On a generalization of the Neukirch-Uchida theorem

In this paper we generalize a part of Neukirch-Uchida theorem for number fields from the birational case to the case of curves $\text{Spec}{\text{caO}}_{K,S}$ with $S$ a stable set of primes of a number field $K$. In particular, such sets can have arbitrarily small (positive) Dirichlet density. The proof consists of two parts: first one establishes a local correspondence at the boundary $S$, which works as in the original proof of Neukirch. But then, in contrast to Neukirchs proof, a direct conclusion via Chebotarev density theorem is not possible, since stable sets are in general too small, and one has to use further arguments.