Vol. 10, No. 1, 2016

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Stable sets of primes in number fields

Alexander Ivanov

Vol. 10 (2016), No. 1, 1–36

We define a new class of sets —stable sets —of primes in number fields. For example, Chebotarev sets PMK(σ), with MK Galois and σ G(MK), 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 SpecOK,S is a K(π,1) (simultaneously for all p).

number field, Galois cohomology, restricted ramification, Dirichlet density
Mathematical Subject Classification 2010
Primary: 11R34
Secondary: 11R45
Received: 23 June 2014
Revised: 7 September 2015
Accepted: 23 October 2015
Published: 14 February 2016
Alexander Ivanov
Zentrum Mathematik
Technischen Universität
Boltzmannstraße 3
D-85747 Garching bei München