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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
##### Keywords:
height; elliptic curves; subvarieties
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##### References:
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