Timeline for Number of elements of "$\mathrm{SL}_n(\mathbb{F}_p^\times)$" mod $p$
Current License: CC BY-SA 3.0
11 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Jan 1, 2015 at 17:38 | answer | added | alpoge | timeline score: 1 | |
| Dec 26, 2014 at 7:34 | vote | accept | alpoge | ||
| Dec 26, 2014 at 1:42 | answer | added | Will Sawin | timeline score: 36 | |
| Dec 25, 2014 at 20:08 | comment | added | alpoge | (Quick typo correction: the $n\to\infty$ in my comment should be $e\to\infty$.) | |
| Dec 25, 2014 at 15:24 | answer | added | Krishanu Sankar | timeline score: 0 | |
| Dec 25, 2014 at 15:01 | comment | added | Richard Stanley | Not relevant to your question, but Exercise 1.182 of Enumerative Combinatorics, vol. 1, 2nd ed., shows that if $f(n,q)$ is the number of matrices in $\mathrm{GL}(n,q)$ with no 0 entries, and $g(n,q)$ is the number of matrices in $\mathrm{GL}(n-1,q)$ with no entry equal to 1, then $f(n,q)=(q-1)^{2n-1}g(n,q)$. A reference for counting matrices in $\mathrm{GL}(n,q)$ with specified entries equal to 0 (though it doesn't seem useful for your question) is arXiv:1011.4539. | |
| Dec 25, 2014 at 11:18 | comment | added | alpoge | Indeed, I should have written more, but it suffices to count points mod $p$ (or even over $\mathbb{F}_{p^e}$ for large $e$ if that's easier) on the hypersurface defined by $\prod_{i,j=1}^n a_{i,j} \det(A)$ in $\mathbb{P}^{n^2-1}$. I know essentially nothing about this --- the little I know would just give me an asymptotic as $n\to \infty$, which says nothing about the answer mod $p$. Feel free to take $p$ very large, by the way --- I'm mainly wondering if e.g. there are infinitely many $p$ for which the count is nonzero mod $p$! | |
| Dec 25, 2014 at 10:35 | answer | added | Geoff Robinson | timeline score: 8 | |
| Dec 25, 2014 at 6:50 | comment | added | user74230 | In the spirit of Geoff's comment, the count is $1/(p-1)$ times the count for ${\rm{GL}}_n$. We know the number matrices with all coordinates nonzero, so it is the same to study the space of matrices where the determinant vanishes, say avoiding the zero matrix. That reduces to the analogue (mod-$p$ count of points with all coordinates nonzero) for the hypersurface of degree $n^2$ in $\mathbf{P}^{n^2-1}$ defined by vanishing of det. Section 4 in Expose XXII in SGA7 is all about that kind of question for general projective hypersurfaces, so ask Nick Katz about this when you next see him. | |
| Dec 25, 2014 at 3:14 | comment | added | Geoff Robinson | A little remark is that the number of such matrices is an integer multiple of $(p-1)^{n-1}.$ The group $T$ of invertible diagonal matrices acts on such matrices by conjugation and only scalar matrices fix anything in the action. | |
| Dec 25, 2014 at 2:02 | history | asked | alpoge | CC BY-SA 3.0 |