Stable sets of primes in number fields

Abstract

We define a new class of sets —stable sets —of primes in number fields. For example, Chebotarev sets $P_{M∕K}\left(\sigma \right)$, with $M∕K$ Galois and $\sigma \in G\left(M∕K\right)$, are very often stable. These sets have positive (but arbitrarily small) Dirichlet density and they generalize sets with density one in the sense that arithmetic theorems such as certain Hasse principles, the Grunwald–Wang theorem, and Riemann’s existence theorem hold for them. Geometrically, this allows us to give examples of infinite sets $S$ with arbitrarily small positive density such that $Spec{\mathcal{O}}_{K,S}$ is a $K\left(\pi ,1\right)$ (simultaneously for all $p$).