Geodesic
On the Quotients of the Maximal Unramified 2-Extension of a Number Field
Documenta mathematica, Tome 23 (2018), pp. 1263-1290
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Let K be a totally imaginary number field. Denote by GKur​(2) the Galois group of the maximal unramified pro-2 extension of K. By using cup-products in étale cohomology of SpecOK​ we study situations where GKur​(2) has no quotient of cohomological dimension 2. For example, in the family of imaginary quadratic fields K, the group GKur​(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).
DOI : 10.4171/dm/647
Classification : 11R29, 11R37
Mots-clés : cohomological dimension, unramified extensions, uniform pro-2 groups 
@article{10_4171_dm_647,
     author = {Christian Maire},
     title = {On the {Quotients} of the {Maximal} {Unramified} {2-Extension} of a {Number} {Field}},
     journal = {Documenta mathematica},
     pages = {1263--1290},
     year = {2018},
     volume = {23},
     doi = {10.4171/dm/647},
     url = {http://geodesic.mathdoc.fr/articles/10.4171/dm/647/}
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Christian Maire. On the Quotients of the Maximal Unramified 2-Extension of a Number Field. Documenta mathematica, Tome 23 (2018), pp. 1263-1290. doi: 10.4171/dm/647

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