On the Quotients of the Maximal Unramified 2-Extension of a Number Field

Abstract

Let $\mathrm{K}$ be a totally imaginary number field. Denote by $G_{\mathrm{K}}^{ur}(2)$ the Galois group of the maximal unramified pro-2 extension of $\mathrm{K}$. By using cup-products in étale cohomology of $\mathrm{Spec}{\mathcal{O}}_{\mathrm{K}}$ we study situations where $G_{\mathrm{K}}^{ur}(2)$ has no quotient of cohomological dimension 2. For example, in the family of imaginary quadratic fields $\mathrm{K}$, the group $G_{\mathrm{K}}^{ur}(2)$ almost never has a quotient of cohomological dimension 2 and of maximal 2-rank. We also give a relation between this question and that of the 4-rank of the class group of $\mathrm{K}$, showing in particular that when ordered by absolute value of the discriminant, more than 99% of imaginary quadratic fields satisfy an alternative (but equivalent) form of the unramified Fontaine–Mazur conjecture (at $p=2$).