# Can someone suggest a path to study Mordell-Weil theorem for someone studying on their own?

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.

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 will of course come back later to read them according to the need.

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 claims (p.117) that his proof of the weak Mordell-Weil is simpler than the standard proof since it avoids translating the "putative finiteness of $E(L)/nE(L)$ into a statement about certain field extensions of $L$." – anon Jun 20 at 12:01
• – user 170039 Jun 20 at 14:01
• Try Cassels lectures on elliptic curves, LMSST 24. – lemiller Jun 21 at 14:03

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 found Milne's treatment of Mordell Weil really clear. – Asvin Jun 19 at 19:33
• Thanks for the answer! I have started going through RPEC but soon after finishing the proof of Mordell-Weil and digesting it well, I hope to be back to AEC or Milne. I do have background in both AG and ANT, so might go through the contents of both them roughly to decide which one will be best suitable for me. I suspect that AEC and Milne prove the same generalization of Mordell-Weil but have different approaches, that is, one with height descent argument and another using group cohomology, am I right – Mojojojo Jun 20 at 14:01
• @Mojojojo I haven't looked at Milne in a while, but I think one pretty much always has to do a height descent at some point in the argument. The proof of MW has two parts. First is "weak" Mordell-Weil, which is proving $E(K)/mE(K)$ is finite. That proof is cleaner if you know and use group cohomology, which is Milne's approach, and I do that in VIII Sec. 2; but I also do it more directly, essentially by unsorting the cohomology, in VIII Sec. 1. But Step 2, namely going from finiteness of $E(K)/m(K)$ to finite generation of $E(K)$, needs a height-type argument AFAIK. – Joe Silverman Jun 20 at 14:53
• That actually clears up things a bit, thanks! – Mojojojo Jun 21 at 12:44

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.

• That's also the historical path. Once you understand how to do it when there's a 2-torsion point (which is a natural generalization of Fermat's original "descent"), and how to generalize to arbitrary number fields (which is a common theme in modern number theory), you automatically get to drop the 2-torsion assumption, because you always have a 2-torsion point over some number field. The group structure (which was obtained relatively late, by Mordell) also explains the apparent miracle that two "descents" get you back to the original curve. – Noam D. Elkies Jun 19 at 20:39
• Thank you for posting the link to your notes! I have started going through $E/\mathbb{Q}$ with a rational 2-torsion point case using Rational Points on Elliptic Curves, have heard some praises about Cassels as well, so I think I might check out your notes as well since they cover the same contents. – Mojojojo Jun 20 at 14:04
• Everyone should read Cassels's historical essay on the genesis of Mordell's theorem (Math. Proc. Cambridge Philos. Soc 1983), and no one should try to learn the theorem by reading Mordell's original 1922 paper. – anon Jun 20 at 14:12