# Remark on function field analogy

## TL;DR

This paper explores the analogy between number fields and function fields, establishing an isomorphism between their Hilbert class fields and using K-theory and birational geometry to construct explicit generators.

## Contribution

It introduces a novel isomorphism linking Hilbert class fields of class number one to function fields over algebraic curves, utilizing K-theory and birational geometry techniques.

## Key findings

- Established an isomorphism between Hilbert class fields and function fields over curves.
- Constructed explicit generators of Hilbert class fields from Drinfeld module torsion submodules.
- Applied K-theory of Serre C*-algebras and birational geometry in the proof.

## Abstract

We study the analogy between number fields and function fields in one variable over finite fields. The main result is an isomorphism between the Hilbert class fields of class number one and a family of the function fields $\mathbf{F}_q(C)$ over a desingularized algebraic curve $C$. Our proof is based on the K-theory of the Serre $C^*$-algebras and birational geometry of the curve $C$. We apply the isomorphism to construct explicit generators of the Hilbert class fields coming from the torsion submodules of the Drinfeld module.