Remark on function field analogy

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.