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238 12. Endomorphisms of Elliptic Curves

(3.3) Definition. Let E and E be two elliptic curves over k.Anisogeny λ : E →
E is a nonzero rational map over k with λ(0) = 0. This means λ is givenbyan

embedding k(E) ← k(E ) which we usually view as an inclusion. The degree of λ,

denoted deg(λ),is[k(E) : k(E )].
In arbitrary characteristic, the degree of a field extension is the product of the
separable degree and the purely inseparable degree so that

deg(λ) = [k(E) : k(E )] = [k(E) : k(E )] s [k(E) : k(E )] i
= deg(λ) s deg(λ) i .

The separable degree deg(λ) s is the order of the group of geometric points in
ker(λ) = λ −1 (0) as a variety while deg(λ) i is related to the scheme theoretical struc-
ture of ker(λ). Multiplication by p in characteristic p is always inseparable.
The condition λ(0) = 0 can be understood in terms of the function field k(E)

k(x , y ), where x , y satisfy a Weierstrass equation. On E the functions x and y ,
i.e., viewed in k(E),musthaveapoleat0for λ(0) = 0 to hold.
In order to see that in isogeny λ is automatically additive and to define the dual

isogeny ˆ λ, we have to look at divisors on E. These are finite formal sums n P P of
points on E over an algebraically closed field k, and the degree of a divisor is given
by

deg n P P = n P , an integer.
∗
If f is in the multiplicative group k(E) , then the divisor ( f ) of zeros and poles is
defined, and, as on the projective line, we have
deg( f ) = 0.
We have three groups Div(E) ⊃ Div 0 (E) ⊃ Div 1 (E) of all divisors, all divisors of
degree 0, and all divisors of fnctions. The sum function s :Div 0 (E) → E(k) where

s( n P P) = n P P in E(k) has kernel Div 1 (E) and is surjective, see 9(3.5) for
the proof over the complex numbers. The function P → P − 0 is a cross-section of
s. Thus P 1 + P 2 + P 3 = 0in E(k) if and only if P 1 + P 2 + P 3 =−3 · 0 is the divisor
of a function.

(3.4) Remarks/Definition. An isogeny λ : E → E defines a group morphism

λ :Div(E) → Div(E ) by λ( n P P) = n P λ(P), and the following diagram is
commutative:
λ

Div 0 (E) −−−−→ Div 0 (E )
 
s s


λ
E(k) −−−−→ E (k)

showing that λ is additive. Further, λ defines λ −1 :Div 0 (E ) → Div 0 (E) by