论文标题
Serre-HazeWinkel本地阶级字段理论和本地Langlands通信的几何证明$ \ operatatorName {gl}（1）$
Serre-Hazewinkel Local Class Field Theory and a Geometric Proof of the Local Langlands Correspondence for $\operatorname{GL}(1)$
论文作者
论文摘要
在此说明论文中，我们提供了$ \ operatatorName {gl} _ {1} $定义在$ p $ -Adic fields $ k $上的本地Langlands通信的几何证明。我们通过重新开发促进群体的理论来做到这一点，并以此来以Serre and Hazewinkel的风格得出本地阶级领域理论。特别是，我们表明，Serre和Hazewinkel的本地阶级字段理论对于均等特征和混合特征超级局部领域都是有效的。最后，我们用它来证明$ k^{\ ast} $的平滑表示类别的等效性，并连续表示$ w_k^{\ text {ab}} $，以推断出本地的langlands对应的$ \ propatatorname {gl} _ {1，k} $。
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 $\operatorname{GL}_{1,K}$.
