Lang conjectured that if \(A\) is an abelian variety defined over an algebraically closed field of characteristic zero, \(X \subset A\) is a closed subvariety and \(\Gamma \subset A\) is a finite rank subgroup then there exist in \(X\) finitely many translates of abelian subvarieties whose union contains \(X \cap \Gamma\). This conjecture is now a theorem due to G. Faltings [cf. his paper in the Barsotti Symposium, Abano Terme 1991, Perspect. Math. 15, 175-182 (1994)]plus previous work of M. Hindry [Invent. Math. 94, No. 3, 575-603 (1988; Zbl 0638.14026)]. The paper under review is devoted to the proof, in the geometric case (i.e., in case \(A\) has trace zero over the algebraic numbers), of a stronger result which says that, under assumptions as above, one has the same conclusion with \(X \cap \Gamma\) replaced by \(X \cap \Gamma^*\) where \(\Gamma^*\) is a substantially bigger group, namely the group of “solutions” to all “algebraic differential equations” satisfied by the points of \(\Gamma\) (with respect to a fixed derivation of the ground field).
The method of proof, based on the Ritt-Kolchin differential algebraic viewpoint and on the big Picard theorem, is new and subsequently leads to an explicit bound for the number of translates of abelian subvarieties appearing in the geometric case of Lang’s conjecture [cf. the author, Duke Math. J. 71, No. 2, 475-499 (1993; Zbl 0812.14029); ibid. 75, No. 3, 639-644 (1994)].