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A generalization of conjectures of Bogomolov and Lang over finitely generated fields. (English) Zbl 1009.11039

Let \((A,L)\) be a polarized Abelian variety defined over a finitely generated field \(K\) over \(\mathbb{Q}\) of transcendence degree \(d\). For every tuple \(\overline{B}:= (B,H_1,\dots, H_d)\), with \(B\) a normal projective scheme over \(\mathbb{Z}\) and \(H_i\) nef and big Hermitian line bundles on \(B\), the author defined in [Invent. Math. 140, 101-142 (2000; Zbl 1007.11042)] a height pairing \(\langle\;,\;\rangle^{\overline{B}_L}: A(\overline{K})\times A(\overline{K})\to \mathbb{R}\). If for \(x_1,\dots, x_l\in A(\overline{K})\) we denote \(\delta_L^{\overline{B}} (x_1,\dots, x_l)= \det(\langle x_i.x_j \rangle_L^{\overline{B}})\), then the main result of the paper is the following theorem: Let \(\Gamma\) be a subgroup of finite rank in \(A(\overline{K})\), let \(X\) be a subvariety of \(A_{\overline{K}}\) and \(\{\gamma_1,\dots, \gamma_n\}\) a basis of \(\Gamma\otimes \mathbb{Q}\). If the set \(\{x\in X(\overline{K})\mid \delta_L^{\overline{B}} (\gamma_1,\dots, \gamma_n,x)\leq \varepsilon\}\) is Zariski dense in \(X\) for every positive number \(\varepsilon\), then \(X\) is a translation of an Abelian subvariety of \(A_{\overline{K}}\) by an element of \(\Gamma_{\text{div}}:= \{x\in A(\overline{K})\mid nx\in \Gamma\) for some positive integer \(n\}\).
This answers a question of B. Poonen in [Invent. Math. 137, 413-425 (1999; Zbl 0995.11040)], who proved an equivalent version of the above theorem for number fields \(K\). The proof essentially follows Poonen’s ideas. The new idea is to remove the measure theoretic argument from the original proof.

MSC:

11G10 Abelian varieties of dimension \(> 1\)
11G50 Heights
14G40 Arithmetic varieties and schemes; Arakelov theory; heights
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References:

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