What a brilliant thread! Elliptic curves, what could be better? It's interesting to hear that you were first introduced to them in a graduate complex analysis course. I'm a cryptography enthusiast and my lecturer told said that elliptic curve cryptography (ECC) was very powerful. I have since decided to pick out lots of number theory/cryptography books from the library which contain elliptic curve sections. So I guess I have been gently introduced through the number theory approach which seems less daunting than say an algebraic geometry or ''graduate complex analysis course'' approach. I have become absolutely obsessed with ECC and am going to do a project on them over the summer (ur guys winter), really amazing stuff!