MA426–ELLIPTIC CURVES
FALL 2014
Exercise Sheet 1
Eexercise 1. (10 points) Let K be a field.
(a) Show that the following map is injective.
φ : K 2 → P2 (K)
(x, y) 7→ [x : y : 1]
Show that Im(φ) = {[x : y : z] ∈ P2 (K) : z 6= 0}.
(b) Show that the map
φ∞ : P1 (K) → P2 (K)
[x : y] 7→ [x : y : 0]
is injective, and describe its image.
(c) Show that we have a disjoint union P2 (K) = Im(φ) t Im(φ∞ ).
Exercise 2. (15 points) For each of the curves below, write down its homogenisation. Find
all the points at infinity over Q.
(a) x + 2y − 1 = 0.
(b) x2 − 3xy + 3y 2 − x − y = 0.
(c) x3 + x2 y − 3xy 2 − 3y 3 + y 2 − y + 5 = 0.
Exercise 3. (15 points)
(a) Show that the cubic Y 2 Z = X 3 + AXZ 2 + BZ 3 , with A, B ∈ Q, is non-singular if and
only if 4A3 + 27B 2 6= 0.
(b) For which values of A ∈ Q is the cubic X 3 + Y 3 + Z 3 − AXY Z = 0 non-singular?
Exercise 4.(20 points) Let K be a field and let n be a positive integer. Let Cn be the
projective diagonal plane cubic X 3 + n Y 3 + n2 Z 3 = 0.
(a) Is Cn a smooth projective curve? Over which fields it is not smooth? Justify your answer.
(b) Let K = Q and let p be a prime, does Cp admits rational points? Justify your answer.
Exercise 5. (20 points) Does the equation 3x2 + 5y 2 − 7z 2 = 0 have integral solutions?
Justify your answer.
Exercise 6. (20 points) Show that the equation y 2 = x3 − 24 has no integral solutions.
Due on 13/10/2014 before 3pm.
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