An arithmetic site for the rings of integers of algebraic number fields
Let \(K\) be an algebraic number field and \(S\) a finite set of primes of \(K\) containing the infinite primes and the primes dividing a given prime number \(p\). Let \(G_S (p)\) be the Galois group of the maximal \(p\)-extension of \(K\) unramified outside \(S\). The author presents a geometric study of the group \(G_S (p)\) and generalizes results of \textit{K. Wingberg} [Invent. Math. 77, 557-584 (1984; Zbl 0563.12008)]. The basis of this study is the construction of a Grothendieck topology (called by the author positive topology) on \(\text{Spec}(O_K)\). There is a natural isomorphism of \(G_S(p)\) onto the pro-\(p\)-fundamental group of the open subscheme \(\text{Spec}(O_{K,S})\) of \(\text{Spec}(O_K)\) furnished with the positive topology. If \(K\) is an abelian number field containing the \(p\)th roots of unity, the author proves the following result, which in fact is valid in much greater generality: Let \(p\neq 2\) and \(G\) the positive pro-\(p\)-fundamental group of \(\text{Spec}(O_K)\). If the \(p\)-genus \(g_p(K)\) is greater than zero, then \(G\) is a Poincaré pro-\(p\)-group of dimension 3 with dualizing module \(\mu_{p^\infty}\). If \(G_p(K)=0\) then \(G\) is a free pro-\(p\)-group of rank \(r\) with \(1\leq r\leq 1+r_2(K)/2\), where \(r_2(K)\) is the number of complex places of \(K\).
Galois group of the maximal \(p\)-extension
scientific article; zbMATH DE number 922463
scientific article