Definitions

# Artin reciprocity law

The Artin reciprocity law, established by Emil Artin in a series of papers (1924; 1927; 1930), is a general theorem in number theory that forms a central part of the global class field theory. The term "reciprocity law" refers to a long line of more concrete number theoretic statements which it generalized, from the quadratic reciprocity law and the reciprocity laws of Eisenstein and Kummer to Hilbert's product formula for the norm symbol. Artin's result provided a partial solution to Hilbert's ninth problem.

## Significance

Artin's reciprocity law implies a description of the abelianization of the absolute Galois group of a global field K which is based on the local–global principle and the use of the Frobenius elements. Together with the Takagi existence theorem, it is used to describe the abelian extensions of K in terms of arithmetic of K and to understand the behavior of the nonarchimedean places in them. Therefore, the Artin reciprocity law can be interpreted as one of the main theorems of the global class field theory.

## Statement and cohomological interpretation

Let LvKv be a Galois extension of local fields with Galois group G. The local reciprocity law describes a canonical isomorphism

$theta_v: K_v^\left\{times\right\}/N_\left\{L_v/K_v\right\}\left(L_v^\left\{times\right\}\right) to G^\left\{text\left\{ab\right\}\right\},$

called the local Artin symbol.

Let LK be a Galois extension of global fields and CL stand for the idèle class group of L. The maps θv for different places v of K can be assembled into a single global symbol map. One of the statements of the Artin reciprocity law is that this results in the canonical isomorphism

$theta: C_K/\left\{N_\left\{L/K\right\}\left(C_L\right)\right\} to text\left\{Gal\right\}\left(L/K\right)^\left\{text\left\{ab\right\}\right\}.$

A cohomological proof of the global reciprocity law can be achieved by first establishing that

$\left(text\left\{Gal\right\}\left(K^\left\{sep\right\}/K\right),varinjlim C_L\right)$

constitutes a class formation in the sense of Artin and Tate. Then one proves that

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

where $hat\left\{H\right\}^\left\{i\right\}$ denote the Tate cohomology groups. Working out the cohomology groups establishes that θ is an isomorphism.

## Alternative statement

An alternative version of the reciprocity law, leading to the Langlands program, connects Artin L-functions associated to abelian extensions of a number field with Hecke L-functions associated to characters of the idèle class group.

A Hecke character (or Größencharakter) of a number field K is defined to be a quasicharacter of the idèle class group of K. Robert Langlands interpreted Hecke characters as automorphic forms on the reductive algebraic group GL(1) over the ring of adeles of K.

Let EK be an abelian Galois extension with Galois group G. Then for any character σ: G  → C× (i.e. one-dimensional complex representation of the group G), there exists a Hecke character χ of K such that

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

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 with χ, Section 7.D of .

## References

• Emil Artin, Über eine neue Art von L-Reihen, Abh. Math. Semin. Univ. Hamburg, 3 (1924), 89–108; Collected Papers, Addison Wesley, 1965, 105–124
• Emil Artin, Beweis des allgemeinen Reziprozitätsgesetzes, Abh. Math. Semin. Univ. Hamburg, 5 (1927), 353–363; Collected Papers, 131–141
• Emil Artin, Idealklassen in Oberkörpern und allgemeines Reziprozitätsgesetzes, Abh. Math. Semin. Univ. Hamburg, 7 (1930), 46–51; Collected Papers, 159–164
• Günther Frey, On the history of the Artin reciprocity law in abelian extensions of algebraic number fields: how Artin was led to his reciprocity law, in The legacy of Niels Henrik Abel, 267–294, Springer-Verlag, Berlin, 2004

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