Basic Understanding of Elliptic curve

There are lot many articles to name. The theory of elliptic curves, have originated a few decades earlier, but has influenced major areas, and became a central part where one can formulate a beautiful conjectures.

To add some references :

1. The Arithmetic of Elliptic curves by J.Silverman.
2. Rational Points on Elliptic Curves by J.Silverman and J.Tate.
3. If you scroll down to the bottom of this page, you exactly find a set of books, that will be very useful for a complete reference.
4. Another fantastic book which gives a very gentle introduction into the subject is Lawrence C. Washington Elliptic Curves Number Theory and Cryptography, Second Edition Discrete Mathematics and Its Applications 2008 which is available online.

And if you search in google, you can in fact find lot many paper presentations given by Prof.Silverman. Do search it.

Thank you.

At least in the field of Cryptography lately in the recent years a lot of progress has been contributed due to progress in Elliptic Curves. The study of Elliptic Curves have been done past few decades as Iyengar pointed out, but the applications to that have been only realized lately.

Read the book http://tinyurl.com/7kjywzh (at least preview the first few pages on line to get an idea). It all depends what you want to achieve. If you just want to read and understand because you have a passion, then keep digging on-line. If not, you have to choose where you are inclined to apply that.

If you are interested in the cryptographical applications of elliptic curves, Washington does a great job of introducing elliptic curves and applying them to cryptography.

Vanstone et al has a book that doesn't spend much time talking about elliptic curves but rather talks about algorithms related to elliptic curve cryptography (ECC) and their implementation.

Finally David Jao has a expository paper on ECC and Menezes an expository paper on pairing-based cryptography that is often implemented in ECC for more effective security.