Unramified 2-extensions of totally imaginary number fields and 2-adic analytic groups

Christian Maire (1)

1 LMB - Laboratoire de Mathématiques de Besançon (UMR 6623)

— 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 étale 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.

Domaines: Théorie des nombres [math.NT]

Christian Maire. Unramified 2-extensions of totally imaginary number fields and 2-adic analytic groups. 2017. ⟨hal-01622008⟩