# zbMATH — the first resource for mathematics

The defect of strong approximation for commutative algebraic groups. (Le défaut d’approximation forte pour les groupes algébriques commutatifs.) (French. English summary) Zbl 1194.14067
Let $$k$$ be a number field. The first main result of the paper under review is the construction of an exact sequence describing the topological closure of the group of $$k$$-rational points of a 1-motive $$H^0(k,M)$$ in its adelic points. The second main result shows that the obstruction for strong approximation to hold for a semiabelian variety is essentially controlled by its algebraic Brauer group. Throughout the paper it is assumed that the Tate-Shafarevich groups of all abelian varieties involved are finite.
For a finite place $$v$$ of $$k$$, let $$\mathcal O_v$$ and $$k_v$$ be the local ring and the local field at $$v$$. For an infinite place $$\mathcal O_v$$ and $$k_v$$ denote both the completion of $$k$$ at $$v$$, and cohomology over the fields of real or complex numbers is understood to be Galois cohomology modified à la Tate. Let $$M$$ be a 1-motive over $$k$$ with dual $$M^\vee$$. The first main theorem states that there is an exact sequence of topological groups $0 \to \overline{H^0(k,M)} \to {\prod_{v}}'H^0(k_v, M) \to H^1(k,M^\vee)^D \to \text{III}^1(k,M) \to 0$ where $$(-)^D$$ stands for Pontryagin dual, and where the product is a restricted product with respect to maps $$H^0(\mathcal O_v, M) \to H^1(k_v, M)$$ for all places $$v$$ of $$k$$. In the case $$M$$ is a semiabelian variety $$G$$ and ignoring the real places of $$k$$, this restricted product is the group of adelic points of $$G$$. Finally, the first group in this sequence is the topological closure of the image of $$H^0(k, M)$$ in the restricted product.
The second main result, an application of the first, concerns the integral Manin obstruction. Let $$X$$ be a flat scheme over the ring of integers of $$k$$, whose generic fibre is a principal homogeneous space under a semiabelian variety $$G$$ over $$k$$. Let $$S$$ be a finite set of places of $$k$$ containig all infinite places and let $$A_S$$ be the ring of $$S$$-adeles and let $$P = (P_v)_{v\in S}$$ be an $$A_S$$-point of $$X$$. Suppose that $$P$$ is orthogonal to the Brauer group $$\mathrm{Br}(X)$$ under the Brauer-Manin pairing. Then there exists an $$\mathcal O_S$$-integral point of $$X$$ which is $$v$$-adically close to $$P$$ if $$v$$ is non-archimedean, and which lies in the same connected component of $$X(k_v)$$ as $$P_v$$ is $$v$$ is archimedean.
In particular, if $$P = (P_v)_{v}$$ is an adelic point of $$X$$ orthogonal to $$\mathrm{Br}(X)$$, then $$X(\mathcal O_k)$$ is nonempty.

##### MSC:
 14L15 Group schemes 12G05 Galois cohomology 11J61 Approximation in non-Archimedean valuations 11R56 Adèle rings and groups
##### Keywords:
strong approximation; Brauer group; 1-motive
Full Text: