Stable sets of primes in number fields

TL;DR

This paper introduces the concept of stable sets of primes in number fields, which generalize sets with density 1 and support various arithmetic and geometric theorems, even with arbitrarily small positive density.

Contribution

It defines stable sets of primes, including Chebotarev sets, and demonstrates their applicability to key arithmetic and geometric theorems in number theory.

Findings

Chebotarev sets are often stable

Stable sets can have arbitrarily small positive density

Arithmetic theorems hold for stable sets

Abstract

We define a new class of sets -- stable sets -- of primes in number fields. For example, Chebotarev sets $P_{M/K}(\sigma)$, with $M/K$ Galois and $\sigma \in \Gal(M/K)$, are very often stable. These sets have positive (but arbitrary small) Dirichlet density and generalize sets with density 1 in the sense that arithmetic theorems like certain Hasse principles, the Grunwald-Wang theorem, the Riemann's existence theorem, etc. hold for them. Geometrically this allows to give examples of infinite sets $S$ with arbitrary small positive density such that $\Spec \mathcal{O}_{K,S}$ is algebraic $K(\pi,1)$ (for all $p$ simultaneous).