Take the 2-minute tour ×
MathOverflow is a question and answer site for professional mathematicians. It's 100% free, no registration required.

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.

share|improve this question
4  
Del Pezzo surfaces are rational and you can work out their zeta function explicitly. This should be in Manin's book on cubic surfaces. –  Felipe Voloch Aug 18 '13 at 12:48
4  
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 $9$). –  ulrich Aug 18 '13 at 12:56
    
Dear Ulrich, is there any reference for that result (besides reading the proof ...)? Thanks! –  Casaubon Aug 18 '13 at 13:43
3  
Like I said, Manin's book. Theorem 27.1 and corollary 27.1.1. –  Felipe Voloch Aug 18 '13 at 15:02
add comment

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Browse other questions tagged or ask your own question.