Ideles, Hecke Characters, and Artin Reciprocity

For a number field $K$, the idele group is $\mathbb I_K:=\prod_v' K_v^\times$ (restricted product with respect to $\mathcal O_v^\times$ at finite $v$), and the idele class group is $C_K:=K^\times\backslash \mathbb I_K$.

A Hecke character (Grössencharakter) is a continuous homomorphism $\chi:C_K\to \mathbb C^\times$; it has an Euler product $L(s,\chi)=\prod_v L_v(s,\chi_v)$.

Artin reciprocity identifies (canonically, up to the standard class field theory normalizations) the abelianized Galois group $\mathrm{Gal}(K^{\mathrm{ab}}/K)$ with a profinite quotient of $C_K$, sending uniformizers to Frobenius elements.

In the letter: this realizes abelian Artin $L$-series as $L$-series of Hecke characters (example (iii)).