Dual Abelian Variety - Construction of The Dual Isogeny

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

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