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European Mathematical Society Publishing House
2016-09-19 17:04:52
Commentarii Mathematici Helvetici
Comment. Math. Helv.
CMH
0010-2571
1420-8946
General
10.4171/CMH
http://www.ems-ph.org/doi/10.4171/CMH
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European Mathematical Society Publishing House
Zuerich, Switzerland
© Swiss Mathematical Society
88
2013
1
Manin obstruction to strong approximation for homogeneous spaces
Mikhail
Borovoi
Tel Aviv University, TEL AVIV, ISRAEL
Cyril
Demarche
Université Pierre et Marie Curie - Paris 6, PARIS CEDEX 05, FRANCE
Manin obstruction, strong approximation, Brauer group, homogeneous spaces, connected algebraic groups
For a homogeneous space $X$ (not necessarily principal) of a connected algebraic group $G$ (not necessarily linear) over a number field $k$, we prove a theorem of strong approximation for the adelic points of $X$ in the Brauer–Manin set. Namely, for an adelic point $x$ of $X$ orthogonal to a certain subgroup (which may contain transcendental elements) of the Brauer group $\operatorname{Br}(X)$ of $X$ with respect to the Manin pairing, we prove a strong approximation property for $x$ away from a finite set $S$ of places of $k$. Our result extends a result of Harari for torsors of semiabelian varieties and a result of Colliot-Thélène and Xu for homogeneous spaces of simply connected semisimple groups, and our proof uses those results.
Algebraic geometry
Number theory
Group theory and generalizations
General
1
54
10.4171/CMH/277
http://www.ems-ph.org/doi/10.4171/CMH/277