# dual isogeny

Given an isogeny $f:E\to {E}^{\prime}$ of elliptic curves^{} of degree $n$, the *dual isogeny* is an isogeny $\widehat{f}:{E}^{\prime}\to E$ of the same degree such that $f\circ \widehat{f}=[n]$. Here $[n]$ denotes the multiplication-by-$n$ isogeny $e\mapsto ne$ which has degree ${n}^{2}$.

Often only the existence of a dual isogeny is needed, but the construction is explicit as

$${E}^{\prime}\to {\mathrm{Div}}^{0}({E}^{\prime})\stackrel{{f}^{*}}{\to}{\mathrm{Div}}^{0}(E)\to E$$ |

where ${\mathrm{Div}}^{0}$ is the group of divisors^{} of degree 0.
To do this, we need maps $E\to {\mathrm{Div}}^{0}(E)$ given by $P\mapsto P-O$ where $O$ is the neutral point of $E$ and ${\mathrm{Div}}^{0}(E)\to E$ given by $\sum {n}_{P}P\mapsto \sum {n}_{P}P$.

To see that $f\circ \widehat{f}=[n]$, note that the original isogeny $f$ can be written as a composite

$$E\to {\mathrm{Div}}^{0}(E)\stackrel{{f}_{*}}{\to}{\mathrm{Div}}^{0}({E}^{\prime})\to {E}^{\prime}$$ |

and that since $f$ is finite of degree $n$, ${f}_{*}{f}^{*}$ is multiplication^{} by $n$ on ${\mathrm{Div}}^{0}({E}^{\prime})$.

Alternatively, we can use the smaller Picard group^{} ${\mathrm{Pic}}^{0}$, a quotient of ${\mathrm{Div}}^{0}$. The map $E\to {\mathrm{Div}}^{0}(E)$ descends to an isomorphism^{}, $E\stackrel{\sim}{\to}{\mathrm{Pic}}^{0}(E)$. The dual isogeny is

$${E}^{\prime}\stackrel{\sim}{\to}{\mathrm{Pic}}^{0}({E}^{\prime})\stackrel{{f}^{*}}{\to}{\mathrm{Pic}}^{0}(E)\stackrel{\sim}{\to}E$$ |

Note that the relation^{} $f\circ \widehat{f}=[n]$ also implies the conjugate relation $\widehat{f}\circ f=[n]$. Indeed, let $\varphi =\widehat{f}\circ f$. Then $\varphi \circ \widehat{f}=\widehat{f}\circ [n]=[n]\circ \widehat{f}$. But $\widehat{f}$ is surjective^{}, so we must have $\varphi =[n]$.

Title | dual isogeny |
---|---|

Canonical name | DualIsogeny |

Date of creation | 2013-03-22 12:52:58 |

Last modified on | 2013-03-22 12:52:58 |

Owner | mathcam (2727) |

Last modified by | mathcam (2727) |

Numerical id | 9 |

Author | mathcam (2727) |

Entry type | Definition |

Classification | msc 14-00 |

Related topic | ArithmeticOfEllipticCurves |