product of (x ’ xQ )2 for all points Q ∈ S with 2Q = ∞ and that

it contains (x ’ xQ ) for each point Q ∈ S with 2Q = ∞. Conclude

that deg q = #C ’ 1.

(c) Show that X ’ x has the form r(x)/q(x) with deg r < deg q.

(d) Use the fact that

r(x)

p(x)

=x+

q(x) q(x)

to prove that deg p = #C. This shows that the isogeny constructed

in Theorem 12.16 is separable.

© 2008 by Taylor & Francis Group, LLC

405

EXERCISES

12.9 Let E be an elliptic curve over a ¬eld 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 ¬elds

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 E1 , E2 be elliptic curves over a ¬eld K and let ± : E1 ’ E2 be an

isogeny such that deg(±) is not divisible by the characteristic of K.

(a) Suppose E3 is an elliptic curve over K and that β1 : E2 ’ E3 and

β2 : E2 ’ E3 are isogenies such that β1 —¦ ± = β2 —¦ ±. Show that

β1 = β2 .

(b) Show that the map ± is the unique isogeny E2 ’ E1 such that

± —¦ ± is multiplication by deg ±.

(c) Let f : A ’ B and g : B ’ C be surjective homomorphisms

of abelian groups. Show that #Ker(g —¦ f ) = #Ker(g)#Ker(f ).

Deduce that deg ± = deg ±.

(d) Show that ± —¦ ± equals multiplication by deg(±) on E2 . (Hint:

[n] —¦ ± = ± —¦ [n] = ± —¦ ± —¦ ±; now use (a).)

(e) Show that ± = ±.

12.11 Consider the elliptic curve E : y 2 = x3 ’ 1 over F7 . It has j-invariant

0.

(a) Show that the 3rd division polynomial (see page 81) is ψ3 (x) =

x(x3 ’ 4).

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

√

C1 = {∞, (0, ±i)}, C2 = {∞, (41/3 , ± 3)},

√ √

C3 = {∞, (2 · 41/3 , ± 3)}, C4 = {∞, (4 · 41/3 , ± 3)},

√

where i = ’1 ∈ F49 . Note that 23 = 1 in F7 , so 2 is a cube root

of unity.

(c) Show that the 3rd modular polynomial satis¬es ¦3 (0, T ) ≡ T (T ’

3)3 (mod 7).

(d) Let ζ : E ’ E by (x, y) ’ (2x, ’y). Then ζ is an endomorphism

of E. Show that C1 is the kernel of the endomorphism 1 + ζ.

Therefore C1 is the kernel of the isogeny 1 + ζ : E ’ E. Since

j(E) = 0, this corresponds to the root T = 0 of ¦3 (0, T ) mod 7.

(e) Let φ = φ7 be the 7th power Frobenius map. Show that φ has the

eigenvalue ’1 on C1 .

© 2008 by Taylor & Francis Group, LLC

406 CHAPTER 12 ISOGENIES

(f) Show that φ(41/3 ) = 41/3 . (Hint: what x satisfy φ(x) = x?) Con-

clude that none of C2 , C3 , C4 is an eigenspace for φ.

(g) Let E1 be an elliptic curve with j = 3 that is 3-isogenous to E (it

exists by Theorem 12.19). Show that there does not exist ν ∈ Z

such that φP = νP for all P in the kernel of the isogeny. This

shows that the restriction j = 0 is needed in Proposition 12.20.

12.12 Let E1 , E2 be elliptic curves de¬ned over Fq and suppose there is an

isogeny ± : E1 ’ E2 of degree N de¬ned over Fq .

qN and let n ≥ 1. Show that ± gives

(a) Let be a prime such that

an isomorphism E1 [ ] E2 [ n ].

n

(b) Use Proposition 4.11 to show that #E1 (Fq ) = #E2 (Fq ).

12.13 Let f : C/L1 ’ C/L2 be a continuous map. This yields a continuous

map f : C ’ C/L2 such that f (z) = f (z mod L1 ). Let f (0) = z0 . Let

z1 ∈ C and choose a path γ(t), 0 ¤ t ¤ 1, from 0 to z1 .

(a) Let 0 ¤ t1 ¤ 1. Show that there exists a complex-valued continuous

function h(t) de¬ned in a small interval containing t1 , say (t1 ’

, t1 + )©[0, 1] for some , such that h(t) mod L2 = f (γ(t)). (Hint:

Represent C/L2 using a translated fundamental parallelogram that

contains f (γ(t1 )) in its interior.)

(b) As t1 runs through [0, 1], the small intervals in part (a) give a

covering of the interval [0, 1]. Since [0, 1] is compact, there is a

(1) (n)

¬nite set of values t1 < · · · < t1 whose intervals I1 , . . . , In cover

all of [0, 1]. Suppose that for some t0 with 0 ¤ t0 < 1, we have

a complex-valued continuous function g(t) on [0, t0 ] such that g(t)

mod L2 = f (γ(t)). Show that if [0, t0 ] © Ij is nonempty, and if h(t)

is the function on Ij constructed in part (a), then there is an ∈ L2

such that g(t) = h(t) ’ for all t ∈ [0, t0 ] © Ij . (Hint: g(t) ’ h(t) is

continuous and L2 is discrete.)

(c) Show that there exists a continuous function g : [0, 1] ’ C such

that g(t) mod L2 = f (γ(t)) for all t ∈ [0, 1].

˜

(d) De¬ne f (z1 ) = g(1), where z1 and g are as above. Show that this

de¬nition is independent of the choice of path γ. (Hint: Deform

one path into another continuously. The value of g(1) can change

only by a lattice point.)

˜ ˜

(e) Show that the construction of f yields a continuous function f :

˜

C ’ C such that f (z mod L1 ) = f (z) mod L2 for all z ∈ C.

12.14 Consider the elliptic curves E1 , E2 in Example 12.5. Use V´lu™s formulas

e

(Section 12.3) to compute the equations of E1 and E2 . Decide which

has j = ’1 and which has j = ’3.

© 2008 by Taylor & Francis Group, LLC