Arithmetic and anabelian theorems for stable sets of primes in number fields

Abstract

In this thesis we study arithmetic and anabelian properties of the Galois group GK,S of the maximal extension of a number field K unramified outside a set of primes S. The work can be divided in two parts: the one part deals with finite and the other with infinite sets S. The main idea of the part dealing with infinite sets S is to introduce a new class of sets of primes in number fields – stable sets. These sets have positive, but arbitrary small Dirichlet density. We give different examples of stable sets. This can be done in a rather explicit way. For example, Chebotarev sets P_{M/K}(\sigma) with M/K finite Galois and σ \in G_{M/K} are often stable. Stable sets generalize in some sense sets of density one. In particular, the most arithmetic results, holding for sets with density one, also hold for them. We generalize certain Hasse principles, Grunwald-Wang theorem, Riemann’s existence theorem and a statement about the (strict) cohomological dimension from density one sets (cf. [NSW] Chapters IX and X) to stable sets. Then we show that curves Spec \mathcal{O}_{K,S} with S stable are often K(\pi, 1) (for p). In particular, this gives many (explicit) examples of sets S of positive, but arbitrary small density, such that Spec \mathcal{O}_{K,S} is an algebraic K(\pi, 1) (for all p simultaneously). Finally, we study anabelian properties of curves Spec \mathcal{O}_{K,S} with S stable. It turns out that it is possible to generalize a part of the birational anabelian theorem of Neukirch-Uchida to stable sets. More precise, we show that if for i = 1, 2, a number field K_i together with a stable set of primes S_i is given, such that K_1 is normal over the rationals, the groups G_{K_1 ,S_1} and G_{K_2 ,S_2} are isomorphic as topological groups and some easy technical conditions are satisfied, then K_1 and K_2 are isomorphic. In the part concerning finite sets S we consider some anabelian properties of the group G_{K,S}. In contrast to the situation with affne hyperbolic curves over finite fields, for which the Isom-form of Grothendieck’s Anabelian Conjecture was proven by A. Tamagawa [Ta] some years ago, very little is known about anabelian properties of G_{K,S} in the number field case. It seems even to be impossible to describe purely group-theoretically (by known methods) the location of the decomposition groups at primes in S inside the group G_{K,S}. We show that this is possible if one has given a bit more information, than simply the group G_{K,S}. We prove that it is equivalent to have the following pieces of information (additionally to G_{K,S}: the location of decomposition groups at primes in S inside G_{K,S}, the p-part of the cyclotomic character for some prime p lying under S, or some further pieces of information. In particular, if \sigma is an isomorphism of G_{K_1 ,S_1} and G_{K_2 ,S_2} preserving the p-cyclotomic character,then one obtains a local correspondence at the boundary, i.e., for primes in S_1 , S_2.