## Integral points on subvarieties of semiabelian varieties. I.(English)Zbl 1011.11040

We cite from the author’s well-written introduction: Let $$X$$ be a closed subvariety of an abelian variety $$A$$, and assume that both are defined over some number field $$k$$. Then a conjecture of S. Lang [Publ. Math., Inst. Hautes Étud. Sci. 6, 319-335 (1960; Zbl 0112.13402)] states that the set of rational points is as small as one might reasonably expect:
Theorem 0.1. The set $$X(k)$$ is contained in a finite union $$\bigcup B_i(k)$$, where each $$B_i$$ is a translated abelian subvariety of $$A$$ contained in $$X$$.
In [Ann. Math. (2) 133, 549-576 (1991; Zbl 0734.14007)] G. Faltings proved this in the special case where $$X\times_k \overline{k}$$ contains no nontrivial translated abelian subvarieties of $$A\times_k \overline{k}$$; the conclusion in that case simplifies to the assertion that $$X(k)$$ is finite. He proved this in general in [Perspect. Math. 15, 175-182 (1994; Zbl 0823.14009)]. This proof is also described in detail in [P. Vojta, Lect. Notes Math. 1553, 164-208 (1993; Zbl 0846.14009)]; we follow the latter exposition closely here.
In this paper we generalize Theorem 0.1 to cover the corresponding statement for integral points on closed subvarieties of semiabelian varieties:
Theorem 0.2. Let $$k$$ be a number field, with ring of integers $$R$$. Let $$S$$ be a finite set of places of $$k$$, containing the set of Archimedean places, and let $$R_S$$ be the localization of $$R$$ away from (non-Archimedean) places in $$S$$. Let $$X$$ be a closed subvariety of a semiabelian variety $$A$$; assume both are defined over $$k$$. Let $${\mathcal X}$$ be a model for $$X$$ over $$\operatorname {Spec}R_S$$. Then the set $${\mathcal X}(R_S)$$ of $$R_S$$-valued points in $${\mathcal X}$$ equals a finite union $$\bigcup{\mathcal B}_i(R_S)$$, where each $${\mathcal B}_i$$ is a subscheme of $${\mathcal X}$$ whose generic fiber $$B_i$$ is a translated semiabelian subvariety of $$A$$.
Another paper will address similar questions for certain open subvarieties of $$A$$.
Theorem 0.2 partially proves a conjecture of S. Lang [Fundamentals of Diophantine geometry. New York, Springer (1983; Zbl 0528.14013), p. 221]: Let $$A$$ be a semiabelian variety, and let $$\Gamma$$ be a finitely generated subgroup of $$A$$. Let $$\overline{\Gamma}$$ be the division group of $$\Gamma$$; i.e., the group of all $$x\in A$$ such that $$nx\in \Gamma$$ for some positive integer $$n$$. Then Lang conjectures that the intersection of $$\overline{\Gamma}$$ with any closed subvariety $$X$$ of $$A$$ is contained in the union of finitely many translated semiabelian subvarieties of $$A$$ contained in $$X$$. Theorem 0.2 does not apply to this more general conjecture, but it is equivalent to a similar statement where one does not take the division group. Indeed, the set of integral points on $$A$$ is a finitely generated group. More recently, M. McQuillan [Invent. Math. 120, 143-159 (1995; Zbl 0848.14022)] has proved Lang’s conjecture in full generality, by using methods of M. Hindry to reduce the general statement to the special case proved here.
I doubt that this result can be generalized to a larger class of group varieties: Consider, for example, Pell’s equation on $$\mathbb{A}^2\cong \mathbb{G}_a^2$$.
By a standard result on subvarieties of abelian varieties (Theorem 4.2), Theorem 0.1 gives an affirmative answer, in the case of subvarieties of abelian varieties, to a question posed by Bombieri: If the variety $$X$$ is of general type, then the set $$X(k)$$ contained in a proper Zariski-closed subset? Similarly, in is the semiabelian case, Theorem 0.2 provides a partial answer to P. Vojta [Diophantine approximations and value distribution theory. (Lect. Notes Math. 1239) Berlin, Springer (1987; Zbl 0609.14011), 4.1.2].
Moreover, by the Kawamata structure theorem (Theorem 4.3), the nontrivial $$B_i$$ occurring in the conclusion of Theorem 0.2 must lie in a proper subvariety which is geometrical in nature. This supports a conjecture of Lang which strengthens the question posed by Bombieri: S. Lang conjectures in [Bull. Am. Math. Soc., New Ser. 14, 159-205 (1986; Zbl 0602.14019)] that if $$X$$ is of general type then the higher-dimensional components of $$X(k)$$ must lie in a subvariety which is independent of $$k$$.
Section 14 proves a corollary of Theorem 0.2 which generalizes P. Vojta [(1987) loc. cit., Theorem 2.4.1]. The proof essentially reduces to showing that the given variety embeds into a semiabelian variety.
Corollary 0.3. Let $$X$$ be a projective variety defined over a number field $$k$$, and let $$\rho$$ denote its Picard number. Let $$D$$ be an effective divisor on $$X$$, also defined over $$k$$, which has at least $$\dim X-h^1 (X,{\mathcal O}_X)+ \rho+1$$ geometrically irreducible components. Then any set of $$D$$-integral points on $$X$$ is not dense in the Zariski topology.

### MSC:

 11G10 Abelian varieties of dimension $$> 1$$ 14G05 Rational points 14K12 Subvarieties of abelian varieties
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