I'm currently studying the $3\times 3$ magic square of squares problem for a research project. The variety is initially given by the intersection of $8$ quadrics in $\mathbb{P}^8$, but via Gröbner basis calculations, one can reduce it to a scheme-theoretic complete intersection of $6$ quadrics in $\mathbb{P}^8$, hence a complete intersection surface. I want to study the $p$-adic points of the locus of "distinct entry squares", i.e. the variety associated to the equations of the $3\times 3$ square of squares with the intersections with the quadrics $X_{i}^2 - X_{j}^2 = 0$ cut out, but am unsure of methods for how to do this. I understand the local to global principle for individual quadrics well, but don't know how it applies to complete intersections of them. Are there any resources on $p$-adic points of complete intersections where I can learn more?
$\begingroup$
$\endgroup$
4
-
1$\begingroup$ 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. $\endgroup$– Jason StarrCommented Oct 21 at 14:59
-
3$\begingroup$ 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). $\endgroup$– R. van Dobben de BruynCommented Oct 21 at 18:11
-
2$\begingroup$ 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. $\endgroup$– R. van Dobben de BruynCommented Oct 21 at 18:19
-
1$\begingroup$ @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.). $\endgroup$– Ben SingerCommented Oct 22 at 15:50
Add a comment
|