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

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.

Tags:

Arithmetic Geometry

Elliptic Curves

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