Timeline for Number of $\mathbb F_p$ points constant mod $p$?
Current License: CC BY-SA 3.0
15 events
| when toggle format | what | by | license | comment | |
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| Jun 26, 2015 at 16:54 | answer | added | FriendlyWendy | timeline score: 5 | |
| Jun 26, 2015 at 14:06 | answer | added | Will Sawin | timeline score: 5 | |
| Jun 26, 2015 at 12:12 | answer | added | Dan Petersen | timeline score: 5 | |
| Jun 26, 2015 at 9:17 | comment | added | Mikhail Bondarko | Yes, Esnault has some papers related to this questions (and they are nice to read); yet they not seem to be really close related. An idea related to her papers: one may hope that a positive answer to my Question 2 (below) is equivalent to the following: the class of $H^{\ast c}(X)$ is congruent to that of a point modulo the Lefschetz class in the Grothendieck ring of effective Hodge structures. | |
| Jun 26, 2015 at 9:09 | history | edited | Mikhail Bondarko |
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| Jun 26, 2015 at 9:09 | answer | added | Mikhail Bondarko | timeline score: 4 | |
| Jun 26, 2015 at 1:26 | comment | added | KConrad | @JasonStarr, whoops. Got it. | |
| Jun 26, 2015 at 1:09 | comment | added | Allen Knutson | I'd certainly like to compute $\#X(\bf F_p)$ but don't expect to have much luck. | |
| Jun 26, 2015 at 0:24 | comment | added | Jason Starr | @KConrad. "If so, 'modulo $q$' is not what you mean (e.g., $q=8$)." I believe that this is what Esnault works with: a congruence for the number of points modulo the cardinality of the finite field. I do realize that the number of points changes when we base change from $\mathbb{F}_q$ to $\mathbb{F}_{q^r}$, but the number is congruent to $1$ modulo $q$, resp. $q^r$. | |
| Jun 25, 2015 at 23:36 | comment | added | Terry Tao | Reminds me a bit of Higman's PORC conjecture, groupprops.subwiki.org/wiki/Higman's_PORC_conjecture, though this conjecture is now believed to be false in general. | |
| Jun 25, 2015 at 20:36 | comment | added | KConrad | Can you work out a complete formula for $\#X({\mathbf F}_p)$ as $p$ varies, e.g., compute the zeta-function of $X_{/{\mathbf F}_p}$? Perhaps the counting formula for $\#X({\mathbf F}_p)$ is a universal polynomial in $p$ (like Hall polynomials). | |
| Jun 25, 2015 at 20:35 | comment | added | KConrad | @JasonStarr, is $\mathbf F_q$ a potentially general finite field and not one of prime order? If so, "modulo $q$" is not what you mean (e.g., $q = 8$). | |
| Jun 25, 2015 at 20:33 | comment | added | Jason Starr | Esnault proves that for a proper, regular variety $X_R$ over a finitely generated $\mathbb{Z}$-module $R$, if the geometric generic fiber has "coniveau 1", e.g., if it is rationally connected, then for every maximal ideal $\mathfrak{m}$ of $R$ with finite residue field $R/\mathfrak{m} \cong \mathbb{F}_q$, the number $X_{R/\mathfrak{m}}(\mathbb{F}_q)$ is congruent to $1$ modulo $q$. She may have considered affine varieties, but I am not aware of that. "Similar" results for affine varieties $Y$ usually have that $Y(\mathbb{F}_q)$ is congruent to $0$, not $1$ (e.g., affine spaces). | |
| Jun 25, 2015 at 20:06 | comment | added | F. C. | Reminds me of work of Dmitry Doryn on "The dual graph polynomials and a 4-face formula". He uses the coefficients of $q^2$. Maybe also look at Helene Esnault's articles ? | |
| Jun 25, 2015 at 19:44 | history | asked | Allen Knutson | CC BY-SA 3.0 |