Obstruction sets and extensions of groups

Francesca Balestrieri

Acta Arithmetica (2016)

  • Volume: 173, Issue: 2, page 151-181
  • ISSN: 0065-1036

Abstract

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Let X be a nice variety over a number field k. We characterise in pure “descent-type” terms some inequivalent obstruction sets refining the inclusion X ( k ) é t , B r ⊂ X ( k ) B r ₁ . In the first part, we apply ideas from the proof of X ( k ) é t , B r = X ( k ) k by Skorobogatov and Demarche to new cases, by proving a comparison theorem for obstruction sets. In the second part, we show that if ⊂ ⊂ k are such that ⊂ E x t ( , k ) , then X ( k ) = X ( k ) . This allows us to conclude, among other things, that X ( k ) é t , B r = X ( k ) k and X ( k ) S o l , B r ₁ = X ( k ) S o l k .

How to cite

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Francesca Balestrieri. "Obstruction sets and extensions of groups." Acta Arithmetica 173.2 (2016): 151-181. <http://eudml.org/doc/279275>.

@article{FrancescaBalestrieri2016,
abstract = {Let X be a nice variety over a number field k. We characterise in pure “descent-type” terms some inequivalent obstruction sets refining the inclusion $X(_k)^\{ét,Br\} ⊂ X(_k)^\{Br₁\}$. In the first part, we apply ideas from the proof of $X(_k)^\{ét,Br\} = X(_k)^\{_k\}$ by Skorobogatov and Demarche to new cases, by proving a comparison theorem for obstruction sets. In the second part, we show that if $ ⊂ ⊂ _k$ are such that $ ⊂ \{Ext\}(, _k)$, then $X(_k)^\{\} = X(_k)^\{\}$. This allows us to conclude, among other things, that $X(_k)^\{ét,Br\} = X(_k)^\{_k\}$ and $X(_k)^\{Sol,Br₁\} = X(_k)^\{Sol_k\}$.},
author = {Francesca Balestrieri},
journal = {Acta Arithmetica},
keywords = {rational points; Brauer-Manin obstruction; Étale-Brauer obstruction; torsors; linear algebraic groups},
language = {eng},
number = {2},
pages = {151-181},
title = {Obstruction sets and extensions of groups},
url = {http://eudml.org/doc/279275},
volume = {173},
year = {2016},
}

TY - JOUR
AU - Francesca Balestrieri
TI - Obstruction sets and extensions of groups
JO - Acta Arithmetica
PY - 2016
VL - 173
IS - 2
SP - 151
EP - 181
AB - Let X be a nice variety over a number field k. We characterise in pure “descent-type” terms some inequivalent obstruction sets refining the inclusion $X(_k)^{ét,Br} ⊂ X(_k)^{Br₁}$. In the first part, we apply ideas from the proof of $X(_k)^{ét,Br} = X(_k)^{_k}$ by Skorobogatov and Demarche to new cases, by proving a comparison theorem for obstruction sets. In the second part, we show that if $ ⊂ ⊂ _k$ are such that $ ⊂ {Ext}(, _k)$, then $X(_k)^{} = X(_k)^{}$. This allows us to conclude, among other things, that $X(_k)^{ét,Br} = X(_k)^{_k}$ and $X(_k)^{Sol,Br₁} = X(_k)^{Sol_k}$.
LA - eng
KW - rational points; Brauer-Manin obstruction; Étale-Brauer obstruction; torsors; linear algebraic groups
UR - http://eudml.org/doc/279275
ER -

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