suggestion about elliptic curves

Question

I have read a little bit of number theory and covered upto Kummer's proof of Fermat's Last Theorem for regular primes. I am familiar with the concepts like disciminant, class number. Could anyone tell me whether I am in a position to start elliptic curves? If yes, then suggest me the most basic book with which I should start with.

Answer 1

Some good books for a first introduction to elliptic curves:

• Silverman--Tate, "Rational points on elliptic curves" (Springer Undergraduate Texts): very nice, fairly gentle
• Silverman, "The Arithmetic of Elliptic Curves" (Springer Graduate Texts #106): a fantastic book but a bit more advanced, the standard text for a masters/beginning graduate level course.
• Cassels, "Lectures on elliptic curves" (LMS Student Texts): very readable, with extremely few prerequisites (e.g. no use of algebraic number fields) but doesn't go as far as the other two above. (It's also substantially cheaper!)

If you're interested in applications to Fermat you might also like Diamond & Shurman's book "A First Course in Modular Forms" which is intended to give enough background on modular forms to appreciate the statement (not the proof!) of the Modularity Theorem for elliptic curves (formerly the Taniyama--Shimura conjecture).