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 ).
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