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

Autor:	Vooys, Geoff
Rok vydání:	2020
Předmět:	Mathematics - Number Theory Mathematics - Algebraic Geometry Mathematics - Representation Theory
Druh dokumentu:	Working Paper
Popis:	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 $\operatorname{GL}_{1,K}$. Comment: 25 pages
Databáze:	arXiv
Externí odkaz:	http://arxiv.org/abs/2011.00025 Zobrazit plný text záznamu View this record from Arxiv