The defect of strong approximation for commutative algebraic groups
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.
scientific article; zbMATH DE number 5529339