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Complements on a theorem of Faltings. (Quelques compléments à une démonstration de Faltings.)(French)Zbl 0823.14010

A former conjecture byS. Lang [cf. Publ. Math., Inst. Hautes Étud. Sci. 6, 27-43 (1960; Zbl 0112.134)] about rational points on arithmetic abelian varieties has recently been confirmed byG. Faltings [in: Memorial Meeting in honour of I. Barsotti, Abano Terme 1991, 175-182 (1994; see the preceding review)]. It states that, for any closed subvariety \(X\) of an abelian variety \(A\) over a number field \(K\), the set \(X(K)\) of K-rational points of \(X\) is contained in a finite union of translates of K-rational abelian subvarieties of \(A\). This theorem of G. Faltings can be generalized to semi-abelian schemes, according to a (yet unpublished) work ofP. Vojta [cf. “Integral points on subvarieties of semi-abelian varieties” (Preprint 1994)]. In the present note, the author uses the methods of Faltings and Vojta for investigating the more general case of families of subvarieties of an abelian variety over a number field. More precisely, let \(A\) be an abelian variety over a number field \(K\) and let \(X\) be a closed subvariety of \(A \times_ K S\), where \(S\) is a quasi-projective scheme over \(K\).
Under these assumptions, and with respect to chosen ample line bundles \(L\) (on a compactification \(\overline {S}\) of \(S\)) and \({\mathfrak L}\) (on \(A\)), a relative version of Faltings’ theorem for the K-rational fibers \(X_ S (K)\), \(s\in S(K)\), is proved. The author’s result is related to a conjecture of P. Vojta, called the “effective Mordell conjecture”, and also to a conjecture due to A. Parshin, L. Szpiro, and L. Moret-Bailly [cf.L. Moret-Bailly in: Les pinceaux de courbes elliptiques, Sémin., Paris 1988, Astérisque 183, 37-58 (1990;Zbl 0727.14015)] concerning the boundedness of height functions on the fiber of families of smooth curves of genus \(g\geq 2\).
With a view to his main theorem, the author proposes a generalization of these conjectures.

MSC:

14G05 Rational points
14K05 Algebraic theory of abelian varieties
14G40 Arithmetic varieties and schemes; Arakelov theory; heights

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