I heard that if $\mathbf F_q$ is a finite field, $X, Y$ are birational smooth proper variety over $\mathbf F_q$, then $\#(X(\mathbf F_q)) = \#(Y(\mathbf F_q))$, and I heard that the proof uses crystalline cohomology. Anyone could explain the proof, or is there a reference, thanks.
Number of rational points over finite fields is birational invariant
Aoi Koshigaya
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