## Counting in a conjecture of Lang. (Décompte dans une conjecture de Lang.)(French)Zbl 0972.11054

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)].

### MSC:

 11G10 Abelian varieties of dimension $$> 1$$ 14G05 Rational points 11D45 Counting solutions of Diophantine equations
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