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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

$\displaystyle y^2 +xy = x^3 -8369487776175x + 9319575518172005625,$

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.




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William A Stein 2001-11-23