Ideles, Hecke characters, and Artin reciprocity

Let $F$ be a number field. Its idele group is the restricted product

\mathbb A_F^\times = \prod_v'F_v^\times

with respect to $\mathcal O_v^\times$ at the finite places, and its idele class group is

C_F=F^\times\backslash\mathbb A_F^\times.

A Hecke character or Grössencharakter is a continuous quasicharacter $\chi:C_F\to\mathbb C^\times$ .

Hecke L-function

Writing $\chi=\bigotimes_v'\chi_v$ as a restricted tensor product, the Hecke $L$ -function has local factors. At an unramified finite place,

L_v(s,\chi_v) = \left(1-\chi_v(\varpi_v)q_v^{-s}\right)^{-1},

with the inverse altered if the local reciprocity map uses the opposite Frobenius convention.

Artin reciprocity

Global class field theory supplies a continuous reciprocity map

\operatorname{Art}_F:C_F \longrightarrow \operatorname{Gal}(F^{\mathrm{ab}}/F).

For a number field it is surjective with kernel the identity component $C_F^\circ$ , so it induces an isomorphism from the profinite completion of $C_F$ to the abelianized absolute Galois group. Some authors send a local uniformizer to arithmetic Frobenius and others to geometric Frobenius.

Finite-order Hecke characters therefore correspond to finite-order one-dimensional Galois characters. Algebraic Hecke characters give compatible one-dimensional $\ell$ -adic characters after choosing coefficient embeddings.

Modern placement

This is the $G=\operatorname{GL}_1$ case of global Langlands reciprocity. Packets are singletons, and the nonabelian complications of endoscopy and multiplicity do not appear.

Relation to the letter

The letter uses reciprocity to reinterpret abelian Artin $L$ -series as Hecke $L$ -series. Its proposed nonabelian correspondences generalize this local-to-global pattern rather than the group structure of $C_F$ itself.

References

John Tate, “Fourier analysis in number fields and Hecke's zeta functions,” in Algebraic Number Theory, 1967.
Jürgen Neukirch, Class Field Theory, Springer, 1986.