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The intersection of a curve with algebraic subgroups in a product of elliptic curves. (English) Zbl 1170.11314
Summary: We consider an irreducible curve \({\mathcal C}\) in \(E^n\), where \(E\) is an elliptic curve and \({\mathcal C}\) and \(E\) are both defined over \(\overline \mathbb{Q}\). Assuming that \({\mathcal C}\) is not contained in any translate of a proper algebraic subgroup of \(E^n\), we show that the points of the union \(\cup{\mathcal C}\cap A(\overline \mathbb{Q})\), where \(A\) ranges over all proper algebraic subgroups of \(E^n\), form a set of bounded canonical height. Furthermore, if \(E\) has Complex Multiplication then the set \(\cup{\mathcal C}\cap A(\overline\mathbb{Q})\), for \(A\) ranging over all algebraic subgroups of \(E^n\) of codimension at least 2, is finite. If \(E\) has no Complex Multiplication then the set \(\cup{\mathcal C}\cap A(\mathbb{Q})\) for \(A\) ranging over all proper algebraic subgroups of \(E^n\) of codimension at least \(\frac n2+2\), is finite.

MSC:
11G10 Abelian varieties of dimension \(> 1\)
11G50 Heights
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