An elliptic curve
is given by a complex polynomial
of degree three
in two variables f(y,x)
that is nonsingular (i.e. there is a well-defined tangent
at every zero of f(x,y)
An example is f(x,y)=y2+y-xy-x3.
The curve is the set of zeros of f(x,y) in C2,
complex 2-space. Actually this isn't quite true. Because parallel lines
don't meet it is necessary to throw in extra points at infinity
and so the elliptic curve is the set of zeroes of this polynomial
in P2(C), the complex projective plane. After we've done this
every line in the plane meets the elliptic curve in exactly three points.
If the coefficients of f(x,y) are rational numbers then it is
an interesting problem in Diophantine number theory to study the rational points
on the curve (i.e. the ones with rational coefficients).
One of the most interesting things about elliptic curves is that the points
of the curve have the structure of a group. This is quite easy to understand,
but you will probably want to draw a diagram. This group law was introduced by
Jacobi in 1835.
First of all, given any two points, P and Q on the curve we can form the
line through the points and this intersects the curve at a uniquely determined third
point. Let's call it pt(P,Q).
You might be a bit concerned about how to define pt(P,P) but this is OK
because the tangent line to the curve at P (which meets the curve at P twice)
intersects the curve in exactly one other point pt(P,P).
This gives us a binary operation
on the curve but it doesn't give us a group. To do this we first need to choose
a point O on the curve, which is going to be the identity element in our
group. Now the line through O and pt(P,Q) again meets the curve
at a unique point, which we call P+Q.
With a little effort one can show that this addition operation makes the curve into a
group with identity O. Since Q+P=P+Q it is an abelian group.
Note that -P is the third point of intersection of the curve with the line
through P and pt(O,O).
For the example given earlier take O to be the point at infinity where
all the vertical lines meet. In that case the point P=(1,1) has order 6.
A quick calculation shows:
In the number theory setting (so we have a curve with rational coefficients) we would take
O to be a rational point and this ensures that the group operation restricts
to the rational points of the curve giving a subgroup.
In 1921 Mordell proved that this subgroup is finitely generated a result that was
assumed, if not proved explicitly, by Poincaré in 1901.