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

TL;DR

This paper investigates the structure of the Galois group of the maximal unramified pro-2 extension of totally imaginary number fields, providing new cases where this group lacks non-trivial uniform analytic quotients, supporting aspects of the Fontaine-Mazur conjecture.

Contribution

It introduces a novel approach comparing étale cohomology and pro-2 group cohomology to identify when G_{ur}^K(2) has no non-trivial uniform analytic quotients, advancing understanding of unramified extensions.

Findings

For imaginary quadratic fields with 2-rank of class group equal to 5, at least 33.12% have G_{ur}^K(2) with no non-trivial uniform analytic quotient.

The method applies cup-product comparisons to establish new cases of the unramified Fontaine-Mazur conjecture.

Provides partial evidence supporting the conjecture in specific families of totally imaginary number fields.

Abstract

- 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.