MA426–ELLIPTIC CURVES
SPRING 2016
Exercise Sheet 5
Exercise 1∗ . Give the proof of Corollary 9.3.3, that is, show that for any integer k ≥ 1 and
for any prime p, the map on the p-adic filtration associated to an elliptic curve E/Q given by
an integral short Weierstrass equation,
Ek /E5k → Z/p4k Z
x
(x, y) 7→ p−k (mod p4k )
y
is an injective group homomorphism.
Exercise 2∗ . Let p be a prime and let E be the elliptic curve defined by the equation
y 2 = x3 + px. Use Lutz-Nagel to find all points of finite order in E(Q).
Exercise 3. (20 points) Prove the following theorem:
Let E be an elliptic curve defined over Q. Let ĥ be the canonical height on E(Q). Then:
(1) ∗ ĥ is well-defined and ĥ(P ) ≥ 0 for all P ∈ E(Q).
(2) ∗ There exists a constant c0 > 0 such that ∀P ∈ E(Q), |h(P ) − ĥ(P )| ≤ c0 .
(3) ∗ ∀c > 0, the set {P ∈ E(Q) : ĥ(P ) ≤ c} is finite.
(4) (5 points) ∀m ≥ 1, ∀P ∈ E(Q), ĥ(mP ) = m2 ĥ(P ).
(5) (10 points) ∀P, Q ∈ E(Q), ĥ(P + Q) + ĥ(P − Q) = 2ĥ(P ) + 2ĥ(Q).
(6) (5 points) ĥ(P ) = 0 ⇐⇒ P has finite order.
Exercise 4. (20 points)
(a) (10 points) Let E1 and E2 be elliptic curves over Fq , the finite field of q elements. Prove
that if E1 is isogenous to E2 then #E1 (Fq ) = #E2 (Fq ). The converse of this is due to Tate
(and you do not have to prove it). Therefore, two elliptic curves over Fq are isogenous if
and only if they have the same number of Fq -rational points.
(b) (10 points) Let E1 and E2 be the elliptic curves over F3 given respectively by y 2 = x3 + 2x
and y 2 = x3 + x. Prove that E1 and E2 are isogenous. Let φ be an isogeny between E1
and E2 whose kernel is contained in E1 (F3 ). Prove that
#(φ(E1 (F3 )) ∩ E2 (F3 )) ≤ 2.
What is the group structure of E1 (F3 )? and of E2 (F3 )? Justify your answer.
Exercise 5. (30 points) Show that 8 is not a congruent number.
Exercise 6. (30 points) Let E/Q be the elliptic curve y 2 = x3 −49x. Find the group structure
of E(Q).
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MA426–ELLIPTIC CURVES
SPRING 2016
Exercise 7∗ . For each of the curves E/Q below, find the group structure of E(Q).
(a) E : y 2 = x(x2 − 3x + 3);
(b) E : y 2 = x(x2 + 3x + 5);
The exercises with ∗ are NOT for credit.
Due on 24/3/2016 before 2pm.
There is a dropbox set by the undergraduate office.
You can also send you solution by email (before 2pm) to
s.anni@warwick.ac.uk adding vandita.patel@warwick.ac.uk in cc.