Class field theory for curves over $p$-adic fields

TL;DR

This paper extends class field theory to algebraic curves over $p$-adic fields, introducing new class groups via algebraic K-theory that approximate their abelian étale fundamental groups.

Contribution

It develops a class field theory for curves over $p$-adic fields, generalizing Saito's unramified theory and defining new class groups using algebraic K-groups.

Findings

Introduces class groups related to abelian étale fundamental groups.

Extends unramified class field theory to ramified cases.

Connects class groups with algebraic K-theory.

Abstract

We develop class field theory of curves over $p$-adic fields which extends the unramified theory of S. Saito. The class groups which approximate abelian \'etale fundamental groups of such curves are introduced in the terms of algebraic $K$-groups by imitating G. Wiesend's class group for curves over finite fields.