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§3. Isogenies in the General Case 239

−1 −1

λ n P P = n P λ (P),
where λ −1 (P) = m(Q 1 +· · · + Q r ) for {Q 1 ,... , Q r }= λ −1 (P) set theoretically

and mr = deg(λ). This induces ˆ λ : E → E the dual isogeny such that the following
diagram is commutative:
λ −1

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

Now the following extension of (1.5).

(3.5) Theorem. The function λ → ˆ λ is a group morphism
Hom(E, E ) → Hom(E , E)

ˆ
satisfying ˆ λ = λ, deg(λ) = deg( ˆ λ), ˆ λλ = nin End(E), and λ ˆ λ = nin End(E ).

For λ in Hom(E, E ) and µ in Hom(E , E ) we have µλ = ˆ λ ˆµ and deg(µλ) =

deg(µ) · deg(λ). The involution λ → ˆ λ of the ring End(E) satisfies ˆn = n where
2
deg(n) = n . The degree functions deg : Hom(E, E ) → Z is a positive quadratic

function.
The involution taking an isogeny to its dual is called the rosati involution on
End(E). The assertion that deg(λ) is a positive quadratic function on Hom(E, E )

means that
(a) deg λ ≥ 0 and by definition deg(λ) = 0 for λ = 0.
2
(b) deg(mλ) = m deg λ,and
(c) deg(λ + µ) = deg(λ) + ( ˆ λµ +ˆµλ) + deg(µ) where (λ, µ) → ˆ λµ +ˆµλ is a
biadditive function defined

Hom(E, E ) × Hom(E, E ) → Z.
The remarks following (1.1) have the following extension.

(3.6) Theorem. Let E be an elliptic curve over a field k.
(1) If k is separably closed and n is prime to the characteristic, then E(k) is divisible
by n, i.e., the map n : E(k) → E(k) is surjective.
(2) If k is separably closed and n is prime to the characteristic, then the subgroup
n E(k) of n-division points is isomorphic to Z/n × Z/n.
(3) If k is algebraically closed of characteristic p > 0,then E(k) is divisible by p.
Moreover, the p-division points form a group p E(k) is isomorphic to either Z/p
or 0.