Algebraische Punkte auf analytischen Untergruppen algebraischer Gruppen. (Algebraic points on analytic subgroups of algebraic groups) (Q1121936)

Let G and \(G'\) be two commutative algebraic groups, which are defined over the field \({\bar {\mathbb{Q}}}\) of algebraic numbers, and of positive dimension. Let \(\phi\) : G\({}'({\mathbb{C}})\to G({\mathbb{C}})\) be an analytic homomorphism, which is defined over \({\bar {\mathbb{Q}}}\) (this means that the induced map \(d\phi\) : \(T_{G'}({\mathbb{C}})\to T_ G({\mathbb{C}})\) on the tangent spaces is compatible with the \({\bar {\mathbb{Q}}}\) structure). If \(\phi (G')({\bar {\mathbb{Q}}})\neq 0\), then there exists an algebraic subgroup H of G, defined over \({\bar {\mathbb{Q}}}\) and of positive dimension, which is contained in \(\phi (G')\). This result extends earlier statements (Schneider, Lang, Baker, Masser,...), and has many important corollaries; some of these corollaries have been already published separately by the author: [J. Reine Angew. Math. 354, 164-174 (1984; Zbl 0543.10025); Invent. Math. 78, 381-391 (1984; Zbl 0584.10022)]. Further connected results are quoted in \textit{D. Bertrand} [Sémin. Bourbaki 1985/86, Exp. 652, Astérisque 145/146, 21-44 (1987; Zbl 0613.14001)]. Apart from Baker's method (which had already been developed on abelian varieties, mainly by Masser, Coates and Lang), the new element in the proof is a multiplicity estimate on group varieties (see the preceding review).