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Nondense subsets of varieties in a power of an elliptic curve. (English) Zbl 1168.14030
Let \(A\) be a \(g\)-dimensional abelian variety over \(\bar{\mathbb{Q}}\), and \(V\) be its proper \(d\)-dimensional irreducible algebraic subvariety. \(V\) is said to be transverse (resp. weak-transverse) if \(V\) is not contained in any translate of a proper algebraic subgroup of \(A\) (resp. in any proper algebraic subgroup of \(A\)). Consider the sets \(S_r(V,F)=V\cap\bigcup_{\mathrm{cod} B \geq r}(B+F)\), where \(B\) runs over all abelian subvarieties of \(A\) of codimension at least \(r\), \(1 \leq r \leq g\), and \(F\) is a subset of \(A\). The author considers the case where \(A = E^g\) is a power of an elliptic curve \(E\). Set \(\mathcal{O}_\varepsilon = \{\xi \in E^g \; : \; \|\xi\| \leq \varepsilon\}\), where \(\varepsilon \geq 0\), \(\| \cdot \|\) is a fixed semi-norm on \(E^g\) induced by the Néron-Tate height on \(E\), \(\Gamma_\varepsilon = \Gamma + \mathcal{O}_\varepsilon\), where \(\Gamma\) is a subgroup of finite rank in \(E^g\). Let \(V \subset E^g\) be an irreducible algebraic subvariety of \(E^g\), and \(V_K = V \cap \mathcal{O}_K\). With this notation the main result of the paper asserts: for every \(K \geq 0\) there exists an effective \(\varepsilon \geq 0\) such that: (i) if \(V\) is weak-transverse, then \(S_{d+1}(V_K, \mathcal{O}_\varepsilon)\) is Zariski nondense in \(V\); (ii) if \(V\) is transverse, then \(S_{d+1}(V_K, \Gamma_\varepsilon)\) is Zariski nondense in \(V\).
The author proves firstly that the assertions (i) and (ii) are equivalent. Then the proof of (ii) consists of three steps. The first two steps allow to avoid \(\Gamma\) and to approximate an algebraic subgroup with a subgroup of bounded degree. The third step shows that certain special sets are Zariski nondense in \(V \times \gamma\), where \(\gamma\) is a maximal free set of the division group of \(\Gamma\). Here the proof is based on an essentially optimal Bogomolov-type bound for the normalized height of a transverse subvariety in \(E^g\). There is a conjecture asserting that there exist \(\varepsilon>0\) and a nonempty Zariski open subset \(V^u\) of \(V\) such that if \(V\) is weak-transverse (resp. transverse), then \(S_{d+1}(V^u, \mathcal{O}_\varepsilon)\) (resp. \(S_{d+1}(V^u,\Gamma_\varepsilon)\)) has bounded height. The paper concludes with the proof of a special case of this conjecture.

14K12 Subvarieties of abelian varieties
11G05 Elliptic curves over global fields
11G50 Heights
14H25 Arithmetic ground fields for curves
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