Let $\Lambda_1$ and $\Lambda_2$ be lattices in $\mathbb C$, and let $\phi\colon \mathbb C/\Lambda_1\rightarrow \mathbb C/\Lambda_2$ be an analytic map. Then we know that $\phi$ is a group homomorphism provided that $\phi(0)=0$.
Suppose now, on the other hand, that we have a group homomorphism $\psi\colon \mathbb C/\Lambda_1\rightarrow \mathbb C/\Lambda_2$. What can we say about $\psi$ from the analytic perspective?
Can we have two elliptic curves over $\mathbb C$ which are isomorphic when viewed solely as abelian groups, but are different analytically (or geometrically)? What about other fields?