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Heights of algebraic points on subvarieties of abelian varieties. (English) Zbl 0897.11020
Let \(A\) be an abelian variety defined over \(\overline{\mathbb{Q}}\). Let \(\widehat h\) be the Néron-Tate height associated to a given very ample symmetric line bundle \({\mathcal L}\) on \(A\) and \(d\) the associated semi-distance. Further, let \(X\) be a closed, geometrically irreducible subvariety of \(A\) defined over \(\overline{\mathbb{Q}}\) and \(X^0\) the set obtained by removing from \(X\) all translates of positive-dimensional abelian subvarieties of \(A\) which are contained in \(X\). Recently, L. Szpiro, E. Ullmo and S. Zhang [Invent. Math. 127, 337-347 (1997)] proved the conjecture proposed by Bogolomov that the set of algebraic points in \(X^0\) is discrete with respect to the semi-distance \(d(P,Q)\). In the present paper, the authors prove a more precise quantitative version of this result, but only for abelian varieties satisfying certain strong conditions. Modulo basic theory of abelian varieties, the proof of the authors is elementary.
More precisely, let \(A\) be defined over a number field \(K\). Assume that either (A1) \(A\) has complex multiplication or (A2) there exists an infinite set \(\Sigma\) of primes such that for \(p\in\Sigma\) and some positive integer \(r\), the isogeny \([p^r]\) on the reduction \(A/\wp\) of \(A\) modulo some prime ideal \(\wp\) on \(K\) dividing \(p\) is equal to the Frobenius map on \(A/\wp\). Let a projective embedding \(j: A\hookrightarrow {\mathbb{P}}^m\) be given, defined by some basis of global sections of the line bundle \({\mathcal L}\) chosen above, and suppose \(X\) has degree \(d\) with respect to \(j\). Then there exist effectively computable constants \(\gamma\) and \(N\), depending on \(j\), \(d\), \(K\), the field of complex multiplication if (A1) holds and the set \(\Sigma\) if (A2) holds, such that for every \(P\in A(\overline{\mathbb{Q}})\), there are at most \(N\) points \(Q\in A(\overline{\mathbb{Q}})\) with \(d(P,Q)\leq\gamma\). The hard core of the proof consists of elementary congruence arguments, but for these to work it is necessary to assume either (A1) or (A2). It should be mentioned that there are analogous results with analogous proofs for algebraic points on linear tori \({\mathbb{G}}_m^n\). The first to prove an analogue of Bogolomov’s conjecture for linear tori was S. Zhang. His results were refined later by several people, among them Bombieri and Zannier and then Schmidt [see the following review Zbl 0897.11021].

MSC:
11G35 Varieties over global fields
14G40 Arithmetic varieties and schemes; Arakelov theory; heights
14K15 Arithmetic ground fields for abelian varieties
14G05 Rational points
11H99 Geometry of numbers
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