Local class field theory

Statement

For a nonarchimedean local field $F$ , local class field theory supplies a canonical topological isomorphism

\operatorname{rec}_F:F^\times\xrightarrow{\sim}W_F^{\mathrm{ab}},

where $W_F$ is the Weil group. Equivalently, after profinite completion it identifies $\widehat{F^\times}$ with the abelianization of the absolute Galois group $G_F^{\mathrm{ab}}$ .

This page uses the geometric Frobenius normalization: a uniformizer maps to geometric Frobenius in $W_F/I_F$ , where $I_F$ is the inertia subgroup. Authors using arithmetic Frobenius take the inverse reciprocity map, so the normalization must be checked in formulas for local factors.

Finite extensions

For every finite abelian extension $L/F$ , reciprocity induces

F^\times/N_{L/F}(L^\times) \xrightarrow{\sim}\operatorname{Gal}(L/F).

Under this isomorphism open finite-index subgroups of $F^\times$ correspond to finite abelian extensions of $F$ . Norm subgroups, ramification groups, and unit filtrations are thereby translated into Galois-theoretic data.

Langlands interpretation

For $G=\operatorname{GL}_1$ , local class field theory is the local Langlands correspondence: continuous characters of $F^\times$ correspond to one-dimensional representations of $W_F$ . It is the abelian prototype for local Langlands. Its global compatibility is recorded by global Artin reciprocity.

References

John Tate, “Number theoretic background,” in Automorphic Forms, Representations and L-Functions, Proceedings of Symposia in Pure Mathematics 33, part 2, 1979.
Jean-Pierre Serre, Local Fields, Graduate Texts in Mathematics 67, Springer, 1979, Chapters XIII–XV.