Math 168: Homework Assignment 2

William Stein

Due: Wednesday, Oct 12, 2005

The SIX problems have equal point value, and multi-part problems are of the same value. You are allowed to use a computer on any problem, as long as you include the exact code used to solve the problem with your solution. Any software systems (e.g., Magma, SAGE, Mathematica, C) are allowed.

1 Announcements

1. Office Hours: Tuesdays 3-5 in my office (AP&M 5111).

2. Section: Thursday 5-6 in my office

2 Problems

1. One rational solution to the equation y^{2} = x^{3} − 2 is (3, 5). Find a rational solution with x ≠ 3 by drawing the tangent line to (3, 5) and computing the second point of intersection.

2. Write down an equation y^{2} = x^{3} + ax + b over a field K such that −16(4a^{3} + 27b^{2}) = 0 (yes equals 0). Precisely what goes wrong when trying to endow the set E(K) = {(x, y) ∈ K × K : y^{2} = x^{3} + ax + b} ∪ {O} with a group structure (in an algebraic way)?

3. Let E be the elliptic curve over the finite field K = 𝔽_{5} defined by the

equation

y^{2} = x^{3} + x + 1.

(a) List all 9 elements of E(K).

(b) What is the structure of E(K), as a product of cyclic groups?

4. Let E be the elliptic curve defined by the equation y^{2} = x^{3} + 1. For each prime p ≥ 5, let N_{p} be the cardinality of the group E(𝔽_{p}) of points on this curve having coordinates in 𝔽_{p}. For example, we have that N_{5} = 6, N_{7} = 12, N_{11} = 12, N_{13} = 12, N_{17} = 18, N_{19} = 12, , N_{23} = 24, and N_{29} = 30 (you do not have to prove this).

(a) 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).

(b) (*) Prove your conjecture.

5. Suppose y^{2} = x^{3} + ax + b with a, b ∈ ℚ defines an elliptic curve. Show that there is another equation Y^{2} = X^{3} + AX + B with A, B ∈ ℤ whose solutions are in bijection with the solutions to y^{2} = x^{3} + ax + b (via a bijection defined by algebraic formulas).

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