# About points on affine curves defined over finite fields

So let $\mathbf{F}_q$ be a finite field with $q$ elements where $q=p^m$,

$p$ a prime number and $m\in\mathbf{Z}_{\geq 1}$. Let $f(x,y)\in \mathbf{F}_q[x,y]$

be a smooth non-constant polynomial and let $A:=\mathbf{F}_q[x,y]/(f)$.

Q: Does there exists an integer $N_0$ (which depends on $A$) such that for $N\geq N_0$ one

may always find a prime ideal $P\subseteq A$ such that $A/P\simeq\mathbf{F}_{q^N}$?

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Well now that I think it is obvious. If $C$ is the corresponding curve then we have an exact formula for $\#C(\mathbb{F}_{q^n})$. It is $q^n+1+(error term)$. Nous using weak Weil estimates we are done. –  Hugo Chapdelaine Feb 9 '11 at 17:44
and I should say that for a proper divisor $d|n$ $\#C(\mathbb{F}_{q^d})=q^d+1+(error term)$ so weak estimates imply that $C(\mathbb{F}_{q^n})-\bigcup_{d|n,d\neq n}C(\mathbb{F}_{q^d})$ is larger than 1. –  Hugo Chapdelaine Feb 9 '11 at 18:23