 The definition of "K"("L") has to be in terms of a group scheme that is a scheme-theoretic stabilizer, and the quotient taken is now a quotient by a subgroup scheme.Given an isogeny

of elliptic curves of degree formula_4, the dual isogeny is an isogeny

of the same degree such that

Here formula_7 denotes the multiplication-by-formula_4 isogeny formula_9 which has degree formula_10

Often only the existence of a dual isogeny is needed, but it can be explicitly given as the composition

where formula_12 is the group of divisors of degree 0.To do this, we need maps formula_13 given by formula_14 where formula_15 is the neutral point of formula_16 and formula_17 given by formula_18

To see that formula_19, note that the original isogeny formula_20 can be written as a composite

and that since formula_20 is of degree formula_4, formula_24 is multiplication by formula_4 on formula_26

Alternatively, we can use the smaller Picard group formula_27, a quotient of formula_28 The map formula_29 descends to an isomorphism, formula_30 The dual isogeny is

Note that the relation formula_19 also implies the conjugate relation formula_33 Indeed, let formula_34 Then formula_35 But formula_36 is surjective, so we must have formula_37

The product of an abelian variety and its dual has a canonical line bundle, called the Poincaré line bundle.