Interesting properties of complex tori as groups and how they affect our understanding of elliptic curves

Question

I'm finishing up a small project I've written up for my final year of college, where I was given the objective to learn about elliptic curves and write a paper on what I've learned. I'm writing for an audience that knows basic commutative algebra and complex analysis (up until just before Riemann surfaces), and I am proving everything by first principles.

How I structured the paper so far is as follows:

1. Defined algebriac curves, (proved Hilbert's Nullstellensatz in the appendix)
2. Proved Bézout's theorem using resultants and derived some properties of cubics such as the Cayley-Bacharach theorem
3. Proved that every non-singular cubic curve in the plane is equivalent to a cubic in Weierstrass form by a change of variables and are therefore elliptic curves
4. Defined the group law on elliptic curves and proved it was associative
5. Defined complex tori, elliptic functions and the Weierstrass-$\wp$ function. Showed that every complex torus is isomorphic to an elliptic curve and vice versa

Now I am exploiting this isomorphism to derive properties of the group law on the elliptic curve. So far I've proved that the group law on an elliptic curve is divisible, and that $E[n]\cong (\mathbb{Z}/n\mathbb{Z})\times (\mathbb{Z}/n\mathbb{Z}).$

My question is this: is there any other interesting facts about the group law that can be derived from this isomorphism? I want to make all of the work put in to read through everything seem more "worthwhile".

Answer 1

You can show using this description the nice description of what the endomorphisms of the group are (either $\mathbb{Z}$ or an order in an imaginary quadratic field, depending on which lattice you chose).