Isogenies - Elliptic Curves: Number Theory and Cryptography - page 404

12.9 Let E be an elliptic curve over a field K and let Q ∈ E ( K ). Translation

by Q gives a map ( x, y ) → ( x, y )+ Q =( f ( x, y ) ,g ( x, y )), and therefore

a homomorphism of fields

σ : K ( x, y ) → K ( x, y ) , x → f ( x, y ) , y → g ( x, y ) .

Show that σ has an inverse and therefore that σ is an automorphism of
K ( x, y ).

12.10 Let E 1 ,E 2 be elliptic curves over a field K and let α : E 1 → E 2 be an
isogeny such that deg( α ) is not divisible by the characteristic of K .

(a) Suppose E 3 is an elliptic curve over K and that β 1 : E 2 → E 3 and
β 2 : E 2 → E 3 are isogenies such that β 1 ◦ α = β 2 ◦ α . Show that

β 1 = β 2 .
(b) Show that the map α is the unique isogeny E 2 → E 1 such that

α is multiplication by deg α .
(c) Let f : A → B and g : B → C be surjective homomorphisms
of abelian groups.

α ◦

Show that #Ker( g ◦ f )=#Ker( g )#Ker( f ).

α =deg α .
(d) Show that α ◦ α equals multiplication by deg( α )on E 2 . Hint:
[ n ] ◦ α = α ◦ [ n ]= α ◦ α ◦ α ;nowuse(a).)
(e) Show that α = α .

Deduce that deg

12.11 Consider the elliptic curve E : y 2 = x 3 − 1over F 7 .Ithas j -invariant

0.

(a) Show that the 3rd division polynomial (see page 81) is ψ 3 ( x )=
x ( x 3 − 4).

(b) Show that the subgroups of order 3 on E are

C 2 = {∞, (4 1 / 3 , ± √ 3) }, C 1 = {∞, (0 , ±i ) },

± √ 3) ± √ 3) 4 1 / 3 , 4 1 / 3 , ,

where i = √ − 1 ∈ F 49 . Note that 2 3 =1in F 7 ,so2isacuberoot
of unity.
(c) Show that the 3rd modular polynomial satisfies Φ 3 (0 ,T ) ≡ T ( T −
3) 3 (mod 7).
(d) Let ζ : E → E by ( x, y ) → (2 x, −y ). Then ζ is an endomorphism
of E . Show that C 1 is the kernel of the endomorphism 1 + ζ .
Therefore C 1 is the kernel of the isogeny 1 + ζ : E → E .Since
j ( E ) = 0, this corresponds to the root T =0ofΦ 3 (0 ,T )mod7.
(e) Let φ = φ 7 be the 7th power Frobenius map. Show that φ has the
eigenvalue − 1on C 1 .

C 3 = {∞ , (2 · } , C 4 = {∞ , (4 · }