Xjyuk

So, I want to read the proof of Mordell-Weil theorem and so, I picked up the book 'Arithmetic of Elliptic Curves' by J. Silverman and J.S. Milne's Elliptic Curves book. But after going through both the books as well as Anthony Knapp's Elliptic curve book....I noticed up one thing, that Silverman takes way longer than Milne or Knapp to reach to Mordell-Weil theorem.

My question is- since I can't read all three books at the same time, can someone point out the differences between the approaches taken by these three texts. I know that Milne's has used group cohomology to prove Mordell-Weil but looking at Silverman I don't think he has used the exact same approach.

Also, what about Knapp's text?

I'm self studying and for this summer, my goal is to read up the proof of a big theorem like Mordell-Weil. But looking at the different books is just spinning my mind. And if Milne's shorter o more readable than Silverman than I would maybe read from it and not Silverman which I'm reading through right now.

In short, can someone also suggest me a path that I should follow to read the proof of Mordell-Weil? I don't want it to be unnecessarily long because at the moment, my focus is the big theorem(Mordell-Weil) and not other things, but I might come back later to read them..

Thank a lot and please feel free to add appropriate tags as I'm not sure if I've added the correct ones.

EDIT: Going through 'Rational points on elliptic curves' by Tate and Silverman, it also discusses Mordell-Weil Theorem in its chapter 3. I guess its proof is not as 'rigorous' as the one in Silverman's bookand is described with far less Algebraic Geometry that is the core of proofs in Silverman. Can someone also comment on the difference between two approaches? I mean, if 'Rational points....' also has a good proof then why do we need to explain everything in Algebraic-geometric language in Silverman's text?

EDIT: I'm sorry if this question is not appropriate for this site as it's not exactly a research question but I posted it few days ago on math stack exchange and it has been unanswered there since.

Milne's and Knapp's books are excellent. As for my AEC, it covers lots of stuff that's not needed if you just want to get to Mordell-Weil. So for AEC, here's a path to MW:

1. If you know some algebraic geometry, skip chapters I and II, refer back as needed.


2. Read Ch. III through Sec. 7


3. Reach Ch. IV through Sec. 7


4. Skip chapter V and VI.


5. Read Ch. VII through Sec 4 (or maybe Sec 5).


6. Then Ch. VIII has a proof of MW, and you can get there with just Secs. 1, 3, 5, and 6.

Then there's lots more info related to MW if you also look at Ch. VIII Sec. 2, and all of Ch. X.

BTW, it's not that RPEC is less rigorous than AEC, it's that it restricts to $mathbb Q$ and tries to be as elementary as possible, which means that it is less general, in particular only completely proving MW for elliptic curves $E/mathbb Q$ have a rational 2-torsion point. Also, by avoiding machinery, the algebra is rather messy and the proof is somwwhat unintuitive, so if you have the background (meaning basic algebraic number theory), I'd recommend one of the other treatments.

I've just finished teaching a Master's course on elliptic curves, where we assume no knowledge of number fields and even avoid Galois theory as far as possible. This makes it hard to consider proving Mordell–Weil in full generality. However, the proof over the rational numbers in the case where the curve has a rational 2-torsion point is accessible without any sophisticated tools. I'd suggest understanding this proof first, even if you later want to understand the full proof. I like Cassels' treatment of this, though some might find his style a bit old-fashioned (so my students tell me). I've written my own notes based on Cassels, available here.

Of course, Galois cohomology, and the extra tools from algebraic number theory needed for finiteness of the Selmer group in the general case, are excellent topics and I'd encourage you to learn them. But sometimes the structure of a proof is easier to see in a special case.