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It is explained in a relatively elementary way in Frances Kirwan's "Complex Algebraic Curves"[1]. The book is pretty self contained, and much more entry level than, say, Silverman.

[1] - https://www.amazon.com/Complex-Algebraic-Mathematical-Societ...

The explanation is in lecture 15. It requires knowledge of some advanced mathematics. Abelian varieties over the complex numbers are analytically isomorphic to a torus.

I would expect him to spell this out in lectures 15 and 16. It does take work. What is surprising is that despite their all looking the same -- certainly they are the same as real manifolds -- there are tons of elliptic curves. The difference is in the complex analytic structure.

They (the space of complex points on any elliptic curve over C) are the same (homeomorphic) as topological spaces. However, they are not the same as complex analytic manifolds, which is a stronger condition (requiring an analytic isomorphism, not just a topological one).

I should have said real manifolds; I thought that was what the parent comment had in mind but the brevity wasn't worth it.

Of course, I've learned a lot about these things from your writings.