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Integral points on abelian surfaces are widely spaced. (English) Zbl 0617.14014

Let K be a number field, S a finite set of places of K containing the infinite places, and \(R_ S\) the ring of S-integers of K. Let A be an abelian variety over K and \(U\subset A\) an affine open subset of A. Main result: if \(\dim (A)=2\), then #\(\{P\in U(R_ S)| \hat h(P)\leq H\}\leq c\cdot \log (H)\), where \(U(R_ S)\) is the set of all S-integral points of U, and \(\hat h\) is a logarithmic canonical height on A corresponding to some ample, symmetric divisor on A. Note that it is well known (by a result of Siegel) that if A is an elliptic curve, then U has only finitely many integral points, and then S. Lang conjectured that the same should be true for abelian varieties of arbitrary dimension.
Reviewer: L.Bădescu

MSC:

14G05 Rational points
14K15 Arithmetic ground fields for abelian varieties
14H52 Elliptic curves
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References:

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