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234 12. Endomorphisms of Elliptic Curves

(1.1) Remarks. A nonzero analytic homomorphism λ : C/L → C/L is surjective,
L /L = L /λL is a finite subgroup of order n = [L : λL], the
and ker(λ) = λ −1 ∼

index of λL in L . Note that nL ⊂ λL ⊂ L and (n/λ) : T = C/L → C/L = T
is an isogeny which composed with λ gives n : T → T , i.e., multiplication by the
integer n.
(1.2) Definition. An isogeny between two complex tori, or elliptic curves, is a
nonzero analytic homomorphism. For complex tori the degree, denoted deg(λ),is

the index [L : λL] for λ : T = C/L → C/L = T . The dual isogeny ˆ λ of λ is
defined to be ˆ λ = n/λ : T → T . The dual of 0 is defined to be 0. See also 9(1, Ex.

3).

(1.3) Proposition. The function λ → ˆ λ is a group morphism Hom(T, T ) →
ˆ

Hom(T , T ) satisfying ˆ λ = λ, deg(λ) = deg( ˆ λ) = n, ˆ λλ = nin End(T ), and λ ˆ λ = n

in End(T ).For λ ∈ Hom(T , T ) and µ ∈ Hom(T, T ) we have (µλ) = ˆ λ ˆµ.The

involution λ → ˆ λ of the ring End(T ), called the Rosati involution, satisfies ˆn = n.
2
Finally, deg(n) = n and the function deg : End(T ) → Z is a positive quadratic
2
function where deg(λ) = λ ˆ λ =|λ| .
Proof. Let a(L) denote the area of a period parallelogram associated with the lat-
2

tice L.Thenwehave a(λL) =|λ| a(L) and for L ⊂ L the index [L : L ] =

a(L )/a(L ). Hence for an isogeny λ : C/L → C/L the degree n = deg(λ) =
2

[L : λL] equals |λ| [a(L)/a(L )] which shows that the dual isogeny is given by the
formula
a(L)
ˆ λ = ¯ λ

a(L )
from which the proposition follows directly.
There is a formula for a(L) when L = Zω 1 + Zω 2 , namely
1
a(L) = |ω 1 ¯ω 2 − ω 2 ¯ω 1 | .
2

If ω 1 ,ω 2 is changed to ω = aω 1 + bω 2 , ω = cω 1 + dω 2 , then
1 2

ab

= det · |ω 1 ¯ω 2 − ω 2 ¯ω 1 | ,
ω ¯ω − ω ¯ω
1 2 2 1
cd
and, therefore, this expression for a(L) is independent of a basis of L. From this
2
formula a(λL) =|λ| a(L). Finally, for L τ we calculate directly that a(L τ ) =
(1/2)|τ −¯τ|=|Im(τ)|, which is height times base, namely the area.
(1.4) Definition. For a natural number N and N-division point on an elliptic curve
E(k) or a complex torus T is a solution x to the equation Nx = 0. The N-division
points of E (resp. T ) form a subgroup denoted by N E (resp. N T ).