Course 428

Elliptic Curves III

Dr Timothy Murphy

Maxwell Theatre Thursday, 4 May 2000 16:00–17:00

Attempt 3 questions. (If you attempt more, only the best 3 will be counted.) All questions carry the same number of marks.

1. Define the height h(P) of a point on the elliptic curve

E(ℚ) : y^{2} = x^{3} + ax^{2} + bx + c (a, b, c ∈ ℤ);

and show that there are only a finite number of points P ∈ E satisfying

h(P) ≤ C

for any constant C. State Mordell’s Theorem; and explain the rôle of the height h(P) in its proof.

2. Suppose E is the elliptic curve

E(ℚ) : y^{2} = x^{3} + ax^{2} + bx (a, b ∈ ℚ).

Outline the proof that E/2E is finite.

3. Solve the equation

x^{3} + 3x^{2} + 1 = 0.

4. Show that the group on the elliptic curve

E(ℚ) : y^{2} = x^{3} − x

is finite, and find its order.