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The intersection of a curve with algebraic subgroups in a product of elliptic curves

Evelina Viada — 2003

Annali della Scuola Normale Superiore di Pisa - Classe di Scienze

We consider an irreducible curve 𝒞 in E n , where E is an elliptic curve and 𝒞 and E are both defined over ¯ . Assuming that 𝒞 is not contained in any translate of a proper algebraic subgroup of E n , we show that the points of the union 𝒞 A ( ¯ ) , 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 𝒞 A ( ¯ ) , for A ranging over all algebraic subgroups of E n of codimension at least 2 , is finite. If E has no Complex Multiplication...

The optimality of the Bounded Height Conjecture

Evelina Viada — 2009

Journal de Théorie des Nombres de Bordeaux

In this article we show that the Bounded Height Conjecture is optimal in the sense that, if V is an irreducible subvariety with empty deprived set in a power of an elliptic curve, then every open subset of V does not have bounded height. The Bounded Height Conjecture is known to hold. We also present some examples and remarks.

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