Timeline for Siegel's theorem on integral points for higher dimensional varieties
Current License: CC BY-SA 3.0
11 events
| when toggle format | what | by | license | comment | |
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| Feb 9, 2017 at 15:19 | review | Close votes | |||
| Feb 9, 2017 at 21:59 | |||||
| Feb 5, 2017 at 18:00 | comment | added | Felipe Voloch | Faltings proved that an affine open subset of an abelian variety has finitely many integral points. There is also a number of papers of Corvaja, Zannier and others proving some finiteness statements in various cases. I also agree that this question lacks focus. | |
| Feb 5, 2017 at 13:35 | comment | added | Kevin Buzzard | @DanielLoughran: oh yes, maybe I meant quartic surface. More generally is that right -- we still don't know of a K3 surface with only finitely many rational points? The story I was trying to recollect was a candidate surface for which computer search up to something like naive height 1000 produced just one point, and then a search just a little further produced another one. | |
| Feb 5, 2017 at 10:45 | comment | added | user19475 | One can consult the last chapter "A higher dimensional Mordell conjecture" of [Cornell-Silverman]. | |
| Feb 5, 2017 at 10:31 | comment | added | Daniel Loughran | I also agree with Kevin that this question lacks focus somewhat. Mathoverflow does not work very well for "what is known" type questions. | |
| Feb 5, 2017 at 10:08 | comment | added | Daniel Loughran | Well the main conjecture is the Lang-Vojta conjecture. This has a few formulations, which are conjecturally equivalent. One version says that any smooth Brody hyperbolic variety has only finitely many integral points (a smooth variety $X$ is called Brody hyperbolic if any holomorphic map $\mathbb{C} \to X(\mathbb{C})$ is constant.) This is known is some special cases, but of course wide open in general. | |
| Feb 5, 2017 at 10:04 | comment | added | user19475 | I am interested in theorems as well as in conjectures. | |
| Feb 5, 2017 at 10:02 | comment | added | Daniel Loughran | @Kevin Buzzard: Did you mean K3 surface, rather than cubic surface? | |
| Feb 5, 2017 at 9:44 | comment | added | Kevin Buzzard | We still don't know if there is a cubic surface with a non-zero finite number of rational points and people don't know whether to believe Lang's conjectures, so it's not clear to me we know much about anything in this generality. Are you interested in weak statements which might hold for all varieties, or stronger statements that only apply to smaller classes? And do you want theorems or conjectures? | |
| Feb 5, 2017 at 8:39 | history | edited | user19475 | CC BY-SA 3.0 |
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| Feb 5, 2017 at 8:29 | history | asked | user19475 | CC BY-SA 3.0 |