Homework Assignment 7: Elliptic Curves I Due Wednesday November 20

William Stein

Math 124  Harvard University  Fall 2002

Instructions: Please work with others, and feel free to use a computer, especially on problem 4. (There are 7 problems.)

1. We call a line in ℂ^{2} rational if it is the set of zeros of an equation ax + by + c = 0 with a, b, c ∈ ℚ.

(a) (1 point) Suppose P and Q are distinct elements of ℚ^{2}. Prove that the unique line in ℂ^{2} that contains P and Q is rational.

(b) (2 points) Suppose that L_{1} and L_{2} are distinct rational lines in ℂ^{2} that intersect. Prove that their intersection is a rational point.

2. (4 points) Let Y ⊂ ℂ^{2} be the set of complex solutions (x, y) to the equation y^{2} = x^{5} + 1. Find (with proof!) the closure of Y in ℙ^{2}.

3. (3 points) Let E be the elliptic curve defined by y^{2} = x^{3} + 1. Find the divisor associated to the rational function (x + 1)/(y − 1).

4. (6 points) Let x and y be indeterminates.

(a) Prove that ℂ[x, y]/(y^{2} −(x^{3} +1)) is integrally closed in ℂ(x)[y]/(y^{2} −(x^{3} +1)). That is, if f(x), g(x) ∈ ℂ(x) are rational functions in x and f(x) + yg(x) satisfies a monic polynomial with coefficients in ℂ(x), then f(x) and g(x) are polynomials.

(b) Prove that ℂ[x, y]/(y^{2} −x^{3}) is not integrally closed in ℂ(x)[y]/(y^{2} −x^{3}). [Hint: Consider t = y/x.]

5. Let E be the elliptic curve defined by y^{2} = x^{3} + x + 1. Consider the points P = (72: − 611: 1), Q = (1/4: − 9/8: 1), and R = (1: \sqrt{3}: 1) on E.

(a) (2 points) Compute the sum of P and Q on E.

(b) (2 points) Find nonzero integers n and m such that nP = mQ.

(c) (2 points) Compute R + R.

(d) (3 points) Is there any integer n such that nR = P? (Hint: Keep in mind the automorphism \sqrt{3} ↦ −\sqrt{3} of ℚ(\sqrt{3}).)

6. Let g(t) be a quartic polynomial with distinct complex roots, and let α be a root of g(t). Let β ≠ 0 be any complex number.

(a) (3 points) Prove that the equations

x = \frac{β}{t − α}  and  y = x^{2}u = \frac{β^{2}u}{(t − α)^{2}}

define an algebraic map from the curve u^{2} = g(t) (in ℂ^{2}) to the curve y^{2} = f(x) (in ℂ^{2}), where f(x) is the cubic polynomial

f(x) = g′(α)βx^{3} + \frac{1}{2}g′′(α)β^{2}x^{2} + \frac{1}{6}g′′′(α)β^{3}x + \frac{1}{24}g′′′′(α)β^{4}.

(You just have to check that if (u, t) satisfies u^{2} = g(t) then (x, y) defined by the above equations satisfies y^{2} = f(x).)

(b) (3 points) Prove that since g has distinct complex roots, f also has distinct roots, so u^{2} = g(t) is an elliptic curve.

7. This problem is about the connection between elliptic curves and ellipses. Let α and β be positive real numbers with β ≤ α, and let C be the ellipse

\frac{x^{2}}{α^{2}} + \frac{y^{2}}{β^{2}} = 1.

(a) (3 points) Prove that the arc length of C is given by the integral

4α ∫_{0}^{π/2} \sqrt{1 − k^{2} sin^{2} θ}dθ

for an appropriate choice of constant k depending on α and β.

(b) (1 point) Check your value for k in (i) by verifying that when α = β, the integral yields the correct value for the arc length of a circle.

(c) (3 points) Prove that the integral in (i) is also equal to

4α ∫_{0}^{1} \sqrt{\frac{1 − k^{2}t^{2}}{1 − t^{2}}} dt = 4α ∫_{0}^{1} \frac{1 − k^{2}t^{2}}{\sqrt{(1 − t^{2})(1 − k^{2}t^{2})}} dt.

(d) (0 points) By Exercise 6, the equation

u^{2} = (1 − t^{2})(1 − k^{2}t^{2})

determines an elliptic curve. Hence the problem of determining the arc length of an ellipse comes down to evaluating the integral

∫_{0}^{1} \frac{1 − k^{2}t^{2}}{u} dt

on the elliptic curve determined by u^{2} = (1 − t^{2})(1 − k^{2}t^{2}).