×

The optimality of the bounded height conjecture. (English) Zbl 1203.11048

Let \(A\) be an abelian variety, and \(V\) a subvariety of \(A\), both defined over \(\overline{{\mathbb Q}}\). A translate of an abelian subvariety of \(A\) is called a coset of \(A\), and a coset of the type \(a+B\) with \(B\) an abelian subvariety of \(A\) and \(a\in A(\overline{{\mathbb Q}})_{\text{tors}}\) a torsion coset. Given a positive integer \(r\), denote by \(S^r(V)\) the intersection of \(V(\overline{{\mathbb Q}})\) with all torsion cosets of \(A\) of codimension at least \(r\). For a subset \(V'\) of \(V\), let \(S^r(V'):=V'\cap S^r (V)\). Bombieri, Masser and Zannier started the study of sets of the type \(S^r(V')\). They formulate the so-called bound height conjecture for subvarieties of tori, whose analogue for abelian varieties reads as follows: Let \(A\) be an abelian variety and \(V\) a \(d\)-dimensional subvariety of \(A\), both defined over \(\overline{{\mathbb Q}}\). Suppose that \(V\) is not contained in a proper coset of \(A\). Let \(V^{\text{oa}}\) be the subset of \(V\) obtained by removing all its anomalous subvarieties, i.e. all positive-dimensional subvarieties \(X\subseteq V\) for which there exists a coset \(H\) of \(A\) such that \(X\subset H\) and \(\text{codim}_A X <\text{codim}_A V +\text{codim}_A H\). Then (the set of points of) \(S^d(V^{\text{oa}})\) has bounded height. This conjecture has been proved in a more general form by P. Habegger [Invent. Math. 176, 405–447 (2009; Zbl 1176.14008)]. By a result of G. Rémond [Comment. Math. Helv. 84, No. 4, 835–863 (2009; Zbl 1227.11078)], the set \(V^{\text{oa}}\) is Zariski open in \(V\), but it may be empty.
In the paper under review, the author shows that in some sense the bounded height conjecture is optimal if \(A=E^g\), where \(E\) is an elliptic curve over \(\overline{{\mathbb Q}}\). Let \(V\) be a \(d\)-dimensional subvariety of \(E^g\) defined over \(\overline{{\mathbb Q}}\). The author proves that \(V^{\text{oa}}\not=\emptyset\) if and only if \(V\) has ‘property (S)’, i.e., for all morphisms \(\phi :\, A\to A\) with \(\dim \phi (A)\geq d\) one has \(\dim \phi (V)=d\). Thus, Habegger’s theorem implies that if \(V\) has property (S) then there is a non-empty Zariski open subset \(V^e\) of \(V\) (i.e., \(V^{\text{oa}}\)) such that \(S_d(V^e)\) has bounded height. In the paper under review, the author proves the following result:
Theorem. If \(V\) does not have property (S) then for every non-empty, Zariski open subset \(U\) of \(V\), the set \(S^d(U)\) does not have bounded height.
The proof is constructive.

MSC:

11G50 Heights
14H52 Elliptic curves
14K12 Subvarieties of abelian varieties

References:

[1] E. Bombieri, D. Masser and U. Zannier, Intersecting a curve with algebraic subgroups of multiplicative groups. Int. Math. Res. Not. 20 (1999), 1119-1140. · Zbl 0938.11031
[2] E. Bombieri, D. Masser and U. Zannier, Anomalous subvarieties - Structure Theorem and applications. Int. Math. Res. Not. 19 (2007), 33 pages. · Zbl 1145.11049
[3] P. Habegger, Bounded height for subvarieties in abelian varieties. Invent. math. 176 (2009), 405-447. · Zbl 1176.14008
[4] G. Rémond, Intersection de sous-groupes et de sous-variétés II. J. Inst. Math. Jussieu 6 (2007), 317-348. · Zbl 1170.11014
[5] G. Rémond, Intersection de sous-groups et de sous-variétés III. To appear in Com. Mat. Helv. · Zbl 1227.11078
[6] G. Rémond and E. Viada, Problème de Mordell-Lang modulo certaines sous-variétés abéliennes. Int. Math. Res. Not. 35 (2003), 1915-1931. · Zbl 1072.11038
[7] E. Viada, The intersection of a curve with algebraic subgroups in a product of elliptic curves. Ann. Scuola Norm. Sup. Pisa cl. Sci. 5 vol. II (2003), 47-75. · Zbl 1170.11314
[8] E. Viada, The intersection of a curve with a union of translated codimension-two subgroups in a power of an elliptic curve, Algebra and Number Theory 3 vol. 2 (2008), 248-298. · Zbl 1168.11024
[9] E. Viada, Non-dense subsets of varieties in a power of an elliptic curve. Int. Math. Res. Not. 7 (2009), 1214-1246. · Zbl 1168.14030
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.