Homework 9: Elliptic Curves
DUE WEDNESDAY, NOVEMBER 28

William Stein

Date: Math 124 $\quad$ HARVARD UNIVERSITY $\quad$ 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 $\Phi$ be the set of the $15$ possible groups of the form $E(\mathbb{Q})_{\tor}$ for $E$ an elliptic curve over  $\mathbb{Q}$ (see Lecture 27). For each group $G\in\Phi$, if possible, find a finite field $k = \mathbb{Z}/p\mathbb{Z}$ and an elliptic curve $E$ over $k$ such that $E(k) \approx G$. (Hint: It is a fact that $\vert p + 1 - \char93 E(\mathbb{Z}/p\mathbb{Z}))\vert \leq 2\sqrt{p}$, 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 $5\leq p<30$, describe the group of points on this curve having coordinates in the finite field $\mathbb{Z}/p\mathbb{Z}$. (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 $p\equiv 2\pmod{3}$, can you see a pattern for the values of $N_p$? Make a general conjecture for the value of $N_p$ when $p\equiv 2\pmod{3}$.

(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 $E(\mathbb{Q})$.

5.

(4 points)

(iv)

Let $E$ be an elliptic curve over the real numbers  $\mathbb{R}$. Prove that $E(\mathbb{R})$ is not a finitely generated abelian group.

(v)

Let $E$ be an elliptic curve over a finite field $k = \mathbb{Z}/p\mathbb{Z}$. Prove that $E(k)$ is a finitely generated abelian group.