Abstract

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 $\bigcup {\mathcal{C}} \cap A$, 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 $\bigcup {\mathcal{C}}\cap A$, 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 $\bigcup {\mathcal{C}} \cap A$ for $A$ ranging over all proper algebraic subgroups of $E^n$ of codimension at least $\frac{n}{2}+2$, is finite