<< стр. 2(всего 2)СОДЕРЖАНИЕ
polynomials with no common factor. Show that q(x) contains the
product of (x в€’ xQ )2 for all points Q в€€ S with 2Q = в€ћ and that
it contains (x в€’ xQ ) for each point Q в€€ S with 2Q = в€ћ. Conclude
that deg q = #C в€’ 1.
(c) Show that X в€’ 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.

В© 2008 by Taylor & Francis Group, LLC
405
EXERCISES

12.9 Let E be an elliptic curve over a п¬Ѓeld K and let Q в€€ E(K). Translation
by Q gives a map (x, y) в†’ (x, y) + Q = (f (x, y), g(x, y)), and therefore
a homomorphism of п¬Ѓelds

Пѓ : K(x, y) в†’ K(x, y), x в†’ f (x, y), y в†’ g(x, y).

Show that Пѓ has an inverse and therefore that Пѓ is an automorphism of
K(x, y).
12.10 Let E1 , E2 be elliptic curves over a п¬Ѓeld K and let О± : E1 в†’ E2 be an
isogeny such that deg(О±) is not divisible by the characteristic of K.
(a) Suppose E3 is an elliptic curve over K and that ОІ1 : E2 в†’ E3 and
ОІ2 : E2 в†’ E3 are isogenies such that ОІ1 в—¦ О± = ОІ2 в—¦ О±. Show that
ОІ1 = ОІ2 .
(b) Show that the map О± is the unique isogeny E2 в†’ E1 such that
О± в—¦ О± is multiplication by deg О±.
(c) Let f : A в†’ B and g : B в†’ C be surjective homomorphisms
of abelian groups. Show that #Ker(g в—¦ f ) = #Ker(g)#Ker(f ).
Deduce that deg О± = deg О±.
(d) Show that О± в—¦ О± equals multiplication by deg(О±) on E2 . (Hint:
[n] в—¦ О± = О± в—¦ [n] = О± в—¦ О± в—¦ О±; now use (a).)
(e) Show that О± = О±.
12.11 Consider the elliptic curve E : y 2 = x3 в€’ 1 over F7 . It has j-invariant
0.
(a) Show that the 3rd division polynomial (see page 81) is П€3 (x) =
x(x3 в€’ 4).
(b) Show that the subgroups of order 3 on E are
в€љ
C1 = {в€ћ, (0, В±i)}, C2 = {в€ћ, (41/3 , В± 3)},
в€љ в€љ
C3 = {в€ћ, (2 В· 41/3 , В± 3)}, C4 = {в€ћ, (4 В· 41/3 , В± 3)},
в€љ
where i = в€’1 в€€ F49 . Note that 23 = 1 in F7 , so 2 is a cube root
of unity.
(c) Show that the 3rd modular polynomial satisп¬Ѓes О¦3 (0, T ) в‰Ў T (T в€’
3)3 (mod 7).
(d) Let О¶ : E в†’ E by (x, y) в†’ (2x, в€’y). Then О¶ is an endomorphism
of E. Show that C1 is the kernel of the endomorphism 1 + О¶.
Therefore C1 is the kernel of the isogeny 1 + О¶ : E в†’ E. Since
j(E) = 0, this corresponds to the root T = 0 of О¦3 (0, T ) mod 7.
(e) Let П† = П†7 be the 7th power Frobenius map. Show that П† has the
eigenvalue в€’1 on C1 .

В© 2008 by Taylor & Francis Group, LLC
406 CHAPTER 12 ISOGENIES

(f) Show that П†(41/3 ) = 41/3 . (Hint: what x satisfy П†(x) = x?) Con-
clude that none of C2 , C3 , C4 is an eigenspace for П†.
(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 ОЅ в€€ Z
such that П†P = ОЅP 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 deп¬Ѓned over Fq and suppose there is an
isogeny О± : E1 в†’ E2 of degree N deп¬Ѓned over Fq .
qN and let n в‰Ґ 1. Show that О± 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 в†’ C/L2 be a continuous map. This yields a continuous
map f : C в†’ C/L2 such that f (z) = f (z mod L1 ). Let f (0) = z0 . Let
z1 в€€ C and choose a path Оі(t), 0 в‰¤ t в‰¤ 1, from 0 to z1 .
(a) Let 0 в‰¤ t1 в‰¤ 1. Show that there exists a complex-valued continuous
function h(t) deп¬Ѓned in a small interval containing t1 , say (t1 в€’
, t1 + )в€©[0, 1] for some , such that h(t) mod L2 = f (Оі(t)). (Hint:
Represent C/L2 using a translated fundamental parallelogram that
contains f (Оі(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)
п¬Ѓnite set of values t1 < В· В· В· < t1 whose intervals I1 , . . . , In cover
all of [0, 1]. Suppose that for some t0 with 0 в‰¤ t0 < 1, we have
a complex-valued continuous function g(t) on [0, t0 ] such that g(t)
mod L2 = f (Оі(t)). Show that if [0, t0 ] в€© Ij is nonempty, and if h(t)
is the function on Ij constructed in part (a), then there is an в€€ L2
such that g(t) = h(t) в€’ for all t в€€ [0, t0 ] в€© Ij . (Hint: g(t) в€’ h(t) is
continuous and L2 is discrete.)
(c) Show that there exists a continuous function g : [0, 1] в†’ C such
that g(t) mod L2 = f (Оі(t)) for all t в€€ [0, 1].
Лњ
(d) Deп¬Ѓne f (z1 ) = g(1), where z1 and g are as above. Show that this
deп¬Ѓnition is independent of the choice of path Оі. (Hint: Deform
one path into another continuously. The value of g(1) can change
only by a lattice point.)
Лњ Лњ
(e) Show that the construction of f yields a continuous function f :
Лњ
C в†’ C such that f (z mod L1 ) = f (z) mod L2 for all z в€€ C.
12.14 Consider the elliptic curves E1 , E2 in Example 12.5. Use VВґluвЂ™s formulas
e
(Section 12.3) to compute the equations of E1 and E2 . Decide which
has j = в€’1 and which has j = в€’3.

В© 2008 by Taylor & Francis Group, LLC

 << стр. 2(всего 2)СОДЕРЖАНИЕ