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The canonical arithmetic height of subvarieties of an abelian variety over a finitely generated field. (English) Zbl 1013.11032
Author’s summary: This paper is the sequel to the author’s paper [Invent. Math. 140, 101-142 (2000; Zbl 1002.11042)]. In this paper, the author defines the canonical height of subvarieties of an abelian variety over a finitely generated field over $$\mathbb{Q}$$ and proves that the canonical height of a subvariety is zero if and only if it is a translation of an abelian subvariety by a torsion point as a generalization of S. Zhang’s result over a number field [Ann. Math. (2) 147, 159-165 (1998; Zbl 0991.11034)].

##### MSC:
 11G50 Heights 11G10 Abelian varieties of dimension $$> 1$$ 11G35 Varieties over global fields 14G40 Arithmetic varieties and schemes; Arakelov theory; heights 14K15 Arithmetic ground fields for abelian varieties
##### Keywords:
canonical height; subvarieties; abelian variety; torsion point
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##### References:
 [1] Iitaka S., GTM pp 76– (1982) [2] Invent. Math. 140 pp 101– (2000) [3] Ann. Math. 147 pp 167– (1998) [4] J. Alg. Geom. 4 pp 281– (1995) [5] Ann. Math. 147 pp 159– (1998)
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