Frobenius endomorphism

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An endomorphism $\phi : X \rightarrow X$ of a scheme $X$ over the finite field $\mathbf{F}_q$ of $q$ elements such that $\phi$ is the identity mapping on $X(\mathbf{F}_q)$, and the mapping of the structure sheaf $\phi^* : \mathcal{O}_X \rightarrow \mathcal{O}_X$ is that of raising to the $q$-th power. The Frobenius endomorphism is a purely-inseparable morphism and has zero differential. For an affine variety $X \subset \mathbf{A}^n$ defined over $\mathbf{F}_q$, the Frobenius endomorphism $\phi$ takes the point $(x_1,\ldots,x_n)$ to $(x_1^q,\ldots,x_n^q)$.

The number of geometric points of $X$ that are defined over $\mathbf{F}_q$ is the same as the number of fixed points of $\phi$, which enables one to use the Lefschetz formula to determine the number of such points.


[1] R. Hartshorne, "Algebraic geometry" , Springer (1977) MR0463157 Zbl 0367.14001


Here, $X(\mathbf{F}_q)$ is the set of $\mathbf{F}_q$-points of $X$, i.e. the set of points of $X$ that are defined over $\mathbf{F}_q$.


[a1] P. Gabriel, "Etude infinitésimal des schémas en groupes" M. Demazure (ed.) A. Grothendieck (ed.) , SGA 3. Exp. VII , Lect. notes in math. , 151 , Springer (1970)
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Frobenius endomorphism. Encyclopedia of Mathematics. URL:
This article was adapted from an original article by L.V. Kuz'min (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article