text{Nm}_{L/K}(C_L)} \simeq \text{Gal}(L/K)^{\text{ab}}$

which seems to be the standard notation of Artin reciprocity.^[2]

A proof of the reciprocity law is provided by showing that $(\text{Gal}(K^{sep}/K),\varinjlim C_L)$ forms a class formation. (Note the analog statement of the local reciprocity law: Replace $C L$ in the first equation by $L^{\times}$ and $C K / Nm L / K (C L)$ in the second by $K^{\times}/{\text{Nm}_{L/K}L^{\times}}$ )

[edit] Alternative version of the theorem (leading to Langlands)

An alternative version (leading to the Langlands program^[3]) connects Artin L-functions associated to abelian extensions of a number field with Hecke L-functions associated to Hecke's grössencharacters of that number field:

Recall that a Hecke character (or Größencharakter) of a number field $K$ is defined to be a quasicharacter of the idèle class group of $K$ , and is the same notion as an automorphic form on GL(1,A_K) (where A_K denotes the adeles of K)^[4].

Let $E / K$ an abelian Galois extension of $K$ with Galois group $G$ . Then for any group character σ: $G$ → C^× (i.e. one-dimensional complex representation of G), 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 ^[4])

[edit] References

^ Hasse, Helmut, History of Class Field Theory in Algebraic Number Theory, Proceedings, Academic Press, 1967, pp. 266-279.
^ Neukirch, Jürgen, Algebraische Zahlentheorie, Springer, 1992, p. 408.
^ Milne, James, Class Field Theory
^ ^a ^b Gelbart, Stephen, Automorphic Forms on Adele Groups, Annals of Mathematics Studies, No. 83, Princeton University Press, 1975. ISBN 0-691-08156-5