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We consider an irreducible curve in , where is an elliptic curve and and are both defined over . Assuming that is not contained in any translate of a proper algebraic subgroup of , we show that the points of the union , where ranges over all proper algebraic subgroups of , form a set of bounded canonical height. Furthermore, if has Complex Multiplication then the set , for ranging over all algebraic subgroups of of codimension at least , is finite. If has no Complex Multiplication...
In this article we show that the Bounded Height Conjecture is optimal in the sense that, if is an irreducible subvariety with empty deprived set in a power of an elliptic curve, then every open subset of does not have bounded height. The Bounded Height Conjecture is known to hold. We also present some examples and remarks.
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