Timeline for Lang's conjecture beyond the curve case
Current License: CC BY-SA 4.0
9 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Dec 28, 2020 at 8:17 | answer | added | Ariyan Javanpeykar | timeline score: 10 | |
| Dec 28, 2020 at 7:44 | comment | added | Ariyan Javanpeykar | @JacksonMorrow The results of Corvaja-Zannier and Levin were subsequently improved by Autissier (as one should probably mention). | |
| Dec 27, 2020 at 13:20 | vote | accept | Stanley Yao Xiao | ||
| Dec 24, 2020 at 14:14 | comment | added | Jackson Morrow | The conjectural equivalence of general type to other notions of hyperbolicity is spelled out in this nice survey of Javanpeykar (arxiv.org/pdf/2002.11981.pdf). If one looks at integral points, then there are results of Corvaja, Levin, and Zannier which tell us that once we remove enough ample divisors then there is no set of integral points which is Zariski dense (see e.g., their paper ``Integral points on threefolds and other varieties"). This work builds on previous work of Corvaja--Zannier where they give a new proof of Siegel's theorem using the Schmidt subspace theorem. | |
| Dec 24, 2020 at 7:48 | history | became hot network question | |||
| Dec 24, 2020 at 1:20 | answer | added | Noam D. Elkies | timeline score: 20 | |
| Dec 24, 2020 at 1:10 | comment | added | R. van Dobben de Bruyn | My understanding is that there is no single smooth projective variety of general type with finite fundamental group (e.g. large degree hypersurfaces) where the conjecture is known to hold or fail. I have also heard the opinion that 'general type' is too weak an assumption and should be replaced by 'hyperbolic' (of some sort), in which case it becomes more natural to look at quasi-projective varieties (and maybe integral points instead of rational?). I hope someone who knows more about this will comment. | |
| Dec 24, 2020 at 1:01 | comment | added | naf | Faltings, in his paper "Diophantine approximation on abelian varieties", has proved another conjecture of Lang which implies that the conjecture you state holds for general type subvarieties of abelian varieties (essentially all subvarieties that are not translates of abelian subvarieties). | |
| Dec 23, 2020 at 23:48 | history | asked | Stanley Yao Xiao | CC BY-SA 4.0 |