Timeline for Understanding Faltings's Theorem
Current License: CC BY-SA 3.0
19 events
| when toggle format | what | by | license | comment | |
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| Jan 5, 2018 at 12:29 | history | edited | GH from MO |
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| Sep 14, 2017 at 15:37 | answer | added | user19475 | timeline score: 1 | |
| Apr 16, 2015 at 17:53 | vote | accept | pjox | ||
| Apr 13, 2015 at 13:25 | answer | added | Joe Silverman | timeline score: 24 | |
| Apr 13, 2015 at 12:37 | comment | added | user74230 | @ulrich: Thanks for mentioning that. I think I had been aware of it at some past time, but I hadn't ever looked into their paper (as my background is closer to the methods used by Faltings), so I had forgotten; oops. | |
| Apr 13, 2015 at 12:20 | answer | added | Robert Wilms | timeline score: 4 | |
| Apr 13, 2015 at 6:18 | comment | added | naf | @user74230 Actually, there is now a proof of the isogeny theorem by Masser and Wustholz. | |
| Apr 12, 2015 at 22:33 | comment | added | user74230 | @PJOX: Some other useful references are the book "Rational Points" co-authored by Faltings (last chapter does the version over finitely generate fields of char. 0) and an Asterisque volume where Gabber's Lemma first appeared (someone else will remember the title). But Cornell/Silverman and the notes I linked should get you quite far. Note that my list of background topics didn't mention "Arakelov theory". What is needed of that is limited enough that it does not require the development of a general "theory" around that stuff. | |
| Apr 12, 2015 at 17:58 | comment | added | pjox | @user74230, I'm aware this is a long-term goal, and I know it is currently impossible for me to understand something as complex as this. That is why I want to read and educate myself, I mean, I've been doing that with algebraic geometry and I want to do it with arithmetic geometry. Also, you are telling me the proof is awesome, I think that's enough reason to want to understand it :-). Oh, and thank you so much for the Stanford notes! that was exactly what I was asking for. If you know other resources, they will also be appreciated. | |
| Apr 12, 2015 at 14:24 | comment | added | user74230 | @PJOX: to understand the proof of the Isogeny Theorem we have only Faltings' (awesome) proof. Arapura is right: it should be a long-term goal, not realistic to fully understand at your stage of education. You also need Neron models, moduli of stable curves (which uses relative coherent duality), SGA7 material on the orthogonality theorem, the finiteness theorem of Borel & Harish-Chandra on arithmetic groups, Tate's work on $p$-divisible groups...you might find math.stanford.edu/~akshay/ntslearn.html a helpful supplement to Cornell/Silverman. | |
| Apr 12, 2015 at 11:54 | comment | added | pjox | @ulrich Yes, I'm aware there's another proof, but I really want to read Faltings's proof. Also that other background material is exactly what I'm asking for. I mean, I have not graduated yet but I have taken a course in classical algebraic geometry, a graduate course in algebraic curves and a graduate course in modern algebraic geometry, I'm reading Hartshorne's and Liu's book simply because I need to remember things and to learn some topics that where not covered on those courses. | |
| Apr 12, 2015 at 9:09 | comment | added | naf | As Donu said, Faltings's proof requires quite a lot of background material. However, there is a simpler proof due to Bombieri in his paper "The Mordell conjecture revisited" (based on ideas of Vojta and Faltings) which might be more accesible. | |
| Apr 12, 2015 at 8:18 | comment | added | Donu Arapura | No, I actually meant that it might be a bit too ambitious at your stage… But anyway, for Faltings you would need an understanding of abelian varieties, moduli spaces, heights etc. None of these topics are covered in Hartshorne. | |
| Apr 12, 2015 at 7:01 | comment | added | pjox | @DonuArapura: Sorry, I didn't understand that last comment. Did you mean that what is covered in Hartshorne's book is enough to start reading Silverman-Cornell? And what do you mean by "Falting's proof requires more…"? | |
| Apr 12, 2015 at 6:44 | comment | added | Donu Arapura | (I mean for someone at your stage. Falting's proof requires more…) | |
| Apr 12, 2015 at 6:42 | comment | added | Donu Arapura | Not to discourage you or anything, but if you just go through Hartshorne or one of the other sources you mention carefully, that seems more than enough. | |
| Apr 12, 2015 at 5:31 | review | Close votes | |||
| Apr 12, 2015 at 11:06 | |||||
| Apr 12, 2015 at 4:56 | review | First posts | |||
| Apr 12, 2015 at 6:48 | |||||
| Apr 12, 2015 at 4:55 | history | asked | pjox | CC BY-SA 3.0 |