 
Summary: Rational torsion points on elliptic curves
Amod Agashe
agashe@math.fsu.edu
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
figure below:
Elliptic curves have been studied for decades, and have recently found applications in cryp
tography.
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
yaxis. Any vertical line (i.e., a line parallel to the yaxis) is considered to pass through this point.
