# dual isogeny

Given an isogeny $f:E\rightarrow E^{\prime}$ of elliptic curves of degree $n$, the dual isogeny is an isogeny $\hat{f}:E^{\prime}\rightarrow E$ of the same degree such that $f\circ\hat{f}=[n]$. 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

 $E^{\prime}\rightarrow\operatorname{Div}^{0}(E^{\prime})\lx@stackrel{{% \scriptstyle f^{*}}}{{\rightarrow}}\operatorname{Div}^{0}(E)\rightarrow E$

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

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

 $E\rightarrow\operatorname{Div}^{0}(E)\lx@stackrel{{\scriptstyle f_{*}}}{{% \rightarrow}}\operatorname{Div}^{0}(E^{\prime})\rightarrow E^{\prime}$

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

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

 $E^{\prime}\lx@stackrel{{\scriptstyle\sim}}{{\rightarrow}}\operatorname{Pic}^{0% }(E^{\prime})\lx@stackrel{{\scriptstyle f^{*}}}{{\rightarrow}}\operatorname{% Pic}^{0}(E)\lx@stackrel{{\scriptstyle\sim}}{{\rightarrow}}E$

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

Title dual isogeny DualIsogeny 2013-03-22 12:52:58 2013-03-22 12:52:58 mathcam (2727) mathcam (2727) 9 mathcam (2727) Definition msc 14-00 ArithmeticOfEllipticCurves