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Mordell-Lang plus Bogomolov. (English) Zbl 0995.11040
We formulate a conjecture for semiabelian varieties that includes both the Mordell-Lang conjecture and the Bogomolov conjecture. Let $$A$$ be a semiabelian variety over a number field $$k$$. Let $$h: A(\overline{k})\to \mathbb R$$ be a canonical height. For $$\varepsilon>0$$, let $$B_\varepsilon= \{z\in A(\overline{k})\mid h(z)< \varepsilon\}$$. Let $$\Gamma$$ be a finitely generated subgroup of $$A(\overline{k})$$, and let $$\Gamma'$$ be its division group. Define $$\Gamma_\varepsilon':= \Gamma'+ B_\varepsilon$$. Let $$X$$ be a geometrically integral closed subvariety of $$A$$. Let $$X_{\overline{k}}$$ denote $$X\times _k \overline{k}$$. Our conjecture is as follows:
if $$X_{\overline{k}}$$ is not a translate of a semiabelian subvariety of $$A_{\overline{k}}$$ by a point in $$\Gamma'$$, then for some $$\varepsilon>0$$, $$X(\overline{k})\cap \Gamma_\varepsilon'$$ is not Zariski dense in $$X$$.
We prove this statement in the case that $$A$$ is isogenous to a product of an abelian variety and a torus.

##### MSC:
 11G35 Varieties over global fields 11G10 Abelian varieties of dimension $$> 1$$ 14G40 Arithmetic varieties and schemes; Arakelov theory; heights 11G50 Heights
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