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Let $X$ be a quasi-projective scheme over $\mathbb{F}_q$. If its base change to $\overline{\mathbb{F}}_q$ is an affine space $\mathbb{A}^n$, then it is known that $|X(\mathbb{F}_q)|=q^n$.

The only proof I have seen, for example, 10.11 in Digne--Michel's book Representations of Finite Groups of Lie type, uses Grothendieck's Lefschetz trace formula on \'etale cohomology.

As it seems this result should be known before the development of \'etale cohomology (please correct me if I am wrong), I would like to know if there is any more elementary proof?

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    $\begingroup$ why do you think this should be known? $\endgroup$ – Will Sawin Apr 8 '17 at 20:24
  • $\begingroup$ @WillSawin I dont have a good math explanation for this, but it would surprise me a bit if this is not the case... $\endgroup$ – user148212 Apr 8 '17 at 22:07
  • $\begingroup$ @WillSawin And if I understand it correctly, by a Galois descent argument, the only such variety $X$ is the affine space $\mathbb{A}^n$ right? $\endgroup$ – user148212 Apr 12 '17 at 14:12
  • $\begingroup$ Possibly, but it's not immediately obvious. One would have to compute Galois cohomology of the full group of automorphisms of $\mathbb A^n$, which is different from the groups of linear and affine automorphisms. $\endgroup$ – Will Sawin Apr 12 '17 at 15:10
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    $\begingroup$ Nope, that is just the group of affine transformations. But there are more, for instance $(x,y) \to (x,y+x^2)$. $\endgroup$ – Will Sawin Apr 12 '17 at 15:53

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