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.


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


[1] G. Faltings, Diophantine approximation on abelian varieties , Ann. of Math. (2) 133 (1991), 549–576. JSTOR: · Zbl 0734.14007 · doi:10.2307/2944319
[2] M. Hindry, Autour d’une conjecture de Serge Lang , Invent. Math. 94 (1988), 575–603. · Zbl 0638.14026 · doi:10.1007/BF01394276
[3] M. McQuillan, Division points on semi-abelian varieties , Invent. Math. 120 (1995), 143–159. · Zbl 0848.14022 · doi:10.1007/BF01241125
[4] A. Moriwaki, Arithmetic height functions over finitely generated fields, Invent. Math. 140 (2000), 101–142. · Zbl 1007.11042 · doi:10.1007/s002220000050
[5] B. Poonen, Mordell-Lang plus Bogomolov, Invent. Math. 137 (1999), 413–425. · Zbl 0995.11040 · doi:10.1007/s002220050331
[6] E. Ullmo, Positivité et discrétion des points algébriques des courbes , Ann. of Math. (2) 147 (1998), 167–179. JSTOR: · Zbl 0934.14013 · doi:10.2307/120987
[7] S.-W. Zhang, Equidistribution of small points on abelian varieties , Ann. of Math. (2) 147 (1998), 159–165. JSTOR: · Zbl 0991.11034 · doi:10.2307/120986
[8] –. –. –. –., Distribution of almost division points , Duke Math. J. 103 (2000), 39–46. · Zbl 0972.11053 · doi:10.1215/S0012-7094-00-10313-4
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.