About points on affine curves defined over finite fields - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-24T16:33:29Z http://mathoverflow.net/feeds/question/54881 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/54881/about-points-on-affine-curves-defined-over-finite-fields About points on affine curves defined over finite fields Hugo Chapdelaine 2011-02-09T13:50:02Z 2011-02-09T14:02:38Z <p>So let $\mathbf{F}_q$ be a finite field with $q$ elements where $q=p^m$, </p> <p>$p$ a prime number and $m\in\mathbf{Z}_{\geq 1}$. Let $f(x,y)\in \mathbf{F}_q[x,y]$</p> <p>be a smooth non-constant polynomial and let $A:=\mathbf{F}_q[x,y]/(f)$. </p> <p>Q: Does there exists an integer $N_0$ (which depends on $A$) such that for $N\geq N_0$ one </p> <p>may always find a prime ideal $P\subseteq A$ such that $A/P\simeq\mathbf{F}_{q^N}$? </p> http://mathoverflow.net/questions/54881/about-points-on-affine-curves-defined-over-finite-fields/54884#54884 Answer by Charles Matthews for About points on affine curves defined over finite fields Charles Matthews 2011-02-09T14:02:38Z 2011-02-09T14:02:38Z <p>The answer is "yes", given what we know about the number of points on the corresponding projective curve (or more accurately a smooth model of it, in case of singularities). Since points at infinity and singular points can be bounded terms of the degree of <em>f</em>, they don't affect the asymptotics here. So the question comes down to the existence of points with a given field of definition, of size a large power of <em>q</em>. The calculation that the main term gives the right answer (>0) can be read off the case of the affine line, where it is certainly true.</p>