18 . 783 Elliptic Curves Spring 2015

In Lecture 1 we defined an elliptic curve as a smooth projective curve of genus 1 with a distinguished rational point. An equivalent definition is that an elliptic curve is an abelian variety of dimension one. An abelian variety is a smooth projective variety that is also a group, where the group operation is defined by rational functions (ratios of polynomials). Remarkably, these constraints force the group to be commutative, which is why they are called abelian varieties. A variety is (roughly speaking) the zero locus of a set of polynomials, subject to an irreducibility condition. The precise definition won’t concern us here, it is enough for us to know that a variety of dimension one is a curve, so an abelian variety of dimension one is a smooth projective curve with a group structure specified by rational functions. We will prove in this lecture that elliptic curves are abelian varieties. In fact the converse holds, every abelian variety of dimension one is an elliptic curve, but we won’t prove this. As mentioned in the first lecture, it is possible to associate an abelian variety to any smooth projective curve; this abelian variety is called the Jacobian of the curve. The dimension of the Jacobian is equal to the genus g of the curve, which means that in general the Jacobian is a much more complicated object than the curve itself (which always has dimension one). Writing explicit equations for the Jacobian as a projective variety is quite complicated, in general, but for elliptic curves, the curve and its Jacobian both have dimension one, and in fact the Jacobian is isomorphic to the curve itself.