# Non-analytic homomorphism between complex tori

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?

• Any two elliptic curves over $\Bbb C$ are isomorphic as real Lie groups. – Angina Seng Nov 7 '18 at 17:48
• 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. – reuns Nov 7 '18 at 18:09

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