Frobenius morphism

Let K be a field of characteristicPlanetmathPlanetmath p>0 and let q=pr. Let C be a curve defined over K contained in N, the projective spaceMathworldPlanetmath of dimension N. Define the homogeneous idealMathworldPlanetmath of C to be (the ideal generated by):

I(C)={fK[X0,,XN]PC,f(P)=0,f is homogeneous}

For fK[X0,,XN], of the form f=iaiX0i0XNiN we define f(q)=iaiqX0i0XNiN. We define a new curve C(q) as the zero setMathworldPlanetmathPlanetmath of the ideal (generated by):

Definition 1.

The qth-power Frobenius morphism is defined to be:


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


This is equivalent to proving that for any gI(C(q)) we have g(ϕ(P))=0. Without loss of generality we can assume that g is a generatorPlanetmathPlanetmathPlanetmathPlanetmathPlanetmath of I(C(q)), i.e. g is of the form g=f(q) for some fI(C). Then:

g(ϕ(P))=f(q)(ϕ(P)) = f(q)([x0q,,xNq])
= (f([x0,,xN]))q,[aq+bq=(a+b)qin characteristic p]
= (f(P))q
= 0,[PC,fI(C)]

as desired.

Example: Suppose E is an elliptic curveMathworldPlanetmath defined over K=𝔽q, the field of pr elements. In this case the Frobenius map is an automorphismPlanetmathPlanetmathPlanetmathPlanetmath of K, therefore


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


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