On a generalization of the Neukirch-Uchida theorem

TL;DR

This paper extends the Neukirch-Uchida theorem to include curves over number fields with stable prime sets of arbitrarily small density, using new methods to handle the limitations of small sets.

Contribution

It generalizes the Neukirch-Uchida theorem from number fields to curves with stable prime sets of small density, introducing novel techniques beyond Chebotarev density arguments.

Findings

Established a local correspondence at the boundary set S.

Extended the theorem to stable sets with arbitrarily small positive density.

Developed new arguments to overcome limitations of small prime sets.

Abstract

In this paper we generalize a part of Neukirch-Uchida theorem for number fields from the birational case to the case of curves $\Spec \caO_{K,S}$ 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.