Short Homework Assignment 8: Elliptic Curves II Due Monday November 25

William Stein

Math 124  Harvard University  Fall 2002

There are only two problems.

1. (2 points) Prove that ℝ/ℤ is not finitely generated.

2. If a, b ∈ ℤ/p and 4a^{3} + 27b^{2} ≠ 0, then MAGMA can easily compute the group E(ℤ/p) associated to the elliptic curve defined by y^{2} = x^{3} + ax + b. Here’s an example:

> AbelianGroup(EllipticCurve([GF(7)|1,3]));
Abelian Group isomorphic to Z/6
Defined on 1 generator
Relations:
6*$.1 = 0

(a) (2 points) Find a prime p and a, b ∈ ℤ/p, so that the group E(ℤ/p) associated to y^{2} = x^{3} + ax + b is not a cyclic group. (Hint: Just try a, b, p at random...)

(b) (5 points) Let Φ be the set of the fifteen possible torsion subgroups of elliptic curves over ℚ. For each group G ∈ Φ, find a prime p and a, b ∈ ℤ/p such that G ≅ E(ℤ/p). Hint: Systematically list the groups that occur. You might find the MAGMA commands illustrated below useful:

      > IsEllipticCurve([GF(3)|0,1]);
      false
      > Invariants(AbelianGroup(EllipticCurve([GF(3)|1,0])));
      [ 4 ]

(c) (2 points) Suppose n > 1 is an integer. Make a guess about whether or not ℤ/n must be isomorphic to a group E(ℤ/p) for some elliptic curve over some finite field ℤ/p. Support your guess with evidence.