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.


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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