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

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 $\bigcup \mathcal{C}\cap A\left(\overline{\mathbb{Q}}\right)$, 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 $\bigcup \mathcal{C}\cap A\left(\overline{\mathbb{Q}}\right)$, for $A$ ranging over all algebraic subgroups of ${E}^{n}$ of codimension at least $2$, is finite. If $E$ has no Complex Multiplication...