MA426–ELLIPTIC CURVES
SPRING 2016
Exercise Sheet 4
Exercise 1∗ . Let E be an elliptic curve, and α ∈ End(E). The endomorphism α satisfies the
quadratic polynomial α2 − tα + d = 0, where t = 2hα, 1i and d = deg(α).
(a) Show that t2 ≤ 4d.
(b) Show that t2 = 4d if and only if α is multiplication-by-m for some m ∈ Z.
Exercise 2∗ . Let E be the elliptic curve y 2 = x3 + x + 1 over F7 , and let ϕ be the Frobenius
endomorphism of E. Find the integer t such that ϕ2 − tϕ + 7 = 0.
Exercise 3∗ . Let E be the elliptic curve y 2 + y = x3 + 2x2 + 3 over F5 , and let ϕ be the
Frobenius endomorphism of E. List the points in E(F5 ) and compute a5 . List the points in
E(F25 ) and compute a25 . What is the relation between a5 and a25 ?
Exercise 4. (30 points) Let q ≥ 3 be a prime power, and n ≥ 1 an integer. Let E : y 2 =
x3 + ax + b, with a, b ∈ Fq , be an elliptic curve. Let φq be the Frobenius endomorphism of E.
We recall that φq satisfies a quadratic polynomial x2 − aq x + q = (x − α)(x − β) = 0, where
α, β are in an imaginary quadratic field.
(a) (5 points) Show that φqn = φnq , and that φnq − 1 is separable. Deduce that #E(Fqn ) =
deg(φnq − 1).
(b) (5 points) Show that x2 − aq x + q divides Q(x) = x4 − (α2 + β 2 )x2 + q 2 . [Hint: first show
that α2 + β 2 is an integer. Can you factor Q(x)?]
(c) (5 points) Show that φq2 satisfies the polynomial x2 − (α2 + β 2 )x + q 2 . Deduce that
#E(Fq2 ) = q 2 + 1 − (α2 + β 2 ).
(d) (5 points) Show that φqn satisfies the polynomial x2 − (αn + β n )x + q n . Deduce that
#E(Fqn ) = q n + 1 − (αn + β n ).
(e) (10 points) Let E be the elliptic curve over F3 given by y 2 = x3 + x2 + x + 1. List the
points in E(F3 ), E(F9 ). Compute #E(F27 ) without listing the points.
Exercise 5. (20 points) Let q ≥ 3 be a prime power. Let E : y 2 = x3 + ax + b, with a, b ∈ Fq ,
be an elliptic curve. Let φq be the Frobenius endomorphism of E. We recall that φq satisfies
a quadratic polynomial x2 − aq x + q = (x − α)(x − β) = 0, where α, β are in an imaginary
quadratic field.
(a) (10 points) Show that ZE (T ), the Z-function associated to E, satisfies the following
equality
!
∞
X
#E(Fqn ) n
qT 2 − aq T + 1
ZE (T ) : = exp
T
=
,
n
(1 − T )(1 − qT )
n=1
P
tn
where #E(Fqn ) = q n + 1 − aqn . [Hint: − log(1 − t) = ∞
n=1 n .]
1
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MA426–ELLIPTIC CURVES
SPRING 2016
(b) (10 points) The zeta function associated to E is defined as
ζE (s) : = ZE (q −s )
for s ∈ C (zeta functions are extremely interesting and important in number theory,
if you want to read about this check http://www.lmfdb.org/L/history). Prove that
ζE (s) = ζE (1 − s) and that if ζE (s) = 0 then <(s) = 21 .
Exercise 6∗ . Let E be a supersingular elliptic curve defined over Fp with p ≥ 5 prime (i.e.
#E(Fp ) = p + 1). Let φp be the Frobenius endomorphism on E.
(a) Show that φ2p + p = 0 as endomorphisms.
(b) Show that φ2p acts as the identity on E[p + 1], and therefore that E[p + 1] ⊂ E(Fp2 ).
(c) Show that E[p + 1] = E(Fp2 ) ∼
= Z/(p + 1)Z × Z/(p + 1)Z.
Exercise 7∗ .
(a) Is there an elliptic curve E over F5 such that #E(F25 ) = 22?
(b) Let p be a prime and let E be an elliptic curve over Fp such that #E(Fq ) = n ∈ Z>0 ,
where q = pe with e ∈ Z>0 . In which interval, depending on n, is the prime power q?
(c) Let q = p2 where p is a prime. Find a p and an equation of an elliptic curve E over Fp
such that #E(Fq ) = 20. Justify your answer.
Exercise 8. (20 points)
(a) (10 points) Let E be an elliptic curve defined over the finite field Fq of q-elements. Prove
that E(Fq ) ∼
= Cn1 ⊕ Cn2 where n1 , n2 ∈ Z≥1 , n1 divides n2 and Ck denotes the cyclic
group of k elements for some integer k.
(b) (10 points) Assume that in (a) we have n1 = n2 = n. Prove that q = n2 +1 or q = n2 ±n+1
or q = (n ± 1)2 . You can assume, without proof, that if E[n] ⊂ E(Fq ) then n divides
q − 1.
Exercise 9∗ . Use either of the Lutz-Nagell Theorem, the Reduction Theorem or both, to
show that the torsion subgroup T of each of the elliptic curves over Q below is as indicated.
(a) E : y 2 = x3 − 2; T ∼
= 0.
(b) E : y 2 = x3 + x + 1; T ∼
= 0.
(c) E : y 2 = x3 + 4x; T ∼
Z/4Z.
=
Exercise 10. (30 points) Use either of the Lutz-Nagell Theorem, the Reduction Theorem or
both, to show that the torsion subgroup T of each of the elliptic curves over Q below is as
indicated.
(a) (15 points) E : y 2 = x3 − 43x + 166; T ∼
= Z/7Z.
(b) (15 points) E : y 2 = x3 − 12987x − 263466; T ∼
= Z/4Z × Z/2Z.
[Hint: you may use the fact P = (−57, 540) is a point of finite order.]
The exercises with
∗
are NOT for credit.
Due on 7/3/2016 before 3pm.
There is a dropbox set by the undergraduate office.