MA 426
THE UNIVERSITY OF WARWICK
FOURTH YEAR EXAMINATION: APRIL 2014
ELLIPTIC CURVES
Time Allowed: 3 hours
Read carefully the instructions on the answer book and make sure that the particulars
required are entered on each answer book.
Calculators are not needed and are not permitted in this examination.
Candidates should answer COMPULSORY QUESTION 1 and THREE QUESTIONS
out of the four optional questions 2, 3, 4 and 5.
The compulsory question is worth 40% of the available marks. Each optional question
is worth 20%.
If you have answered more than the compulsory Question 1 and three optional questions,
you will only be given credit for your QUESTION 1 and THREE OTHER best answers.
The numbers in the margin indicate approximately how many marks are available for
each part of a question.
COMPULSORY QUESTION
1. Let K be a field of characteristic different from 2, 3, and K its algebraic closure.
a) Let C and C 0 be two curves of degree n and n0 in the projective plane P2 (K).
State the Strong Bézout Theorem for C and C 0 .
[2]
b) Let E : Y 2 = X 3 + AX + B be a short Weierstrass cubic curve defined over K.
Show that every line L ⊂ P2 (K) intersects E(K) in three points counted with
multiplicity.
[2]
c) Give the definition of the group law on E. Show that it is commutative.
[6]
d) Let Pi = (xi , yi ) ∈ E(K), i = 1, 2, be such that the line P1 P2 is non-vertical;
and set P3 = (x3 , y3 ) = P1 + P2 . Show that if P1 , P2 are defined over K, then
P3 is also defined over K. Deduce that E(K) is a subgroup of E(K).
[10]
e) Let E : Y 2 = X 3 − 2X + 5, and P = (2, 3). Find Q = 2P .
[8]
1
Question 1 continued overleaf
MA 426
f) Let E be as in b) and P = (x, y) ∈ E(K). Show that P is a 2-torsion point if
and only if y = 0 and x3 + Ax + B = 0. Deduce that there are at most four
2-torsion points counted with multiplicities. Find the 2-torsion points of the
curve E : Y 2 = X 3 − 5X over Q. Which ones are defined over Q?
[12]
OPTIONAL QUESTIONS
2. Let q ≥ 3 be a prime power, and n ≥ 1 an integer. Let Fq be the finite field with q
elements, and E : Y 2 = X 3 + aX + b, an elliptic curve defined over Fq . Let φq be the
Frobenius endomorphism of E. We recall that φq satisifies 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.
[3]
b) Show that E(Fqn ) = ker(φnq − 1). Deduce that #E(Fqn ) = deg(φnq − 1).
[4]
c) Show that Q(x) = x4 − (α2 + β 2 )x2 + q 2 ∈ Z[x] and is divisible by x2 − aq x + q.
[4]
d) Show that φq2 satisfies the polynomial x2 − (α2 + β 2 )x + q 2 . Deduce that
#E(Fq2 ) = q 2 + 1 − (α2 + β 2 ).
[5]
e) Let p = 5, and consider the curve E : Y 2 = X 3 + 3X + 2 over F5 . Find a such
that φ25 − aφ5 + 5 = 0. Deduce that #E(F25 ) = 35.
[4]
3. Let E : Y 2 = X 3 + AX + B, with A, B ∈ Z, be an elliptic curve. Let p be an odd
prime that does not divide the discriminant ∆ = −4A3 − 27B 2 .
a) Let P = (x, y) ∈ E(Q) be a rational point. Show that there exist a, b, c ∈ Z
such that gcd(a, c) = gcd(b, c) = 1, c > 0, and x = a/c2 , y = b/c3 .
[3]
b) Define the reduction map rp : E(Q) → E(Fp ). Describe the kernel of rp . [You
may assume that rp is a group homomorphism.]
[4]
c) Show that the restriction of rp to the torsion subgroup E(Q)tors is injective.
[4]
d) For the curve E : Y 2 = X 3 + 2X + 3, show that E(Q)tors has order 2.
[7]
e) Show that the curve in d) has a point of infinite order.
[2]
2
CONTINUED
MA 426
4. Let E : Y 2 = X(X 2 + AX + B), with A, B ∈ Z, be an elliptic curve.
5.
a) Define the descent map δ : E(Q) → Q× /(Q× )2 . Describe the image of δ, and
deduce that it is finite. [NO proof is required!]
[4]
b) State the Mordell-Weil Theorem for E.
[2]
c) Consider the curve E : Y 2 = X(X 2 − 3X − 5), and let r be the rank of E(Q).
Show that r = 1. [You may assume that E(Q)tors = Z/2Z.]
[14]
a) Show that 2 is not a congruence number.
[15]
b) Is 50 is a congruence number? Explain.
[5]
3
END