MA426–ELLIPTIC CURVES
SPRING 2016
Exercise Sheet 1
Exercise 1∗ . 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)
(a) (2 points) If w2 + aw + b = 0 is a quadratic polynomial with a known root r, give a
formula for the other root using only addition and subtraction with r, a, b, no quadratic
formula.
(b) (5 points) Let C be the unit circle C : u2 + v 2 = 1 in the real plane. Let Lt : v = t(u + 1)
be a line with slope t through the point P = (−1, 0). Using (a), compute the intersection
of C and Lt . What happens if you take the limit for t → ∞?
(c) (8 points) Let Pt be the intersection point, depending on the parameter t, found in (b).
Setting t = m
n , for relatively prime positive integers m and n, express the condition that
Pt lies in the first quadrant in terms of inequalities on n and m. Deduce, by clearing
denominators, that every primitive Pythagorean triple has the form
(n2 − m2 , 2nm, n2 + m2 ),
with n and m not both odd (a primitive Pythagorean triple is a triple of relatively prime
positive integers which satistfy Pythagoras theorem).
Exercise 3. (15 points) For each of the curves below, write down its homogenisation. Find
all the points at infinity over Q.
(a) (5 points) 2x + 3y − 1 = 0.
(b) (5 points) x2 − 6xy + 5y 2 − 3x + 5y − 10 = 0.
(c) (5 points) x3 − 3x2 y − 7xy 2 + 21y 3 + x − 5x2 + y 2 − y + 5 = 0.
Exercise 4. (20 points)
(a) (10 points) Let A, B ∈ Q. Show that the projective plane cubic Y 2 Z = X 3 +AXZ 2 +BZ 3
is non-singular if and only if 4A3 + 27B 2 6= 0.
(b) (10 points) For which values of A ∈ Q is the projective cubic X 3 + Y 3 + Z 3 + 3AXY Z = 0
non-singular?
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MA426–ELLIPTIC CURVES
SPRING 2016
Exercise 5. (25 points) Let K be a field and let n be a positive integer. Let Cn be the
projective plane cubic X 3 + Y 3 + n2 Z 3 − 3n2 XZ 2 = 0 over K. Is Cn a smooth projective
curve? Over which fields it is not smooth? Justify your answer.
Exercise 6∗ . Does the equation 3x2 + 2y 2 − 2z 2 = 0 have integral solutions? Justify your
answer.
Exercise 7∗ . Show that the equation y 2 = x3 − 5 has no integral solutions.
Exercise 8. (25 points) Show that the equation y 2 = x3 + 46 has no integral solutions.
Exercise 9∗ . Show that the only integral solutions to the equation y 2 = x3 + 16 are (0, ±4).
Exercise 10∗ . (only if you are familiar with algebraic number theory) Show that the only
integral solutions to the equation y 2 = x3 − 2 are (3, ±5).
The exercises with
∗
are NOT for credit.
Due on 25/01/2016 before 3pm.
There is a dropbox set by the undergraduate office.