Homework 8: Elliptic Curves
DUE WEDNESDAY, NOVEMBER 21

William Stein

Date: Math 124 $\quad$ HARVARD UNIVERSITY $\quad$ Fall 2001

There are six problems. Some of the problems involve tedious algebra, and in all such cases you are allowed to do that algebra using, e.g., PARI as long as you explain how you used the system to do the algebra. Problems 4, 5, and 6 are from Silverman and Tate's book Rational Points on Elliptic Curves.

1.

(3 points) Consider the elliptic curve $y^2 + xy + y = x^3$ over $\mathbb{Q}$. Find a linear change of variables that transforms this curve into a curve of the form $Y^2 = X^3+a X + b$ for rational numbers $a$ and $b$.

2.

(6 points) Let $E$ be the elliptic curve over the finite field $K=\mathbb{Z}/5\mathbb{Z}$ defined by the equation

$$y^2 = x^3 + x +1.$$

(i)

List all $9$ elements of $E(K)$.

(ii)

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

3.

(8 points) Let $E$ be an elliptic curve over  $\mathbb{Q}$. Define a binary operation $\boxplus$ on $E$ as follows:

$$P \boxplus Q = -(P+Q).$$

Thus the $\boxplus$ of $P$ and $Q$ is the third point of intersection of the line through $P$ and $Q$ with $E$.

(iii)

Lists the axiom(s) of a group that fail for $E(\mathbb{R})$ equipped with this binary operation. (The group axioms are ``identity'', ``inverses'', and ``associativity''.)

(iv)

Under what conditions on $E(\mathbb{Q})$ does this binary operation define a group structure on $E(\mathbb{Q})$? (E.g., when $E(\mathbb{Q})=\{\mathcal{O}\}$ this binary operation does define a group.)

4.

(6 points) Let $g(t)$ be a quartic polynomial with distinct (complex) roots, and let $\alpha$ be a root of $g(t)$. Let $\beta\neq 0$ be any number.

(v)

Prove that the equations

$$x = \frac{\beta}{t-\alpha}, \qquad y = x^2 u = \frac{\beta^2 u}{(t-\alpha)^2}$$

give an ``algebraic transformation'' between the curve $u^2=g(t)$ and the curve $y^2=f(x)$, where $f(x)$ is the cubic polynomial

$$f(x) = g'(\alpha) \beta x^3 + \frac{1}{2} g''(\alpha) \beta^2 x^2 + \frac{1}{6}g'''(\alpha) \beta^3 x + \frac{1}{24} g''''(\alpha)\beta^4.$$

(vi)

Prove that if $g$ has distinct (complex) roots, then $f$ also has distinct roots, and so $u^2=g(t)$ is an elliptic curve.

5.

(8 points) In this problem you will finally find out exactly why elliptic curves are called ``elliptic curves''! Let $0<\beta\leq \alpha$, and let $C$ be the ellipse

$$\frac{x^2}{\alpha^2} + \frac{y^2}{\beta^2} = 1.$$

(vii)

Prove that the arc length of $C$ is given by the integral

$$4\alpha\int_{0}^{\pi/2} \sqrt{1-k^2\sin^2\theta} d\theta$$

for an appropriate choice of constant $k$ depending on $\alpha$ and $\beta$.

(viii)

Check your value for $k$ in (i) by verifying that when $\alpha=\beta$, the integral yields the correct value for the arc length of a circle.

(ix)

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

$$4\alpha\int_0^1 \sqrt{\frac{1-k^2t^2}{1-t^2}} dt = 4\alpha\int_0^1\frac{1-k^2t^2}{\sqrt{(1-t^2)(1-k^2t^2)}}dt.$$

(x)

Prove that if the ellipse $E$ is not a circle, then the equation

$$u^2 = (1-t^2)(1-k^2t^2)$$

defines an elliptic curve (cf. the previous exercise). Hence the problem of determining the arc length of an ellipse comes down to evaluating the integral

$$\int_0^1 \frac{1-k^2t^2}{u} dt$$

on the ``elliptic'' curve $u^2=(1-t^2)(1-k^2t^2)$.

6.

(8 points) Suppose that $P=(x,y)$ is a point on the cubic curve

$$y^2 = x^3 + ax + b.$$

(xi)

Verify that the $x$ coordinate of the point $2P$ is given by the duplication formula

$$x(2P) = \frac{x^4 - 2ax^2 -8bx +a^2}{4y^2}.$$

(xii)

Derive a similar formula for the $y$ coordinate of $2P$ in terms of $x$ and $y$.

(xiii)

Find a polynomial in $x$ whose roots are the $x$-coordinates of the points $P=(x,y)$ satisfying $3P=\mathcal{O}$. [Hint: The relation $3P=\mathcal{O}$ can also be written $2P=-P$.]

(xiv)

For the particular curve $y^2 = x^3 + 1$, solve the equation in (iii) to find all of the points satisfying $3P=\mathcal{O}$. Note that you will have to use complex numbers.