Stable sets of primes in number fields (Q5963074)

The author introduces and studies a new class of sets of primes in algebraic number fields. If \(S\) is a set of primes in a field \(K\), then it is called a \(\lambda\)-\textit{stable set} for an extension \({\mathcal L}/K\) (where \(\lambda>1\)) if for some \(S_0\subset S\), a finite extension \(L_0/K\) contained in \({\mathcal L}\) and all finite extensions \(L/L_0\) with \(L\subset{\mathcal L}\) the Dirichlet density \(\delta_L(S_0)\) of the pullback of \(S_0\) to \(L\) lies in \([a,\lambda a]\) for some positive \(a\). The field \(L_0\) is called a \(\lambda\)-\textit{stabilizing field for \(S\) for} \({\mathcal L}/K\). A set \(S\) is called \textit{sharply \(p\)-stable} for \({\mathcal L}/K\) (where \(p\) is a rational prime) if either \(\zeta_p\in{\mathcal L}\) and \(S\) is \(p\)-stable for \({\mathcal L}/K\) or \(\zeta_p\not\in{\mathcal L}\) and for some \(\lambda>1\) \(S\) is \(\lambda\)-stable for \({\mathcal L}(\zeta_p)/K\).NEWLINENEWLINEMany examples are given illustrating the introduced classes of sets of primes, and in particular the question of stability of Čebotarev sets is studied.NEWLINENEWLINELet \({\mathcal L}/K\) be a Galois extension with group \(G\), let \(A\) be a finite \(G\)-module, let \(p\) be the smallest prime divisor of \(A\) and let \(S\) be a set of primes in \(K\) which is \(p\)-stable for \({\mathcal L}/K\) with the stabilizing field \(L\). The main result (Theorem 4.1) states that the Šafarevič-Tate group (the group of global cohomology classes vanishing locally at primes from \(S\)) is a subgroup of \(H_*^1(L(A)/L,A)\), where \(L(A)\) is the trivializing extension of \(A\), and for a \(G\)-module \(B\) \(H_*^1(G,B)\) denotes the subgroup of \(H^1(G,B)\) formed by classes whose restrictions to cyclic subgroups of \(G\) vanish.NEWLINENEWLINEThis result is then applied to obtain generalizations to the case of stable sets of the Grunwald-Wang theorem in cohomological form and related results. The proofs are mostly modelled after the arguments presented in the case of sets of density one in Chap. IX of the book [\textit{J. Neukirch} et al., Cohomology of number fields. 2nd ed. Berlin: Springer (2008; Zbl 1136.11001)] by Neukirch, Schmidt and Wingberg.