Computer Assignments-Summer 2003 REU

The remainder of the REU will be devoted to your own computer-aided investigations and discussions with Jim Carlson. However, you should all do the following ``mandatory'' assignment.

1a. (Mandatory) Find the torsion points of a given elliptic curve. That is, given a triple of integers:

$$(a,b,c)$$

satisfying $\Delta \ne 0$ your program should output the set of all torsion points (including $0$). Either your program or you should then decide what the group of torsion points is. Hint: It is either:

$${\bf Z}/d{\bf Z}, \ 1 \le d \le 12, d \ne 11 \ \mbox{or}\ {\bf Z}/2{\bf Z}\times {\bf Z}/d{\bf Z}, d = 2,4,6 \ \mbox{or} \ 8$$

so unless the number of torsion points is $4,8$ or $12$, the group is determined.

1b. (Mandatory) Find as many elliptic curves as you can (at least one!) with each of the torsion groups listed above. Which group is the ``rarest''?

You should also be attempting one or more of the following projects, or a project of your own design:

2a. Distribution of numbers of points (mod $p$). We remarked in class that:

$$\vert\char93 C({\bf F}_p) - p\vert \le 2\sqrt p$$

for any elliptic curve. Write a program that takes as input $(a,b,c)$ and produces as output all the primes $5 \le p < 1000$ (or more if you want) that don't divide $2\Delta$, together with $\char93 C({\bf F}_p)$.

2b. What do you notice about these numbers? Try this for rank $0$ curves (such as $y^2 = x^3 + x$) and for curves of rank $> 0$ (such as $y^2 = x^3 - 5x$). Do you notice a difference? Do you believe that the numbers of points mod $p$ and the ranks are related?

3. Continue the search for big integer points on curves with small $(a,b,c)$!

4. Search for elliptic curves of large rank. That is, fashion a search for curves of the form $y^2 = x^3 + ax^2 + bx$ of ranks $\ge 1,2,3,4,...$. I'm curious to see how big a rank you can find!