Geodesic
On the Quotients of the Maximal Unramified 2-Extension of a Number Field
Documenta mathematica, Tome 23 (2018), pp. 1263-1290.

Voir la notice de l'article provenant de la source Electronic Library of Mathematics

Let $\mathrm{K}$ be a totally imaginary number field. Denote by $G_{\mathrm{K}}^{ur}(2)$ the Galois group of the maximal unramified pro-2 extension of $\mathrm{K}$. By using cup-products in étale cohomology of $\mathrm{Spec}\Cal O_{\mathrm{K}}$ we study situations where $G_{\mathrm{K}}^{ur}(2)$ has no quotient of cohomological dimension 2. For example, in the family of imaginary quadratic fields $\mathrm{K}$, the group $G_{\mathrm{K}}^{ur}(2)$ 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 $\mathrm{K}$, showing in particular that when ordered by absolute value of the discriminant, more than 99% of imaginary (respectively, real) quadratic fields satisfy an alternative (but equivalent) form of the unramified Fontaine-Mazur conjecture (at $p=2$).
Classification : 11R37, 11R29
Keywords: unramified extensions, cohomological dimension, uniform pro-2 groups 
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     author = {Maire, Christian},
     title = {On the {Quotients} of the {Maximal} {Unramified} {2-Extension} of a {Number} {Field}},
     journal = {Documenta mathematica},
     pages = {1263--1290},
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     language = {en},
     url = {http://geodesic.mathdoc.fr/item/DOCMA_2018__23__a25/}
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Maire, Christian. On the Quotients of the Maximal Unramified 2-Extension of a Number Field. Documenta mathematica, Tome 23 (2018), pp. 1263-1290. http://geodesic.mathdoc.fr/item/DOCMA_2018__23__a25/