Sur la dimension cohomologique des pro- p -extensions des corps de nombres

Christian Maire[1]

  • [1] GRIMM Université Toulouse le Mirail 5, allées A. Machado 31058 Toulouse cédex

Journal de Théorie des Nombres de Bordeaux (2005)

  • Volume: 17, Issue: 2, page 575-606
  • ISSN: 1246-7405

Abstract

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In this paper, we study the cohomological dimension of groups G S : = Gal ( 𝕂 S / 𝕂 ) , where 𝕂 S is the maximal pro- p -extension of a number field 𝕂 , unramified outside a finite set S of places of 𝕂 . This dimension is well-understood only when S contains all places above p ; in the case where only some of the places above p are contained in S , one can still obtain some results if 𝕂 S / 𝕂 contains at least one ℤ p -extension 𝕂 ∞ / 𝕂 . Indeed, in that case, the study of the ℤ p [ [ Gal ( 𝕂 ∞ / 𝕂 ) ] ] -module Gal ( 𝕂 S / 𝕂 ∞ ) a b allows one to give sufficient conditions for the pro- p -group Gal ( 𝕂 S / 𝕂 ∞ ) to be free. Under the latter condition, the dimension of G S is at most 2. Here, we develop an explicit strategy for realizing these conditions so as to produce numerical examples for which we effectively compute this cohomological dimension.

How to cite

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Maire, Christian. "Sur la dimension cohomologique des pro-$p$-extensions des corps de nombres." Journal de Théorie des Nombres de Bordeaux 17.2 (2005): 575-606. <http://eudml.org/doc/249422>.

@article{Maire2005,
abstract = {In this paper, we study the cohomological dimension of groups $G_S:=\operatorname\{Gal\}(\mathbb\{K\}_S/\mathbb\{K\})$, where $\mathbb\{K\}_S$ is the maximal pro-$p$-extension of a number field $\mathbb\{K\}$, unramified outside a finite set $S$ of places of $\mathbb\{K\}$. This dimension is well-understood only when $S$ contains all places above $p$; in the case where only some of the places above $p$ are contained in $S$, one can still obtain some results if $\mathbb\{K\}_S/\mathbb\{K\}$ contains at least one $\mathbb\{Z\}_p$-extension $\mathbb\{K\}_\infty /\mathbb\{K\}$. Indeed, in that case, the study of the $\mathbb\{Z\}_p[[\operatorname\{Gal\}(\mathbb\{K\}_\infty /\mathbb\{K\})]]$-module $\operatorname\{Gal\}(\mathbb\{K\}_S/\mathbb\{K\}_\infty )^\{ab\}$ allows one to give sufficient conditions for the pro-$p$-group $\operatorname\{Gal\}(\mathbb\{K\}_S/\mathbb\{K\}_\infty )$ to be free. Under the latter condition, the dimension of $G_S$ is at most 2. Here, we develop an explicit strategy for realizing these conditions so as to produce numerical examples for which we effectively compute this cohomological dimension.},
affiliation = {GRIMM Université Toulouse le Mirail 5, allées A. Machado 31058 Toulouse cédex},
author = {Maire, Christian},
journal = {Journal de Théorie des Nombres de Bordeaux},
keywords = {conditions of Leopoldt type; Spiegelungssatz; -extensions; Gras logarithm},
language = {eng},
number = {2},
pages = {575-606},
publisher = {Université Bordeaux 1},
title = {Sur la dimension cohomologique des pro-$p$-extensions des corps de nombres},
url = {http://eudml.org/doc/249422},
volume = {17},
year = {2005},
}

TY - JOUR
AU - Maire, Christian
TI - Sur la dimension cohomologique des pro-$p$-extensions des corps de nombres
JO - Journal de Théorie des Nombres de Bordeaux
PY - 2005
PB - Université Bordeaux 1
VL - 17
IS - 2
SP - 575
EP - 606
AB - In this paper, we study the cohomological dimension of groups $G_S:=\operatorname{Gal}(\mathbb{K}_S/\mathbb{K})$, where $\mathbb{K}_S$ is the maximal pro-$p$-extension of a number field $\mathbb{K}$, unramified outside a finite set $S$ of places of $\mathbb{K}$. This dimension is well-understood only when $S$ contains all places above $p$; in the case where only some of the places above $p$ are contained in $S$, one can still obtain some results if $\mathbb{K}_S/\mathbb{K}$ contains at least one $\mathbb{Z}_p$-extension $\mathbb{K}_\infty /\mathbb{K}$. Indeed, in that case, the study of the $\mathbb{Z}_p[[\operatorname{Gal}(\mathbb{K}_\infty /\mathbb{K})]]$-module $\operatorname{Gal}(\mathbb{K}_S/\mathbb{K}_\infty )^{ab}$ allows one to give sufficient conditions for the pro-$p$-group $\operatorname{Gal}(\mathbb{K}_S/\mathbb{K}_\infty )$ to be free. Under the latter condition, the dimension of $G_S$ is at most 2. Here, we develop an explicit strategy for realizing these conditions so as to produce numerical examples for which we effectively compute this cohomological dimension.
LA - eng
KW - conditions of Leopoldt type; Spiegelungssatz; -extensions; Gras logarithm
UR - http://eudml.org/doc/249422
ER -

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