## Distribution of almost division points.(English)Zbl 0972.11053

In the paper under review, the author proves the following result, generalising earlier work [see S. Zhang, Doc. Math., J. DMV, ICM Berlin 1998, Vol. II, 217-225 (1998; Zbl 0912.14008); Ann. Math. (2) 147, 159-165 (1998; Zbl 0991.11034)].
Let $$A$$ be an abelian variety over a number field $$K$$, and let $$(x_n)$$ be a sequence of distinct points in $$A(\overline{K})$$. Assume this is a sequence of almost division points, meaning that $\lim_{n \to \infty} \sup_{\sigma \in G} \|x_n^\sigma - x_n\|= 0,$ where $$G$$ is the absolute Galois group of $$K$$ and $$\|\cdot \|$$ is the square root of some Néron-Tate height on $$A$$. Then, replacing $$(x_n)$$ by a subsequence, there is an abelian subvariety $$C$$ of $$A$$, a point $$b \in A({\mathbb C})/C({\mathbb C})$$ and a finite subset $$T$$ of the torsion subgroup of $$A/C$$ such that the measures $$\delta x_n^G$$ (probability measure uniformly distributed over the points in the orbit $$x_n^G$$) on $$A({\mathbb C})$$ converge to the measure $$d\mu = |T|^{-1} \sum_{t \in T} \delta_{\pi^{-1}(b+t)}$$, where $$\pi : A \to A/C$$ and $$\delta_{\pi^{-1}(b+t)}$$ is the $$C({\mathbb C})$$-invariant probability measure supported on the fiber $$\pi^{-1}(b+t)$$.
As an application, the author proves that for a subvariety $$X$$ of $$A$$ that is not a translate of an abelian subvariety, the set of points in $$X(\overline{K})$$ that are sufficiently near to $$A(K) \otimes {\mathbb R}$$ (with respect to a Néron-Tate height pairing on $$A(\overline{K}) \otimes {\mathbb R}$$) is not Zariski-dense in $$X$$.

### MSC:

 11G10 Abelian varieties of dimension $$> 1$$ 14K15 Arithmetic ground fields for abelian varieties 11G50 Heights

### Citations:

Zbl 0912.14008; Zbl 0991.11034
Full Text:

### References:

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