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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?

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    $\begingroup$ Any two elliptic curves over $\Bbb C$ are isomorphic as real Lie groups. $\endgroup$
    – Angina Seng
    Nov 7, 2018 at 17:48
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    $\begingroup$ Concretely let $\Lambda_1 = \mathbb{Z}+i\mathbb{Z},\Lambda_2 = u\mathbb{Z}+v\mathbb{Z}$ and send $x+iy+\Lambda_1$ to $xu+yv + \Lambda_2$ (isomorphism) or $(ax+by)u+(cy+dz)v + \Lambda_2$ (homomorphism). The map is real analytic. $\endgroup$
    – reuns
    Nov 7, 2018 at 18:09

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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

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  • $\begingroup$ So for conformal homomorphisms, $\Lambda_1$ is a sublattice of $a \Lambda_2$ $\endgroup$
    – reuns
    Nov 7, 2018 at 18:10

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