Class field theory for open curves over local fields

TL;DR

This paper extends class field theory to open curves over local fields by analyzing the reciprocity map, its kernel, cokernel, and dual, establishing a correspondence between abelian coverings and subgroups of the idèle class group.

Contribution

It introduces a framework for class field theory on open curves over local fields, detailing the reciprocity map and its properties, and generalizing classical results.

Findings

Kernel and cokernel of the reciprocity map are explicitly determined.

A one-to-one correspondence between abelian coverings and subgroups of the idèle class group is established.

The Pontrjagin dual of the reciprocity map is characterized.

Abstract

Theory for open curves over a local field. After introducing the reciprocity map, we determine the kernel and the cokernel of this map. In addition to this, the Pontrjagin dual of the reciprocity map is also investigated. This gives the one to one correspondence between the set of finite abelian \'etale coverings and the set of finite index open subgroups of the id\`ele class group as in the classical class field theory under some assumptions.