Homework 9: Elliptic Curves

Due Wednesday, November 28

William Stein

Math 124 Harvard University Fall 2001

There are 5 problems. Choose 4 of the 5 problems and clearly indicate which ones you will be graded on (as usual, your score will be a fraction between 0 and 1). As usual, you may use PARI for any of them, as long as you explain what you are doing. Work in groups.

1. (10 points) Let Φ be the set of the 15 possible groups of the form E(ℚ)_{tor} for E an elliptic curve over ℚ (see Lecture 27). For each group G ∈ Φ, if possible, find a finite field k = ℤ/pℤ and an elliptic curve E over k such that E(k) ≈ G. (Hint: It is a fact that |p + 1 − #E(ℤ/pℤ))| ≤ 2\sqrt{p}, so you only have to try finitely many p to show that a group G does not occur as the group of points on an elliptic curve over a finite field.)

2. (6 points) Many number theorists, such as myself one week ago, incorrectly think that Lutz-Nagell works well in practice. Describe the steps you would take if you were to use the Lutz-Nagell theorem (Lecture 27) to compute the torsion subgroup of the elliptic curve E defined by the equation

y^{2} + xy = x^{3} − 8369487776175x + 9319575518172005625,

then tell me why it would be very time consuming to actually carry these steps out. Find the torsion subgroup of E using the elltors command in PARI. Does elltors use the Lutz-Nagell algorithm by default?

3. (6 points) Let E be the elliptic curve defined by the equation y^{2} = x^{3} + 1.

(i) For each prime p with 5 ≤ p < 30, describe the group of points on this curve having coordinates in the finite field ℤ/pℤ. (You can just give the order of each group.)

(ii) For each prime in (i), let N_{p} be the number of points in the group. (Don’t forget the point infinity.) For the set of primes satisfying p ≡ 2 (mod 3), can you see a pattern for the values of N_{p}? Make a general conjecture for the value of N_{p} when p ≡ 2 (mod 3).

(iii) Prove your conjecture.

4. (6 points) Let p be a prime and let E be the elliptic curve defined by the equation y^{2} = x^{3} + px. Use Lutz-Nagel to find all points of finite order in E(ℚ).

5. (4 points)

(i) Let E be an elliptic curve over the real numbers ℝ. Prove that E(ℝ) is not a finitely generated abelian group.

(ii) Let E be an elliptic curve over a finite field k = ℤ/pℤ. Prove that E(k) is a finitely generated abelian group.