dual isogeny
Given an isogeny of elliptic curves of degree , the dual isogeny is an isogeny of the same degree such that . Here denotes the multiplication-by- isogeny which has degree .
Often only the existence of a dual isogeny is needed, but the construction is explicit as
where is the group of divisors of degree 0. To do this, we need maps given by where is the neutral point of and given by .
To see that , note that the original isogeny can be written as a composite
and that since is finite of degree , is multiplication by on .
Alternatively, we can use the smaller Picard group , a quotient of . The map descends to an isomorphism, . The dual isogeny is
Note that the relation also implies the conjugate relation . Indeed, let . Then . But is surjective, so we must have .
Title | dual isogeny |
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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 |