Let \(A\) be an abelian variety over a number field, \(X\) an abelian subvariety and \(\Gamma\) a subgroup of finite rank of \(A(\overline{\mathbb Q})\). Lang’s conjecture alluded to in the title is now a consequence of Falting’s theorem [G. Faltings, The general case of S. Lang’s conjecture. Perspect. Math. 15, 175–182 (1994; Zbl 0823.14009); see also M. Hindry, Autour d’une conjecture de Serge Lang. Invent. Math. 94, 575–603 (1988; Zbl 0638.14026)]: there exist a positive integer \(S\), elements \(x_{1},\ldots,x_{S}\) in \(X(\overline{\mathbb Q})\cap\Gamma\), and Abelian subvarieties \(B_{1},\ldots,B_{S}\) of \(A\) such that \(x_{i}+B_{i}\subset X\) for \(1\leq i\leq S\) and \[ X(\overline{\mathbb Q})\cap\Gamma= \bigcup_{i=1}^{S} \Gamma\cap(x_{i}+B_{i})(\overline{\mathbb Q}). \] In the paper under review an effective upper bound for \(S\) is achieved: \[ S\leq (c_{1} D)^{(r+1)g^{5m^{2}}}, \] where \(r\) is the rank of \(\Gamma\), \(m-1\) the dimension of \(X\), \(D\) the degree of \(X\) with respect to a symmetric ample invertible sheaf \({\mathcal L}\) on \(A\) and \(c_{1}\) depends only on \(A\) and \({\mathcal L}\).
A more explicit statement is given for the special case of rational points on curves: given a smooth projective curve \({\mathcal C}\) of genus \(g\geq 2\) on a number field \(K\), the number of rational points on \({\mathcal C}\) is bounded by \[ c_{2}^{7(r+1)g^{3}}, \] where \(r\) is the rank of \(J(K)\), \(J\) is the Jacobian of \({\mathcal C}\), and the constant \(c_{2}\) depends only on \(J\times_{K}\overline{\mathbb Q}\) and on an invertible sheaf \({\mathcal L}\) on \(J\times_{K}\overline{\mathbb Q}\) which corresponds to a symmetric translate of the theta divisor.
The constants \(c_{1}\) and \(c_{2}\) are effective: the tools for computing them explicitly are indicated by means of results due to S. David and P. Philippon [Minorations des hauteurs normalisées des sous-variétés de variétés abéliennes. II. Prépublications de l’Institut de Mathématiques de Jussieu, 277, 64 pp. (2001)]. Two of the main tools of the proof are an explicit version, due to the author [G. Rémond, Inégalité de Vojta en dimension supérieure. Ann. Scuola Norm. Sup. Pisa (4) 29, 101–151 (2000; Zbl 0972.11052)] of an inequality of P. Vojta [Siegel’s theorem in the compact case. Ann. Math. (2) 133, 509–548 (1991; Zbl 0774.14019)] and a refinement of an inequality of D. Mumford [A remark on Mordell’s conjecture. Am. J. Math. 87, 1007–1016 (1965; Zbl 0151.27301)].