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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
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References:
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