Elliptic Curves

Abstract

The theory of elliptic curves is one of the most beautiful and important theories in mathematics. There is no question about this. I know several ways to introduce elliptic curves, as you can find in many books and each method has advantages and disadvantages:

• (1) If I follow the history of mathematics, when I first attempt to compute the arc length of an ellipse, I find that this integral, which is called an elliptic integral, cannot be evaluated by trigonometric functions, and so on, as Gauss did. This method is good if you would like to know how discoveries were made, but it takes time and you may feel it is difficult.

• (2) I can present the theory of 1-dimensional complex analytic varieties, called Riemann surfaces, define an invariant called the genus, and introduce elliptic curves as compact Riemann surfaces of genus one. Such a presentation is good for learning modern theory of Riemann surfaces, but again it takes too much time.

• (3) I can define an elliptic curve as a double cover of the projective line branching at four distinct points. I shall come back to this point of view later.

• (4) I can define an elliptic curve as a non-singular cubic plane curve, but this it is too geometry-oriented for this book.