5
$\begingroup$

Let $X$ be variety in $\mathbb{P}^N$ over $\mathbb{F}_q$ of dimension $n$ and degree $d$. By the Lang-Weil bounds $ |\# X(\mathbb{F}_q) - q^n| \le (d-1)(d-2)q^{n-1/2} + Cq^{n-1}$for a constant $C$ depending on $n$, $d$ and $N$.

Are there any bounds on $C$? Can you improve the estimate in special situations? I am interested in the case of del Pezzo surfaces where I want to use to test for local solubility.

Improve this question
$\endgroup$
4
  • 4
    $\begingroup$ Del Pezzo surfaces are rational and you can work out their zeta function explicitly. This should be in Manin's book on cubic surfaces. $\endgroup$
    – Felipe Voloch
    Commented Aug 18, 2013 at 12:48
  • 5
    $\begingroup$ In general, one gets much better bounds for smooth projective vareties using Deligne's proof of the Weil conjecture. For a del Pezzo surface $X$ you get that the error term is bounded by $bq + 1$ where $b$ is the second Betti number of $X$, which is the number of points of $\mathbb{P}^2$ blown up to get $X$ (over an algebraic closure of $\mathbb{F}_q)$ plus $1$ (so between $1$ and $9$). $\endgroup$
    – naf
    Commented Aug 18, 2013 at 12:56
  • $\begingroup$ Dear Ulrich, is there any reference for that result (besides reading the proof ...)? Thanks! $\endgroup$
    – Casaubon
    Commented Aug 18, 2013 at 13:43
  • 3
    $\begingroup$ Like I said, Manin's book. Theorem 27.1 and corollary 27.1.1. $\endgroup$
    – Felipe Voloch
    Commented Aug 18, 2013 at 15:02

1 Answer 1

Reset to default
7
$\begingroup$

I hope nobody minds me answering this (very old) question.

Me and my collaborators (Barinder Banwait and Francesc Fité) succeeded in completely answering this question in the paper:

Del Pezzo surfaces over finite fields and their Frobenius traces (https://arxiv.org/abs/1606.00300).

Specifically, let $S$ be a del Pezzo surface of degree $d$ over a finite field of size $q$. Then $$\#S(\mathbb{F}_q) = q^2 + aq + 1$$ for some $a \in \mathbb{Z}$. Then in our paper we completely classified which values of $a$ can occur for fixed $d$ and $q$.

For example for cubic surfaces we have $a \in \{-2,-1,0,1,2,3,4,5,7\}$, and each value occurs over every finite field, except for $a = 7$ which does not occur when $q = 2,3,5$.

Other degrees $d$ are similar, but the list of admissible values of $a$ and the number of exceptional cases in small finite fields is quite long.

Improve this answer
$\endgroup$

You must log in to answer this question.

Not the answer you're looking for? Browse other questions tagged .