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dual isogeny
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(Definition)
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Given an isogeny
 of elliptic curves of degree  , the dual isogeny is an isogeny
 of the same degree such that
![$ f \circ \hat{f} = [n]$](http://images.planetmath.org:8080/cache/objects/3226/l2h/img4.png "$ f \circ \hat{f} = [n]$") . Here ![$ [n]$](http://images.planetmath.org:8080/cache/objects/3226/l2h/img5.png "$ [n]$") denotes the multiplication-by- isogeny
 which has degree  .
Often only the existence of a dual isogeny is needed, but the construction is explicit as
where
 is the group of divisors of degree 0. To do this, we need maps
 given by
 where  is the neutral point of  and
 given by
 .
To see that
![$ f \circ \hat{f} = [n]$](http://images.planetmath.org:8080/cache/objects/3226/l2h/img17.png "$ f \circ \hat{f} = [n]$") , note that the original isogeny  can be written as a composite
and that since  is finite of degree  ,  is multiplication by  on
 .
Alternatively, we can use the smaller Picard group
 , a quotient of
 . The map
 descends to an isomorphism,
 . The dual isogeny is
Note that the relation
![$ f \circ \hat{f} = [n]$](http://images.planetmath.org:8080/cache/objects/3226/l2h/img30.png "$ f \circ \hat{f} = [n]$") also implies the conjugate relation
![$ \hat{f} \circ f = [n]$](http://images.planetmath.org:8080/cache/objects/3226/l2h/img31.png "$ \hat{f} \circ f = [n]$") . Indeed, let
 . Then
![$ \phi \circ \hat{f} = \hat{f} \circ [n] = [n] \circ \hat{f}$](http://images.planetmath.org:8080/cache/objects/3226/l2h/img33.png "$ \phi \circ \hat{f} = \hat{f} \circ [n] = [n] \circ \hat{f}$") . But  is surjective, so we must have
![$ \phi = [n]$](http://images.planetmath.org:8080/cache/objects/3226/l2h/img35.png "$ \phi = [n]$") .
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"dual isogeny" is owned by mathcam. [ full author list (2) | owner history (1) ]
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(view preamble)
Cross-references: surjective, conjugate, implies, relation, isomorphism, quotient, Picard group, multiplication, finite, composite, point, maps, divisors, group, degree, elliptic curves, isogeny
There are 2 references to this entry.
This is version 6 of dual isogeny, born on 2002-07-29, modified 2006-03-03.
Object id is 3226, canonical name is DualIsogeny2.
Accessed 2468 times total.
Classification:
| AMS MSC: | 14-00 (Algebraic geometry :: General reference works ) |
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Pending Errata and Addenda
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