dual isogeny

Given an isogeny f:EE of elliptic curvesMathworldPlanetmath of degree n, the dual isogeny is an isogeny f^:EE of the same degree such that ff^=[n]. Here [n] denotes the multiplication-by-n isogeny ene which has degree n2.

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


where Div0 is the group of divisorsMathworldPlanetmathPlanetmathPlanetmath of degree 0. To do this, we need maps EDiv0(E) given by PP-O where O is the neutral point of E and Div0(E)E given by nPPnPP.

To see that ff^=[n], note that the original isogeny f can be written as a composite


and that since f is finite of degree n, f*f* is multiplicationPlanetmathPlanetmath by n on Div0(E).

Alternatively, we can use the smaller Picard groupMathworldPlanetmath Pic0, a quotient of Div0. The map EDiv0(E) descends to an isomorphismMathworldPlanetmathPlanetmathPlanetmathPlanetmathPlanetmathPlanetmath, EPic0(E). The dual isogeny is


Note that the relationMathworldPlanetmathPlanetmath ff^=[n] also implies the conjugate relation f^f=[n]. Indeed, let ϕ=f^f. Then ϕf^=f^[n]=[n]f^. But f^ is surjectivePlanetmathPlanetmath, so we must have ϕ=[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