Artin Reciprocity Law - Alternative Statement

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: GC× (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.

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