# 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):

$$I(C)=\{f\in K[{X}_{0},\mathrm{\dots},{X}_{N}]\mid \forall P\in C,f(P)=0,f\text{is homogeneous}\}$$ |

For $f\in K[{X}_{0},\mathrm{\dots},{X}_{N}]$, of the form $f={\sum}_{i}{a}_{i}{X}_{0}^{{i}_{0}}\mathrm{\dots}{X}_{N}^{{i}_{N}}$ we define ${f}^{(q)}={\sum}_{i}{a}_{i}^{q}{X}_{0}^{{i}_{0}}\mathrm{\dots}{X}_{N}^{{i}_{N}}$. 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 ${q}^{t\mathit{}h}$-power Frobenius morphism is defined to be:

$$\varphi :C\to {C}^{(q)}$$ |

$$\varphi ([{x}_{0},\mathrm{\dots},{x}_{N}])=[{x}_{0}^{q},\mathrm{\dots}{x}_{N}^{q}]$$ |

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

$$P=[{x}_{0},\mathrm{\dots},{x}_{N}]\in C\Rightarrow \varphi (P)=[{x}_{0}^{q},\mathrm{\dots}{x}_{N}^{q}]\in {C}^{(q)}$$ |

This is equivalent to
proving that for any $g\in I({C}^{(q)})$ we have $g(\varphi (P))=0$.
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:

$g(\varphi (P))={f}^{(q)}(\varphi (P))$ | $=$ | ${f}^{(q)}([{x}_{0}^{q},\mathrm{\dots},{x}_{N}^{q}])$ | ||

$=$ | ${(f([{x}_{0},\mathrm{\dots},{x}_{N}]))}^{q},[{a}^{q}+{b}^{q}={(a+b)}^{q}\text{in characteristic}p]$ | |||

$=$ | ${(f(P))}^{q}$ | |||

$=$ | $0,[P\in C,f\in I(C)]$ |

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 |