Timeline for Rational points over completions of a number field
Current License: CC BY-SA 3.0
11 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| May 20, 2011 at 7:00 | comment | added | ACL | @Kevin: Yes Lang-Weil applies uniformly: for any $n$, $d$ and $D$, there exists a real number $C(n,d,D)$ such that if $X$ is geometrically integral, of dimension $d$, and defined by polynomials of degree $D$ in the affine space $A^n$ over a finite field $k$ with $q$ elements, then $|\# X(k)-q^d|\leq C(n,d,D) q^{d-1/2}$. | |
| May 14, 2011 at 0:45 | vote | accept | Mikhail Borovoi | ||
| May 13, 2011 at 21:13 | comment | added | Felipe Voloch | @shenghao: What Pete said. @Pete: I was kidding. | |
| May 13, 2011 at 20:26 | comment | added | Pete L. Clark | @all: I think applying Deligne (presumably this means his proof of the Riemann hypothesis for varieties over finite fields) is big overkill here. See my answer below. | |
| May 13, 2011 at 20:23 | answer | added | Pete L. Clark | timeline score: 13 | |
| May 13, 2011 at 20:18 | comment | added | shenghao | @Felipe Voloch: sorry but, may I ask which result of Deligne? Thanks. | |
| May 13, 2011 at 19:09 | comment | added | Felipe Voloch | @Kevin: Lang-Weil does apply but, if you are bothered by it, apply Deligne instead. | |
| May 13, 2011 at 18:35 | comment | added | Kevin Buzzard | @Charles: doesn't Lang-Weil say something about points over bigger and bigger finite fields of the same characteristic? Does it also work if you're changing the characteristic? | |
| May 13, 2011 at 18:32 | comment | added | Kevin Buzzard | The reductions are almost all smooth if $X$ is smooth, basically because singularity means that some matrix of partial derivatives has zero det, but this can only happen mod $p$ for finitely many $p$. Note that you do need geom irred though, because e.g. the spectrum of a number field $L$ Galois over $k$ doesn't have $k_v$-points when $v$ doesn't split. | |
| May 13, 2011 at 18:09 | comment | added | Charles Matthews | This ought to be the Lang-Weil estimate to get points over almost all residue fields, then Hensel's lemma. So if the reductions are almost all smooth, where is the obstruction? | |
| May 13, 2011 at 17:32 | history | asked | Mikhail Borovoi | CC BY-SA 3.0 |