# What is the amount of abstract algebra needed to study elliptic curves?

To be more specific, how much is needed to understand the book 'Rational points on elliptic curves' by Silverman?

## migrated from mathoverflow.netJul 29 '13 at 19:52

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• A good handle at least on rings and modules is probably required. You will definitely need rings for the algebraic geometry side of it. – rfauffar Jul 29 '13 at 19:10
• Thanks. Would you recommend having experience with complex analysis? – Jonas Jul 29 '13 at 19:12
• Complex analysis is definitely good, but I'm not sure it would be a requirement. – rfauffar Jul 29 '13 at 20:15

This book has six chapters, and one needs perhaps different prerequisites for each chapter. For chapter $I$ and $II$, on geometry of cubic curves and points of finite order, just basic algebra is enough, together with plane curves (affine and projective). For elliptic curves over the complex numbers, a bit of complex analysis is required. For chapter $III$ and $IV$ on the group of rational points and cubic curves over finite fields, more algebra is needed (for example, for the proof of Mordell's theorem). For chapters $V$ and $VI$ on Integer points on cubic curves, and Complex multiplication, Galois representations, algebraic number theory, and algebraic geometry would be helpful.