PlanetMath: Frobenius morphism

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Frobenius morphism

(Definition)

Let be a field of characteristic and let . Let be a curve defined over contained in $\mathbb{P}^N$ , the projective space of dimension . Define the homogeneous ideal of to be (the ideal generated by):

$\displaystyle I(C)=\{f\in K[X_0,...,X_N] \mid \forall P \in C,\quad f(P)=0,\quad f$ is homogeneous $\displaystyle \}$

For $f\in K[X_0,...,X_N]$ , of the form $f=\sum_i a_iX_0^{i_0}...X_N^{i_N}$ we define $f^{(q)}=\sum_i a_i^qX_0^{i_0}...X_N^{i_N}$ . We define a new curve $C^{(q)}$ as the zero set of the ideal (generated by):

$\displaystyle I(C^{(q)})=\{f^{(q)}\mid f\in I(C)\}$

Definition 1 The $q^{th}$ -power Frobenius morphism is defined to be:

$\displaystyle \phi\colon C\to C^{(q)}$

$\displaystyle \phi([x_0,...,x_N])=[x_0^q,...x_N^q]$

In order to check that the Frobenius morphism is well defined we need to prove that

$\displaystyle P=[x_0,...,x_N]\in C \Rightarrow \phi(P)=[x_0^q,...x_N^q]\in C^{(q)}$

This is equivalent to proving that for any $g \in I(C^{(q)})$ we have $g(\phi(P))=0$ . Without loss of generality we can assume that

is a generator of $I(C^{(q)})$ , i.e.

is of the form $g=f^{(q)}$ for some $f\in I(C)$ . Then:

$\displaystyle g(\phi(P))=f^{(q)}(\phi(P))$	$\displaystyle =$	$\displaystyle f^{(q)}([x_0^q,...,x_N^q])$
	$\displaystyle =$	$\displaystyle (f([x_0,...,x_N]))^q,\quad [a^q+b^q=(a+b)^q$ in characteristic $\displaystyle ]$
	$\displaystyle =$	$\displaystyle (f(P))^q$
	$\displaystyle =$	$\displaystyle 0,\quad [P\in C, f\in I(C)]$

as desired.

Example: Suppose is an elliptic curve defined over $K=\mathbb{F}_q$ , the field of elements. In this case the Frobenius map is an automorphism of , therefore