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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.

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You want $X$ to be geometrically irreducible here.

This is a theorem of Lang and Weil, proven well before the Weil conjectures. It relies only on Weil's proof of the Riemann hypothesis for curves.

The Riemann hypothesis for curves was given an elementary proof by Bombieri and Stepanov.

Combining these should give the elementary proof you seek.

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  • $\begingroup$ Many thanks. Fixed the missing condition. The Lang-Weil estimate seems to be much stronger than what I asked. Also the Riemann hypothesis for curves is not very elementary for my taste. In fact I do not have an elementary answer to my question even for the case of a curve. Is Riemann hypothesis necessary in the latter case? $\endgroup$
    – asv
    Commented Sep 20, 2020 at 13:21
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    $\begingroup$ @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. $\endgroup$
    – Will Sawin
    Commented Sep 20, 2020 at 13:54

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