MA426–ELLIPTIC CURVES
FALL 2014
Exercise Sheet 4
Exercise 1. (5 points) 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.
(c) Assume that E is given by the equation y 2 = x3 +x+1 over F7 , and let ϕ be its Frobenius
endomorphism of E. Find the integer t such that ϕ2 − tϕ + 7 = 0.
Exercise 2. (15 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) Show that φqn = φnq , and that φnq − 1 is separable.
(b) Show that E(Fqn ) = ker(φnq − 1). Deduce that #E(Fqn ) = deg(φnq − 1).
(c) 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)?]
(d) Show that φq2 satisfies the polynomial x2 − (α2 + β 2 )x + q 2 . Deduce that
#E(Fq2 ) = q 2 + 1 − (α2 + β 2 ).
(e) Show that φqn satisfies the polynomial x2 − (αn + β n )x + q n . Deduce that
#E(Fqn ) = q n + 1 − (αn + β n ).
(f) Let E be the elliptic curve over F3 given by y 2 + y = x3 + 2x. List the points in E(F3 ),
E(F9 ). Compute #E(F27 ) without listing the points.
Exercise 3. (20 points) Consider the elliptic curve E : y 2 + y = x3 over F2 . Let u, s, t ∈ F2
satisfy respectively u3 = 1, s4 + s = 0 and t2 + t = s6 .
(a) Show that
ϕ(x, y) = (u2 x + s2 , y + u2 sx + t)
is an automorphism of E.
(b) Show that every automorphism arises this way and that #Aut(E) = 24.
[Hint: in any characteristic (hence, also in characteristic 2), an automorphism α is an
endomorphism of degree 1 which has an inverse, i.e. an isomorphism of the elliptic curve
onto itself. In general, an automorphism is a map of the form α(x, y) = (ax+b, cy+dx+e)
for all (x, y) ∈ E(K) where a, b, c, d, e ∈ K which is invertible: this gives restrictions on
the coefficients a, b, c, d, e.]
(c) Show that if ψ ∈ Aut(E) then either ψ 2 = ±1 or ψ 3 = ±1.
(d) Show that Aut(E) is non-abelian.
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MA426–ELLIPTIC CURVES
FALL 2014
Exercise 4. (10 points) 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 5. (25 points)
(a) How many elliptic curves in short Weierstrass equation do exist over F5 ?
(b) Is there an elliptic curve E over F5 such that #E(F25 ) = 24?
(c) 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?
(d) 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 6. (25 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) 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.
=
(d) E : y 2 = x3 − 43x + 166; T ∼
= Z/7Z.
(e) 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.]
Due on 20/11/2014 before 2pm.

Assignment 1

Homework 6 Solution - Han

Arithmetic and Algebraic Geometry 2015