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arXiv · 1309.2800

Stable sets of primes in number fields

Alexander Ivanov
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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 $σ\in \Gal(M/K)$, 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 $K(π,1)$ (for all $p$ simultaneous).

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Alexander Ivanov. 2015-10-30. Stable sets of primes in number fields. https://doi.org/10.2140/ant.2016.10.1

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