Suppose I have an entire function $f : \mathbb{C} \longrightarrow \mathbb{C}^n$ for $n \geq 1$.
Let $C$ be the curve $f(\mathbb{C})$ in $\mathbb{C}^n$.
Let $\Lambda$ be a lattice in $\mathbb{C}^n$ (free $\mathbb{Z}$-submodule of rank $2n$) and assume that $\mathbb{C}^n/\Lambda$ is algebraic - it is an abelian variety.
Suppose that $C$ is stable by $\Lambda$ i.e. $C + \lambda = C$ for all $\lambda$ in $\Lambda$.
What conditions that imposes on $f$? I think it should impose that $f$ is a linear function but I have no proof.
This question is probably trivial to an expert in complex analysis..