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Comment by ice109 | Hacker News Reader

the profusion of posted slides on the internet is a detriment to researchers everywhere. so many times i've googled something technical (CS, math, engineering) and what percolates to the top are lecture slides, which are almost useless for actual in depth learning.

edit: spoke too soon - only the first link is slides.

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When I was doing research projects for my undergraduate degree we primarily used the web differently. 0. First thing we would search is the Arxiv front end http://front.math.ucdavis.edu/ . Usually we would have a researcher in mind. If we didn't then we would look at Wikipedia. 1. We would use wikipedia to get general knowledge and information about the topic. Sometimes lecture notes and presentations would be looked at too but we would eventually get to papers. 2. If the topic was relevant after getting the gist we would look at the citations on wikipedia and track down authors. 3. Books we would obtain electronically. Papers would be tracked down on Arxiv. 4. Rinse and repeat if you didn't understand a topic that was cited in any of the other steps.

It believe mathematicians will continue to favor boards. Sitting in the audience I greatly prefer this; however, such lectures have less chance of being posted at all.

For this topic we are lucky to have Silverman's book [1], which everyone seems to like.

[1] https://www.math.brown.edu/~jhs/AECHome.html

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)?

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.

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...

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