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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
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