The Artin reciprocity law, which was 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 global class field theory.
[1] 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.
stand for the idèle class group of
One of the statements of the Artin reciprocity law is that there is a canonical isomorphism called the global symbol map[2][3] where
is defined by assembling the maps called the local Artin symbol, the local reciprocity map or the norm residue symbol[4][5] for different places
This is the content of the local reciprocity law, a main theorem of local class field theory.
A cohomological proof of the global reciprocity law can be achieved by first establishing that constitutes a class formation in the sense of Artin and Tate.
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 Hasse 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 the 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 global class field theory.
[7] Two years after the publication of his general reciprocity law in 1927, Artin rediscovered the transfer homomorphism of I. Schur and used the reciprocity law to translate the principalization problem for ideal classes of algebraic number fields into the group theoretic task of determining the kernels of transfers of finite non-abelian groups.
[8] (See math.stackexchange.com for an explanation of some of the terms used here) The definition of the Artin map for a finite abelian extension L/K of global fields (such as a finite abelian extension of
) has a concrete description in terms of prime ideals and Frobenius elements.
are equal in Gal(L/K) since the latter group is abelian.
is canonically isomorphic to the Galois group of the extension of residue fields
There is therefore a canonically defined Frobenius element in Gal(L/K) denoted by
If Δ denotes the relative discriminant of L/K, the Artin symbol (or Artin map, or (global) reciprocity map) of L/K is defined on the group of prime-to-Δ fractional ideals,
, by linearity: The Artin reciprocity law (or global reciprocity law) states that there is a modulus c of K such that the Artin map induces an isomorphism where Kc,1 is the ray modulo c, NL/K is the norm map associated to L/K and
The smallest defining modulus is called the conductor of L/K and typically denoted
The Artin map is then defined on primes p that do not divide Δ by where
This shows that a prime p is split or inert in L according to whether
by sending σ to aσ given by the rule The conductor of
is (m)∞,[11] and the Artin map on a prime-to-m ideal (n) is simply n (mod m) in
A basic property of the Artin symbol says that for every prime-to-ℓ ideal (n) When n = p, this shows that
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.
[13] 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.[14] Let
(i.e. one-dimensional complex representation of the group G), there exists a Hecke character
of K such that 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.
[14] The formulation of the Artin reciprocity law as an equality of L-functions allows formulation of a generalisation to n-dimensional representations, though a direct correspondence is still lacking.