Arithmetic of 0-cycles on varieties defined over number fields

Yongqi Liang

Annales scientifiques de l'École Normale Supérieure (2013)

  • Volume: 46, Issue: 1, page 35-56
  • ISSN: 0012-9593

Abstract

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Let X be a rationally connected algebraic variety, defined over a number field k . We find a relation between the arithmetic of rational points on  X and the arithmetic of zero-cycles. More precisely, we consider the following statements: (1) the Brauer-Manin obstruction is the only obstruction to weak approximation for  K -rational points on  X K for all finite extensions K / k ; (2) the Brauer-Manin obstruction is the only obstruction to weak approximation in some sense that we define for zero-cycles of degree 1 on  X K for all finite extensions K / k ; (3) the sequence lim ← n C H 0 ( X K ) / n → ∏ w ∈ Ω K lim ← n C H 0 ' ( X K w ) / n → Hom ( Br ( X K ) , ℚ / ℤ ) is exact for all finite extensions K / k . We prove that (1) implies (2), and that (2) and (3) are equivalent. We also prove a similar implication for the Hasse principle. As an application, we prove the exactness of the sequence above for smooth compactifications of certain homogeneous spaces of linear algebraic groups.

How to cite

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Liang, Yongqi. "Arithmetic of 0-cycles on varieties defined over number fields." Annales scientifiques de l'École Normale Supérieure 46.1 (2013): 35-56. <http://eudml.org/doc/272186>.

@article{Liang2013,
abstract = {Let $X$ be a rationally connected algebraic variety, defined over a number field $k.$ We find a relation between the arithmetic of rational points on $X$ and the arithmetic of zero-cycles. More precisely, we consider the following statements: (1) the Brauer-Manin obstruction is the only obstruction to weak approximation for $K$-rational points on $X_K$ for all finite extensions $K/k;$ (2) the Brauer-Manin obstruction is the only obstruction to weak approximation in some sense that we define for zero-cycles of degree $1$ on $X_K$ for all finite extensions $K/k;$ (3) the sequence\[\varprojlim \_n CH\_0(X\_K)/n\rightarrow \prod \_\{w\in \Omega \_K\}\varprojlim \_nCH\_0^\{\prime \}(X\_\{K\_w\})/n\rightarrow \mathrm \{Hom\}(\mathrm \{Br\}(X\_K),\mathbb \{Q\}/\mathbb \{Z\})\]is exact for all finite extensions $K/k.$ We prove that (1) implies (2), and that (2) and (3) are equivalent. We also prove a similar implication for the Hasse principle. As an application, we prove the exactness of the sequence above for smooth compactifications of certain homogeneous spaces of linear algebraic groups.},
author = {Liang, Yongqi},
journal = {Annales scientifiques de l'École Normale Supérieure},
keywords = {zero-cycles; Hasse principle; weak approximation; Brauer-Manin obstruction; rationally connected varieties; homogeneous spaces},
language = {eng},
number = {1},
pages = {35-56},
publisher = {Société mathématique de France},
title = {Arithmetic of 0-cycles on varieties defined over number fields},
url = {http://eudml.org/doc/272186},
volume = {46},
year = {2013},
}

TY - JOUR
AU - Liang, Yongqi
TI - Arithmetic of 0-cycles on varieties defined over number fields
JO - Annales scientifiques de l'École Normale Supérieure
PY - 2013
PB - Société mathématique de France
VL - 46
IS - 1
SP - 35
EP - 56
AB - Let $X$ be a rationally connected algebraic variety, defined over a number field $k.$ We find a relation between the arithmetic of rational points on $X$ and the arithmetic of zero-cycles. More precisely, we consider the following statements: (1) the Brauer-Manin obstruction is the only obstruction to weak approximation for $K$-rational points on $X_K$ for all finite extensions $K/k;$ (2) the Brauer-Manin obstruction is the only obstruction to weak approximation in some sense that we define for zero-cycles of degree $1$ on $X_K$ for all finite extensions $K/k;$ (3) the sequence\[\varprojlim _n CH_0(X_K)/n\rightarrow \prod _{w\in \Omega _K}\varprojlim _nCH_0^{\prime }(X_{K_w})/n\rightarrow \mathrm {Hom}(\mathrm {Br}(X_K),\mathbb {Q}/\mathbb {Z})\]is exact for all finite extensions $K/k.$ We prove that (1) implies (2), and that (2) and (3) are equivalent. We also prove a similar implication for the Hasse principle. As an application, we prove the exactness of the sequence above for smooth compactifications of certain homogeneous spaces of linear algebraic groups.
LA - eng
KW - zero-cycles; Hasse principle; weak approximation; Brauer-Manin obstruction; rationally connected varieties; homogeneous spaces
UR - http://eudml.org/doc/272186
ER -

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