# An idelic quotient related to Weil reciprocity and the Picard group

## TL;DR

This paper explores the relationship between Weil reciprocity, the Picard group, and idele class groups in algebraic curves over perfect fields, revealing how topological subgroups encode arithmetic properties and inform field extensions.

## Contribution

It introduces a new topological subgroup of the idele class group and demonstrates its role in capturing arithmetic and geometric properties of the base field and curve.

## Key findings

- Topological subgroup encodes base field and Picard group properties
- Applications to extensions of the function field
- Enhanced understanding of arithmetic via idele class groups

## Abstract

This paper studies the function field of an algebraic curve over an arbitrary perfect field by using the Weil reciprocity law and topologies on the adele ring. A topological subgroup of the idele class group is introduced and it is shown how it encodes arithmetic properties of the base field and of the Picard group of the curve. These results are applied to study extensions of the function field.