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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.

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.

Reviewer: Yuichiro Takeda (Kyushu)