3 ms·
can you explain to me why an elliptic curve over C is a quotient manifold (like the torus on page 7 of the first set of slides)?
by ice109 9y ago
can you explain to me why an elliptic curve over C is a quotient manifold (like the torus on page 7 of the first set of slides)?
- yequalsx 9y agoThe explanation is in lecture 15. It requires knowledge of some advanced mathematics. Abelian varieties over the complex numbers are analytically isomorphic to a torus.
- stablemap 9y agoI 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.
- williamstein 9y agoThey (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).
- stablemap 9y agoI 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.
- xyzzyz 9y agoIt 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-Society-Student/dp/0521423538 https://www.amazon.com/Complex-Algebraic-Mathematical-Societ...
- deleted 9y ago[deleted]