Serre-Hazewinkel Local Class Field Theory and a Geometric Proof of the Local Langlands Correspondence for $\operatorname{GL}(1)$

TL;DR

This paper offers a geometric proof of the local Langlands correspondence for GL(1) over p-adic fields, utilizing proalgebraic groups to connect local class field theory with representation theory.

Contribution

It develops a geometric approach to local class field theory and provides a new proof of the local Langlands correspondence for GL(1) over p-adic fields.

Findings

Validates Serre-Hazewinkel local class field theory for all ultrametric local fields.

Establishes an equivalence between smooth characters of K* and Galois representations.

Provides a geometric proof of the local Langlands correspondence for GL(1).

Abstract

In this expository paper we provide a geometric proof of the local Langlands Correspondence for the groups $\operatorname{GL}_{1}$ defined over $p$-adic fields $K$. We do this by redeveloping the theory of proalgebraic groups and use this to derive local class field theory in the style of Serre and Hazewinkel. In particular, we show that the local class field theory of Serre and Hazewinkel is valid for both equal characteristic and mixed characteristic ultrametric local fields. Finally, we use this to prove an equivalence of the categories of smooth representations of $K^{\ast}$ with continuous representations of $W_K^{\text{Ab}}$ in order to deduce the Local Langlands Correspondence for $\operatorname{GL}_{1,K}$.