Timeline for $p$-adic points of open subschemes of complete intersections
Current License: CC BY-SA 4.0
9 events
| when toggle format | what | by | license | comment | |
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| Oct 22 at 15:50 | comment | added | Ben Singer | @R.vanDobbendeBruyn This is precisely why I want to study this problem! I'm hoping to procure existence of $p$-adic points for every prime on this quasi-projective subvariety as a first step, then move on to other methods of analyzing rational points on it (e.g. Brauer-Manin, calculating its Hodge groups to give data about maps from rational curves, etc.). | |
| Oct 22 at 0:49 | history | edited | LSpice | CC BY-SA 4.0 |
Typo
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| Oct 21 at 18:19 | comment | added | R. van Dobben de Bruyn | Methods for failure of the local-global principle have been used successfully to prove non-existence of rational points on smooth projective varieties. But for quasi-projective varieties (e.g. obtained by removing all trivial solutions from a projective variety), these methods typically only address integer solutions. That may seem like enough, but a condition like $x_i \neq x_j$ is translated to geometry by adjoining a variable $y$ with $y(x_i-x_j) = 1$, so an integral point means that $x_i-x_j$ is an invertible integer (i.e. $\pm 1$), which was not the original problem. | |
| Oct 21 at 18:11 | comment | added | R. van Dobben de Bruyn | You are probably aware of this, but let me warn you that the full statement on non-existence of solutions with pairwise distinct coordinates is out of reach with current methods. See also my answer here. Of course it can still be interesting to get some sort of grip on the problem, but the presence of trivial solutions makes it very hard (e.g. this will imply, by Hensel's lemma, existence of loads of $p$-adic solutions, even at primes of bad reduction because you can reduce modulo a power of the prime). | |
| Oct 21 at 15:15 | history | edited | Daniele Tampieri | CC BY-SA 4.0 |
Minor formatting and Math Jaxing
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| Oct 21 at 14:59 | comment | added | Jason Starr | Welcome new contributor. Is your complete intersection smooth over the ring of $p$-adic integers (on the open subset you are studying)? If so, then Hensel’s Lemma reduces your problem to studying points over the (finite) residue field. | |
| Oct 21 at 14:39 | history | edited | Ben Singer | CC BY-SA 4.0 |
added 24 characters in body
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| S Oct 21 at 14:38 | review | First questions | |||
| Oct 21 at 15:15 | |||||
| S Oct 21 at 14:38 | history | asked | Ben Singer | CC BY-SA 4.0 |