Pries: 405 Number Theory, Spring 2012. Homework 8.
Due: Friday 4/20.
Elliptic curves continued
Read: handout from Silverman.
Complete 4 of the following problems.
1. Consider the point P = (2 , 3) on the elliptic curve E : y 2 order 6 in E (
Q
).
= x 3 + 1. Show that P has
2. Let t be a rational number such that t = 0 and t = 1 / 4. Consider the point P = ( t, t ) on the elliptic curve E : y 2 = x 3 − (2 t − 1) x 2 + t 2 x . Show that P has order 4 in E (
Q
).
Hint: you only need to compute 2 P .
3. The Nagell-Lutz theorem states the following: Let a and b be integers such that D =
− 4 a
3 − 27 b
2
= 0. If P = ( x, y ) is a point of finite order on the smooth elliptic curve
E : y 2 = x 3 + ax + b , then x and y are integers and either y = 0 or y divides D .
If p is an odd prime, use the Nagell-Lutz theorem to find all the points of finite order on y 2 = x 3 + px .
4. Let C
1 and C
2 be the cubic curves with the following equations:
C
1
: x
3
+ 2 y
3 − x − 2 y = 0 , C
2
: 2 x
3 − y
3 − 2 x + y = 0 .
(a) Find the 9 points of intersection of C
1 and C
2
.
(b) Let { (0 , 0) , P
1
, . . . , P
8
} be the nine points from part (a). Let C be an arbitrary cubic curve:
C : a
3 , 0 x
3
+ a
2 , 0 x
2
+ a
1 , 0 x + a
0 , 0
+ a
0 , 3 y
3
+ a
0 , 2 y
2
+ a
0 , 1 y + a
1 , 2 xy
2
+ a
2 , 1 x
2 y + a
1 , 1 xy = 0 .
If P
1
, . . . , P
8 are all on C , show that (0 , 0) is on C also. Hint: Concretely, each point gives a linear relation between the coefficients a i.j
and you need to show that a
0 , 0
= 0.
Remark: The problem above is related to showing that the group law on an elliptic curve is associative. It is also an example of Bezout’s theorem.
5. Let L =
Z
[ i ] = { a + bi | a, b ∈
Z
} . Show that g
3
= 0 where g
3
= 140
X w ∈ L,w =0
1 w 6
.
This is the main step in showing that
C
/L ' E : y 2 = x 3 − x . Hint: add up the terms in groups of 4, i.e., w is a + bi , i ( a + bi ), − ( a + bi ), and − i ( a + bi ).
I’m stuck on a line; I’m stuck on a curve - Peter Krebs
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