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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
. 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
.
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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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