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

Viada, Evelina

Annali della Scuola Normale Superiore di Pisa - Classe di Scienze, Série 5, Tome 2 (2003) no. 1, pp. 47-75.

We consider an irreducible curve $𝒞$ in $E^n$, where $E$ is an elliptic curve and $𝒞$ and $E$ are both defined over $\overline{\mathbb{Q}}$. 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 $\bigcup 𝒞\cap A\left(\overline{\mathbb{Q}}\right)$, 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 $\bigcup 𝒞\cap A\left(\overline{\mathbb{Q}}\right)$, 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 $\bigcup 𝒞\cap A\left(\overline{\mathbb{Q}}\right)$ for $A$ ranging over all proper algebraic subgroups of $E^n$ of codimension at least $\frac{n}{2}+2$, is finite.

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