What will be a good source for learning elliptic curves and what viewpoints can I adopt?

Question

I'm attending a research seminar on elliptic curves in my university where the professor is currently presenting a proof of Mordell's theorem (for elliptic curves over $\mathbb Q)$. The professor says that the subject of elliptic curves can be studied in various settings-for extensions of $\mathbb Q$, for finite fields and even in the context of abelian varieties. However, I've very little knowledge of these topics. I do know basic algebraic number theory and commutative algebra, but would appreciate references for looking at the subject of elliptic curves in these diverse settings.

Answer 1

I think that the best alternative to Silverman's book is:

• Dale Husemoller's "Elliptic Curves".

It is a bit softer, but still comprehensive and rigorous.

It also includes a brief introduction to research topics: Birch-Swinnerton-Dyer, modularity, Calabi-Yau varieties, cryptography, topological modular forms... Also, the references are quite extensive.

Answer 2

An easier read than Silverman's Arithmetic is Silverman-Tate's Rational Points on Elliptic Curves which is aimed at the undergraduate level but still touches on more advanced topics like Galois representations which is heavily researched, along with looking at elliptic curves over finite fields.

This also includes a chapter on complex multiplication and an appendix on projective geometry. The appendix is worthwhile learning since this is what connects to the field of algebraic geometry which is the study of polynomials (aka varieties) and is what you would need to learn to move on to the more general abelian varieties.

After you go through Silverman-Tate, I suggest learning some algebraic geometry in conjunction with Silverman's Arithmetic as it will make your life easier and you can then study elliptic curves over other number fields (the Mordell-Weil theorem still holds here) and also local fields.

Husemoller's book I would say is harder than Silverman's Arithmetic as it covers most of what Silverman does but is a bit more terse and is easier to get lost in.