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

2017/10/25 by Christian Maire, Maire, Christian
Mathematics · #Advanced Mathematical Identities #Algebraic Geometry and Number Theory #Analytic Number Theory Research #FOS: Mathematics #Number Theory (math.NT) #advanced mathematical theories

paper · pdf · doi:10.48550/arxiv.1710.09217

openalex publication_date 2017/10/25 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

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 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.

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