Integral points on subvarieties of semiabelian varieties. I.

*(English)*Zbl 1011.11040We 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.

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.

Reviewer: O.Ninnemann (Berlin)