The Geometry of Elliptic Curves

Abstract

Elliptic curves, our principal object of study in this book, are curves of genus 1 having a specified basepoint. Our ultimate goal, as the title of the book indicates, is to study the arithmetic properties of these curves. In other words, we will be interested in analyzing their points defined over arithmetically interesting fields, such as finite fields, local (p-adic) fields, and global (number) fields. Before doing so, however, we are well-advised to study the properties of these curves in the simpler situation of an algebraically closed field (i.e. their geometry). This reflects the general principle in Diophantine geometry that in attempting to study any significant problem, it is essential to have a thorough understanding of the geometry before one can hope to make progress on the number theory. It is the purpose of this chapter to make an intensive study of the geometry of elliptic curves over arbitrary algebraically closed fields. (The particular case of the complex numbers is studied in more detail in chapter VI.)