Timeline for Proofs of Mordell-Weil theorem
Current License: CC BY-SA 3.0
9 events
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| Jan 6, 2017 at 4:54 | comment | added | Pete L. Clark | @Niels: I do think it's minor. In fact, here are my notes for the course: math.uga.edu/~pete/EllipticCurves.pdf. In Section 9.6 I give Cassels's argument, adapted to any p-adic field (and even slightly more generally). I think it's a nice argument. That said, I am certainly a fan of Cohen's exposition as well, and it's nice to have a more formal reference for this argument. | |
| Jan 5, 2017 at 13:35 | comment | added | Niels | @Pete L. Clark : thank you for pointing to Cassels LEC. One minor issue is that the proof of the key fact is performed only over $\mathbb Q_p$. I found the same proof worked out for a general local field in : Henri Cohen Number Theory Volume I: Tools and Diophantine Equations GTM 239 §7.3.6-7. I even found the exposition somewhat better. | |
| Dec 18, 2016 at 12:40 | history | edited | Niels | CC BY-SA 3.0 |
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| Oct 29, 2012 at 18:54 | comment | added | Pete L. Clark | Especially, there is a part of the proof of Mordell-Weil which is traditionally proved using aspects of the reduction theory of elliptic curves over local fields. (There are other ways, e.g. Chevalley-Weil, but I decided to bypass them for various reasons.) Silverman devotes an entire chapter to elliptic curves over local fields and another entire chapter to formal groups, to prove the key fact that the kernel of reduction contains no torsion of order prime to the residue characteristic. Cassels gives a simply beautiful proof of this that takes about two pages. | |
| Oct 29, 2012 at 18:51 | comment | added | Pete L. Clark | A belated comment: I am currently teaching a course on elliptic curves, primarily out of Silverman's first text (which is, of course, wonderful). I used Cassels' LEC as a supplementary text, and I have to say that the proof of Mordell-Weil (for elliptic curves over number fields) is where LEC really shines. He makes a beeline to Mordell-Weil and gives a simple, but not overly computational proof, in an impressively short span of pages. | |
| Aug 9, 2011 at 4:41 | history | edited | Chandan Singh Dalawat | CC BY-SA 3.0 |
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| Aug 9, 2011 at 4:31 | history | edited | Chandan Singh Dalawat | CC BY-SA 3.0 |
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| Aug 8, 2011 at 7:16 | history | edited | Chandan Singh Dalawat | CC BY-SA 3.0 |
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| Aug 8, 2011 at 7:05 | history | answered | Chandan Singh Dalawat | CC BY-SA 3.0 |