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Arithmetic height functions over finitely generated fields. (English) Zbl 1007.11042
Author’s summary: Let \(K\) be a finitely generated field over \(\mathbb{Q}\), \(X\) a projective variety over \(K\), and \(L\) an ample line bundle on \(X\). In this paper, we propose a new height function \[ h_L^{\text{arith}}: X(\overline{\mathbb{Q}})\to \mathbb{R} \] of arithmetic type. The first main result of this paper is the following:
Theorem 1 (Northcott’s theorem). For any numbers \(M\) and any positive integers \(e\), the set \[ \{P\in X(\overline{K})\mid h_L^{\text{arith}} (P)\leq M,\;[K(P):K]\leq e\} \] is finite.
If \(A\) is an Abelian variety over \(K\) and \(L\) is a symmetric ample line bundle on \(A\), then the canonical height function \[ \widehat{h}_L^{\text{arith}}: A(\overline{\mathbb{Q}})\to \mathbb{R} \] is given as a quadratic form. The second main result is the following solution of Bogomolov’s conjecture, which is a generalization of results due to E. Ullmo [Ann. Math. (2) 147, 167-179 (1998; Zbl 0934.14013)] and S. Zhang, Ann. Math. (2) 147, 159-165 (1998; Zbl 0991.11034)].
Theorem 2 (Bogomolov’s conjecture). Let \(X\) be a subvariety of \(A_{\overline{K}}\). If the set \[ \{P\in X(\overline{K})\mid \widehat{h}_L^{\text{arith}} (P)\leq \varepsilon\} \] is Zariski dense in \(X\) for any positive numbers \(\varepsilon\), then \(X\) is a translation of an Abelian subvariety of \(A_{\overline{K}}\) by a torsion point.

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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