Elliptic Curves

The set of points (x,y) satisfying the quadratic eqution y = ax^2 + bx + c is a parabola. Replacing x with x/\sqrt{a} this becomes y = x^2 + 2px + q where 2p = b/sqrt{a} and q = c. Since (x + p)^2 = x^2 + 2px + p^2 we have y = (x + p)^2 - (p^2 - q) = (x + p + \sqrt{p^2 - 1})(x + p - \sqrt{p^2 - q} = (x - r)(x - s) where r = -p - \sqrt{p^2 - q} and s = -p + \sqrt{p^2 - q}. Of course we could also take where r = -p + \sqrt{p^2 - q} and s = -p - \sqrt{p^2 - q} and get the same equation. There is a two element group acting on the set of roots that leaves the quadratic equation invariant. The identity is (r,s)\mapsto (r,s) and the other element of the group is (r,s)\mapsto (s,r).

The fundamental theorem of algebra states for any polynomial y = x^n + \sum_{j<n} a_j x^j there exist n roots r_j\in\bm{C}, j < n, with y = \Pi_{j<n} (x - r_j). The symmetric group of order n acting on the roots leaves the polynomial invariant.

An elliptic curve is the set of points (x,y) satifying y^2 = x^3 + a x^2 + b x + c. Note that an elliptic curve is not an ellipse.

Exercise. Find d such that X = x - d satisfies y^2 = X^3 + AX + B for some A,B.

We will use the form y^2 = x^3 + ax + b for elliptic curves in what follows.

Exercise. If P = (x,y) is on the curve then so is -P = (x, -y).

Hint: (-y)^2 = y^2.

The first remarkable fact about elliptic curves is that an addition can be defined for any two points P and Q on the curve. In general the line determined by P and Q intersects the curve at a third point R. Define P + Q = -R. Since the line determined by P and Q is the same as the line determined by Q and P we have Q + P = -R so addition is abelian.

Exercise. If Q = -P show only P and -P are the only points on the curve determined by the line containing P and -P.

If Q = -P we define P + Q = P + (-P) = 0.

If P = (x_P,y_P) and Q = (x_Q,y_Q) then R = (x,y) satisfies y = m(x - x_P) + y_P where m = (y_P - y_Q)/(x_P - x_Q) so (m(x - x_P) + y_P)^2 = x^2 + ax + b.