Suppose we have an holomorphic function $ f : \frac{\mathbb{C}}{\Lambda} \mapsto \frac{\mathbb{C}}{\Lambda} $ where $\Lambda$ is a lattice. Is it always possible to find another function $\psi : \mathbb{C} \mapsto \mathbb{C}$ which extend $f$, i.e. such that $ \forall x \in \mathbb{C}, \, \overline{\psi(x)} = f(\overline x) $ or, at least, which extend $f$ on a neighbourhood of each point.
I met this issue when I was reading "rational points on elliptic curves" of Silverman & Tate. To give the context, we have an elliptic curve $C(\mathbb C)$ and an endomorphism of $C(\mathbb C)$, and because we know there exists an isomorphism $ \frac{\mathbb C}{\Lambda} \rightarrow C(\mathbb C)$ for a lattice $\Lambda$ (using Weierstrass $\wp$ function), we finally find an "holomorphic" (I don't really know what holomorphic on $\frac{\mathbb C}{\Lambda}$ mean) function $f$, and we use a function like $\psi$ and complex analysis to discover that $ f : z \in \frac{\mathbb C}{\Lambda} \mapsto cz $ with $c \in \mathbb C$. (Yes, it explain the name "complex multiplication")
You can find an article which explain this part of the book at : CMpaper