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polynomials with no common factor. Show that q(x) contains the
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

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