Summary: Rational torsion points on elliptic curves
An elliptic curve is the set of solutions to an equation of the form y2 = x3 + ax + b, where a
and b are rational numbers such that 4a3 + 27b2 = 0. For example, if we take a = -1 and b = 0,
then we get the equation y2 = x3 - x, whose graph consists of the two thick curved pieces in the
Elliptic curves have been studied for decades, and have recently found applications in cryp-
Our interest is in the set of points on an elliptic curve whose coordinates are rational numbers;
such points are called rational points on the elliptic curve. For example, (1, 0) is a rational point
on the curve y2 = x3 - x sketched in the figure above.
One of the nice properties of elliptic curves is that given two points P and Q on the curve,
we can find a third point, called the "sum" of P and Q, and denoted P + Q, as follows. We first
join P and Q by a straight line (if P = Q, we take the tangent at P = Q). This straight line will
intersect the curve at another point; call it R. Then we draw a vertical line through R, which will
interesect the curve at yet another point; we define P + Q to be this latter point. This process is
illustrated in the figure above. If P and Q are rational points, then so is P + Q.
There is a special point on the curve called the zero element which lies infinitely high up on the
y-axis. Any vertical line (i.e., a line parallel to the y-axis) is considered to pass through this point.