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 E⁄K 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
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.
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.
Read more about this topic: Artin Reciprocity Law
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