ON THE QUOTIENTS OF THE MAXIMAL UNRAMIFIED 2-EXTENSION OF A NUMBER FIELD

Let K be a totally imaginary number field. Denote by G(K)(ur )(2) the Galois group of the maximal unramified pro-2 extension of K. By using cup-products in etale cohomology of SpecO(K) we study situations where G(K)(ur)(2) has no quotient of cohomological dimension 2. For example, in the family of imaginary quadratic fields K, the group G(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 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).