Commentarii Mathematici Helvetici

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Volume 88, Issue 1, 2013, pp. 1–54
DOI: 10.4171/CMH/277

Manin obstruction to strong approximation for homogeneous spaces

Mikhail Borovoi[1] and Cyril Demarche[2]

(1) Raymond and Beverly Sackler School of Mathematical Sciences, Tel Aviv University, 69978, Tel Aviv, Israel
(2) Institut de Mathématiques de Jussieu (IMJ), Université Pierre et Marie Curie - Paris 6, 4, place Jussieu, 75252, Paris CEDEX 05, France

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.

Keywords: Manin obstruction, strong approximation, Brauer group, homogeneous spaces, connected algebraic groups

Borovoi Mikhail, Demarche Cyril: Manin obstruction to strong approximation for homogeneous spaces. Comment. Math. Helv. 88 (2013), 1-54. doi: 10.4171/CMH/277