# Asymptotic estimate of the number of points of variety over finite field

EDIT: Let $$X$$ be a geometrically irreducible $$n$$-dimensional variety over finite field $$\mathbb{F}_{q_0}$$. Let $$\mathbb{F}_q$$ denote any finite extension of $$\mathbb{F}_{q_0}$$.

It is known (e.g. follows from the Weil conjectures) that $$\frac{|X(\mathbb{F}_q)|}{q^n}\to 1 \mbox{ as } q\to\infty.$$ I am wondering if there is an elementary proof of this fact.

You want $$X$$ to be geometrically irreducible here.
• @makt For $X$ of dimension $1$, one can prove the weaker estimate by analytic number theory methods a la the usual proof of the prime number theorem over the integers (see Proposition 5.13 of Rosen's Number Theory in Function Fields). Maybe there is an analogue of the Erdős-Selberg elementary proof using Selberg's symmetry formula as well. – Will Sawin Sep 20 at 13:54