Stable sets of primes in number fields

Ivanov, Alexander

doi:10.2140/ant.2016.10.1

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Stable sets of primes in number fields

Alexander Ivanov

Vol. 10 (2016), No. 1, 1–36

DOI: 10.2140/ant.2016.10.1

Abstract

We define a new class of sets —stable sets —of primes in number fields. For example, Chebotarev sets $P_{M ∕ K} (σ)$ , with $M ∕ K$ Galois and $σ \in G (M ∕ K)$ , 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 $Spec O_{K, S}$ is a $K (π, 1)$ (simultaneously for all $p$ ).

Keywords

number field, Galois cohomology, restricted ramification, Dirichlet density

Mathematical Subject Classification 2010

Primary: 11R34

Secondary: 11R45

Milestones

Received: 23 June 2014

Revised: 7 September 2015

Accepted: 23 October 2015

Published: 14 February 2016

Authors

Alexander Ivanov
Zentrum Mathematik Technischen Universität Boltzmannstraße 3 D-85747 Garching bei München Germany

Vol. 10, No. 1, 2016