ILLUSTRATED NOTES ON ELLIPTIC CURVES
The first image is called elliptic curve spiral. Each pixel at
position (x,y) is colored according to then number of solutions of the
equation T^2 = S^3+yS+x. The image is obtained by generating a sequence
of planes paramaterised by values of S. A dot is placed at (x,y) in
each S-plane whenever S^3+yS+x is a perfect square . The resulting
image is obtained by adding the pixels of each S-plane, and colorizing.
The image summarizes basic information for about quarter of a million
elliptic curves.
The first time I calculated this image, back in 2004,I was unable
to view it. The calculation took several hours, mostly because I was
using a slow algorithm. After a long wait I typed in the command
"display ec0.xpm &" and the computer froze. There was no response from
either the keyboard and mouse and even the kill combination CTRL-ALT-DE
failed. I had to switch off the power and reboot. The small memory size
of the computer along with the 2004 versions of ImageMacick and Linux
were not up to displaying the image. As a result I called the first
version of the image "paralyze.gif"
In 2009 energy saving features in Linux make it hard to run programs
for several hours, because the machine hibernates after quite a short
period of user inactivity. Faster machines and better programming
techniques make this point counting calculation reasonably fast even
for a larger image size.
Each 3x3 tile in the image must contain one blank point. This can
easily be proved by looking at the set of all curves t^2=s^3+sx+y (mod
3).
The vertical bars to the right correspond to values x = n^2 or the
square numbers. They express the fact that t^2=s^3+as+n^2 always has a
point (n,0), whatever the value of a. The slanting lines satisfy the
equations sy+x = t^2-s^3 and sy-x = t^2-s^3, or y= +- x/k +
(t^2-k^3/k). When k>1 the pixels on the sloping line become rather
spread out, compared to points that would be placed using a computer
line drawing algorithm. Areas of black correspond to elliptic curves
where there are no integer solutions for the checked range. A frequency
count for vertical lines shows sharp spikes at the points x=n^2.
The equations X^3+aX+b = 0 may be solved by the Newton Raphson
process. This gives a sequence
X[n+1] <- (2X[n]^3-b)/(3X[n]^2+a)
The fractal like image colours each pixel according to the rate of
convergence, for each equation with paramaters (a,b)=(y,x).
There are many questions arising from collections of curves like this.
In the 1920's Mordell wrote a whole book about an equation such as
y^2=x^3+6 which apparently has no integer solutions at all. It was
known since the early 1800s that points on these curves formed a group
under a law of composition which is simple to describe, but hard to
calculate. In the theorem named after him, Mordell proved that the
rational points on such a curve formed an Abelian group with a torsion
group T and r copies of the integers (Z). A subgroup T of an abelian
group T is called a torsion group if and only if for each t in T there
exists a number n = order(t) such that n.t = 0. n.t is the element
obtained by adding t to itself n times. The number 'r' is called the
rank of an elliptic curve. The calculation of the group for a curve is
tricky. The same applies when the curve is in a large finite field. In
the latter case group addition may be used as a basis for cryptographic
systems which avoid the RSA algorithm of Rivest, Shamir, Adelman.
Elliptic curve cryptography was all the rage for early mobile phones
systems in the 1990s. Another thing that has stimulated research in
this area was Wiles search for a proof of Fermat's Last Theorem (FLT).
Such a proof was found, but the mathemtics is not very accessible to
beginners.
FURTHER READING
Simon Singh's book on Fermat's Last Theorem remains one of the
best introductions to the subject. Improvements and
added features to image generating code, along with most of this essay
are based on 'Elliptic Curves over Q' by Steven Finch (2005).