MA 426
THE UNIVERSITY OF WARWICK
FOURTH YEAR EXAMINATION: APRIL 2015
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 E be an elliptic curve over Q and let P = (u, v) be a Q-rational point on E.
a) Let E be given by a long Weierstrass equation:
E : y 2 + a1 xy + a3 y = x3 + a2 x2 + a4 x + a6 .
Show that −P = (u, −v − a1 u − a3 ) and that points of order two satisfy the
equation 2y + a1 x + a3 = 0.
[4]
b) Let E : y 2 = x3 + ax + b be given by a short Weierstrass equation. Find a
polynomial φ(x) such that the following holds: P is a point of order 3 if and
only if φ(u) = 0. Justify your answer.
[4]
c) Let E be the elliptic curve y 2 = x3 + x + 2. Given the points Q = (−1, 0) and
R = (1, 2) in E(Q), find Q + R and 2R. What are the orders of Q and of R?
Compute the torsion subgroup over Q.
[10]
1
Question 1 continued overleaf
MA 426
d) Define the canonical height ĥ on E(Q) in terms of the naive height. State,
without proof, the parallelogram identity for the canonical height on rational
points.
[4]
e) Prove that ĥ(mP ) = m2 ĥ(P ) for all integers m ≥ 1 and for all rational points
P . Show that ĥ(P ) = 0 if and only if P is a torsion point, you may assume
without proof that there are only finetely many points of canonical height zero.
[8]
f) State the Mordell-Weil Theorem, without proof. Why is the torsion subgroup
a finite group?
[4]
g) Let E be the elliptic curve y 2 = x3 + cx + 1, where c is an odd integer. Is the
rank of E over Q positive? Justify your answer.
[6]
OPTIONAL QUESTIONS
2.
a) Let E be an elliptic curve defined over a field K. Define the n-torsion subgroup
of E(K), where K is an algebraic closure of K. Characterize, without proof, its
group structure.
b) 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.
c) Let E be an elliptic curve defined over Fq . Prove that if E(Fq ) ∼
= Cn ⊕ Cn for
3.
[4]
[4]
some integer n, then 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.
[8]
d) Let E over F13 be a supersingular elliptic curve. What is the group structure
of E(F13 )? What is the order of E(F132 )? Justify your answers.
[4]
a) Let E be an elliptic curve defined over the finite field Fq of q-elements. State,
without proof, Hasse’s inequality.
[2]
b) Let E be an elliptic curve defined over the finite field Fq , such that #E(Fq ) = n.
In which interval, depending on n, is the prime power q?
[4]
c) Let q = pe where p is an odd prime and e ≥ 2 is an integer. Find an example
of a p and the equation of an elliptic curve E over Fp such that #E(Fq ) = 16.
Justify your answer.
[8]
d) Let E be the elliptic curve over Fp given by y 2 = x3 + x. Let i be a square root
of −1 in Fp and let α be the automorphism of E given by α(x, y) = (−x, iy).
Which quadratic equation does α satisfy? What is the degree of α + 3? and
the degree of α2 + 3? When do the Frobenius endomorphism of E and −α
commute? Justify your answers.
2
CONTINUED
[6]
MA 426
4. Let Ea /Q be the elliptic curve y 2 = x3 + ax, with a ∈ Z.
a) Show that there exists P ∈ Ea (Q) such that 2P = (0, 0) if and only if a = 4d4 ,
for some integer d.
[Hint: the elliptic curve Ea is 2-isogenous to the elliptic curve Ea0 , given by the
equation y 02 = x0 (x02 − 4a), where the 2-isogeny φ : Ea → Ea0 is given by
 2 y(x2 −a) y

if P = (x, y) 6= 0, (0, 0)
 x2 , x2
φ(P ) :=


0
if P = 0 or (0, 0). ]
[10]
5.
b) Are the elliptic curves Ea : y 2 = x3 + ax and E2a : y 2 = x3 + 2ax isomorphic
over Q? Are they isomorphic over Q? Justify your answers.
[2]
c) Find the group structure of Ea (Q) for a = −4 and for a = −64.
[8]
a) Let E/Q be the elliptic curve y 2 = x3 + ax2 + bx with a, b ∈ Z. Let T = (0, 0)
and let P = (u, v) ∈ E(Q) be a torsion point of order larger than 2. Show that
P + T is a torsion point. Show that u divides b and u + a + ub is a square.
[6]
b) Let E/Q be the elliptic curve y 2 = x3 + px, where p is a prime. Let T = (0, 0).
(i) Show that E(Q)tors = {0, T }.
[4]
(ii) Prove that the rank r of E over Q is at most 2. In particular, show that if
p ≡ 3 mod 4 then r ≤ 1 and if p ≡ 7 mod 16 then r = 0.
[Hint: 5 is not a square mod8. The 4-th power of an odd number is congruent to 1 mod 16.]
[10]
3
END