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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
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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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