Lecture 2

Last time we have seen that the proof of Fermat’s Last Theorem for the exponent 4 provides us with two elliptic curves (y = x + x and y = x − 4x) in the guise of the quartic equations (X + Y 4 = Z and X − 4Y 4 = Z). Moreover, in the actual proof we used connections between these quartic curves and the ‘underlying conic’, namely the quadratic curves x+ y = z and x− 4y = z, which actually both represent the unit circle after a little change of coordinates. What we will do today is sketch the basic theory of elliptic curves, and at the same time compare it with the much simpler theory of conics.