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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.

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