Construction of The Dual Isogeny
Often only the existence of a dual isogeny is needed, but it can be explicitly given as the composition
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
Read more about this topic: Dual Abelian Variety
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