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 asked the same question on mathoverflow and have got some really helpful answers there, if anyone is interested, please check those out here: https://mathoverflow.net/questions/334366/can-someone-suggest-a-path-to-study-mordell-weil-theorem-for-someone-studying-on