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Equidistribution of small points on abelian varieties. (English) Zbl 0991.11034

The author proves the equidistribution theorem conjectured in the paper by L. Szpiro, E. Ullmo and the author [Invent. Math. 127, No. 2, 337-347 (1997; Zbl 0991.11035)].
Let \(K\) be an algebraic number field and \(\overline{K}\) its algebraic closure. For an abelian variety \(A\) defined over \(K\), we denote by \(h(x)\) the Neron-Tate height of \(x\in A(\overline{K})\) with respect to a symmetric and ample line bundle on \(A\). A subvariety \(X\) of \(A\) is called torsion if \(X\) is a translate of an abelian subvariety by a torsion point. A series of distinct points \(\{x_n\}\) in \(A(\overline{K})\) is small if \(h(x_n)\) converges to zero and strict if no subsequence is contained in a proper torsion subvariety.
The main theorem in this paper is as follows: If \(\{x_n\}\) is a small and strict sequence, then the sequence of orbits \(\mathcal O(x_n)\) in the associated complex torus \(A_{\sigma }(\mathbb{C})\) determined by an embedding \(\sigma \colon K\to \mathbb{C}\) admits the equidistribution property, which means that for any continuous function \(f\) on \(A_{\sigma }(\mathbb{C})\), \[ \frac{1}{\#\mathcal O(x_n)}\sum_{y\in \mathcal O(x_n)}f(y) \] converges to the integral of \(f\) by the Haar measure of total volume \(1\).
This theorem easily follows from the generalized Bogomolov conjecture: If a subvariety \(X\) of \(A\) is not torsion, then for a sufficiently small \(\varepsilon >0\) the set \(\left\{x\in X(\overline{K}); h(x)\leq \varepsilon \right\}\) is not Zariski dense. E. Ullmo [Ann. Math. (2) 147, 167-179 (1998; Zbl 0934.14013)] proved the original Bogomolov conjecture, which is concerned with the case of a curve of genus \(\geq 2\) embedded in its Jacobian. In this paper the author applies Ullmo’s idea to solve the generalized conjecture.

MSC:

11G50 Heights
11G10 Abelian varieties of dimension \(> 1\)
14K15 Arithmetic ground fields for abelian varieties
14G05 Rational points
14G40 Arithmetic varieties and schemes; Arakelov theory; heights
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