MA426–ELLIPTIC CURVES
SPRING 2016
Exercise Sheet 2
Exercise 1∗ . Over Q, consider the projective curve C1 : X 2 + Y 2 − Z 2 = 0 and the line
L : Y − 5Z = 0. Show that every intersection point does not belong to the line at ∞. Find
all the points in C1 ∩ L and compute their multiplicities.
Exercise 2∗ . Over Q, consider the projective curves C1 : X 2 + Y 2 − Z 2 = 0 and C2 :
X 2 + Y 2 − 9Z 2 = 0. Show that the intersection points belong to the line at ∞. Find all the
points in C1 ∩ C2 and compute their multiplicities.
Exercise 3. (15 points) Over Q, consider the projective curves C1 : X 2 + Y 2 − Z 2 = 0 and
C3 : X 2 − Y Z + Z 2 = 0. Sketch the plane conics obtained dehomogenizing the projective
conics with respect to Z. Find all the points in C1 ∩ C3 and compute their multiplicities.
Exercise 4. (15 points) Prove the following theorem (Theorem 2.2.2 in the lecture notes):
Theorem Let f, g ∈ K[x] be two polynomials of degree m and n respectively, given by
m
n
X
X
f (x) =
ai xi , g(x) =
bj xj .
i=0
j=0
Then f and g have a common factor which is non-constant if and only if R(f, g) = 0.
Hint: use Lemma 2.2.1 in the lecture notes and linear algebra.
Exercise 5. (20 points) Let E be an elliptic curve defined over a field K. Let P = (x1 , y1 )
be a K-rational point on E.
(a) (10 points) Let E : y 2 = x3 + ax + b = f (x) be given by a short Weierstrass equation.
Show that P is a point of order 2, i.e. P 6= ∞ and 2P = P ⊕ P = ∞, if and only if x1 is
a root of the cubic polynomial f (x).
(b) (10 points) 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 = (x1 , −y1 − a1 x1 − a3 ). Show that points of order
2 satisfy 2y + a1 x + a3 = 0.
Exercise 6∗ . Let E be an elliptic curve defined over a field K. Let P = (x1 , y1 ) be a
K-rational point on E.
(a) Let E : y 2 = x3 + ax + b be given by a short Weierstrass equation. Show that P is a
point of order 3, i.e. P 6= ∞ and 3P = P ⊕ P ⊕ P = ∞, if and only if the tangent line at
P intersects E with multiplicity 3.
(b) Let E : y 2 = x3 + ax + b be given by a short Weierstrass equation. Using (a), find a
polynomial φ(x) such that P is a point of order 3 if and only if φ(x1 ) = 0.
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MA426–ELLIPTIC CURVES
SPRING 2016
Exercise 7. (20 points) For each of the curves below, find the sum of the given points.
(a) (10 points) E : y 2 = x3 + 17 over Q and P = (−1, 4), Q = (2, 5);
(b) (10 points) E : y 2 + xy − 5y = x3 + 2x + 7 over F11 and P = Q = (0, 1).
Exercise 8∗ . For each of the following curves over Q, find all points of order 2 over Q, i.e.
all points P with P 6= ∞ and 2P = P ⊕ P = ∞ and coordinates in Q. What are their fields
of definition? i.e. what is the smallest field containing the coordinates?
(a) y 2 = x3 − x,
(b) y 2 = x3 + x2 + x + 1,
(c) y 2 = x3 − x2 − 3x + 2.
Exercise 9. (30 points) Let En : y 2 = x(x2 − n2 ) be the congruence number curve for n ∈ N.
Let P = (x1 , y1 ) be a rational point of En with y1 > 0, and set Q = 2P = (x2 , y2 ).
(a) (10 points) Show that
2
2
x1 + n2
x2 =
.
2y1
(b) (10 points) Show that x2 ± n are both squares. Find a, b, c > 0 in terms of x1 , y1 , n such
that a2 + b2 = c2 .
n2
(c) (10 points) Prove that En1 ∼
= En2 over Q if and only if n21 is a 4-th power in Q.
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Exercise 10∗ .
(a) Find a factor of 21 using the Elliptic Curve Method: set a = 2 and B = 3.
(b) Find a factor of 33 using the Elliptic Curve Method: set a = 4 and B = 3.
The exercises with
∗
are NOT for credit.
Due on 4/2/2016 before 2pm.
There is a dropbox set by the undergraduate office.