Local class field theory via Lubin-Tate theory

[摘要] This is an exposition of the explicit approach to Local Class Field Theorydue to J. Tate and J. Lubin. We mainly follow the treatment given in [15]and [25]. We start with an informal introduction to p-adic numbers. Wethen review the standard theory of valuedelds and completion of those elds. The complete discrete valuedelds withnite residueeld knownas localelds are our main focus. Number theoretical aspects for local elds are considered. The standard facts about Hensel's lemma, Galois andrami cation theory for localelds are treated. This being done, we continueour discussion by introducing the key notion of relative Lubin-Tate formalgroups and modules. The torsion part of a relative Lubin-Tate module isthen used to generate a tower of totally rami ed abelian extensions of a local eld. Composing this tower with the maximal unrami ed extension givesthe maximal abelian extension: this is the local Kronecker-Weber theorem.What remains then is to state and prove the theorems for explicit local class eld theory and end our discussion.

[发布机构] Stellenbosch University