Frobenius morphism

Let $K$ be a field of characteristic $p>0$ and let $q=p^{r}$ . Let $C$ be a curve defined over $K$ contained in $\mathbb{P}^{N}$ , the projective space of dimension $N$ . Define the homogeneous ideal of $C$ to be (the ideal generated by):

$I(C)=\{f\in K[X_{0},...,X_{N}]\mid\forall P\in C,\quad f(P)=0,\quad f\text{ is% homogeneous}\}$

For $f\in K[X_{0},...,X_{N}]$ , of the form $f=\sum_{i}a_{i}X_{0}^{i_{0}}...X_{N}^{i_{N}}$ we define $f^{(q)}=\sum_{i}a_{i}^{q}X_{0}^{i_{0}}...X_{N}^{i_{N}}$ . We define a new curve $C^{(q)}$ as the zero set of the ideal (generated by):

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

Definition 1.

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

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

$\phi([x_{0},...,x_{N}])=[x_{0}^{q},...x_{N}^{q}]$

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

$P=[x_{0},...,x_{N}]\in C\Rightarrow\phi(P)=[x_{0}^{q},...x_{N}^{q}]\in C^{(q)}$

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 $g$ is a generator of $I(C^{(q)})$ , i.e. $g$ 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}\text{in % characteristic $p$}]$
	$\displaystyle=$	$\displaystyle(f(P))^{q}$
	$\displaystyle=$	$\displaystyle 0,\quad[P\in C,f\in I(C)]$

as desired.

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

$E=E^{(q)}$

Hence the Frobenius morphism is an endomorphism (or isogeny) of the elliptic curve.

References

1 Joseph H. Silverman, The Arithmetic of Elliptic Curves. Springer-Verlag, New York, 1986.

Title	Frobenius morphism
Canonical name	FrobeniusMorphism
Date of creation	2013-03-22 13:51:45
Last modified on	2013-03-22 13:51:45
Owner	alozano (2414)
Last modified by	alozano (2414)
Numerical id	4
Author	alozano (2414)
Entry type	Definition
Classification	msc 14H37
Related topic	FrobeniusAutomorphism
Related topic	FrobeniusMap
Related topic	ArithmeticOfEllipticCurves
Defines	Frobenius morphism