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?
Here is a complex analysis viewpoint which may only be a partial answer. A conformal homeomorphism of complex tori requires that $\Lambda_1 = a \Lambda_2$ for some $a\in\mathbb{C}^\times$. As far as I am aware, this is clear from the fact that (and here we just look at the map $\mathbb{C}\to\mathbb{C}$) a conformal map must be angle-preserving (so think what goes on at the corners). Milne's notes on modular forms have this, Cor 3.5 in the below.
http://www.jmilne.org/math/CourseNotes/mf.html