  - Let K be a totally imaginary number field. Denote by G ur K (2) the Galois
group of the maximal unramified pro-2 extension of K. By comparing cup-products
in {\'e}tale cohomology of SpecO K and cohomology of uniform pro-2 groups, we
obtain situations where G ur K (2) has no non-trivial uniform analytic
quotient, proving some new special cases of the unramified Fontaine-Mazur
conjecture. For example, in the family of imaginary quadratic fields K for
which the 2-rank of the class group is equal to 5, we obtain that for at least
33.12% of such K, the group G ur K (2) has no non-trivial uniform analytic
quotient.
