Today we begin by talking about elliptic curves. We define an elliptic curve to be an abstract nonsingular genus one curve over a field with a given rational point. I’m told the theory of genus one curves without rational points is perhaps even more interesting than that of elliptic curves, but that may or may not be a topic for the future(when I know more about them)
Usually when one talks about elliptic curves, they talk about them in terms of Weierstrass equations, and it was shown that we can do so in a previous post. As such, we speak equally of an elliptic curve as an abstract nonsingular complete curve of genus one with a distinguished rational point and as the zero set of a homogeneous degree 3 polynomial which is unique up to a very specific set of coordinate changes, which we will detail in further posts, particularly when it comes to the reduction map for elliptic curves.
We now show that the chord and tangent group law given in the earlier post on elliptic curves coincides with the group of divisors of degree 0 mod the principal divisors, which we denote .
Let (i.e. the cubic flex of the Weierstrass equation, although we could take it to be any point on the curve whatsoever) be the distinguished rational point of . We make a map from by . This is in fact a group homomorphism. Let be rational points on and denote the line in connecting . Let denote the third point of intersection with dictated by Bezout’s Theorem. Let be the line in connecting and call the similar third point of intersection (the addition of the two points in the tangent-and-chord group law). It suffices then to show that and are equivalent up to the divisor of a rational function and we give it explicitly. Since are lines, they are the zero sets of linear forms and . Thus gives a well-defined rational function on . The divisor of this function is . This shows that and thus our map is a group homomorphism.
This homomorphism is surjective. If we let be a divisor of degree 0, then is a divisor of degree 1. Since has genus 1, by Riemann-Roch, the -dimension of is 1, and so we can pick so that . Thus we consider the divisor of in the sense of this post, . Since , is an effective divisor of degree 1 and is thus 1 times a point . Hence is equivalent to .
Finally we check injectivity. Suppose there were two rational points on , and such that a rational function with the property . Then is an effective divisor of degree 1, so is effective and thus is a nonzero constant function and for all points . Therefore , i.e. and cannot be distinct.
This (as with most things with elliptic curves) is the base case of a general phenomenon with other varieties. Let be a curve of genus with a rational point . The Jacobian variety of is a -dimensional variety with a group structure isomorphic to . This is variety can be constructed over any arbitrary field if one is willing to sit down and ask some difficult questions such as, “Why should copies of the curve modded out by the symmetric group be a nonsingular variety of dimension ?” and “Why should a functor corresponding to points of that variety up to some sort of equivalence be representable?” I will not attempt to answer these questions here today, but will instead refer the reader to chapter 3 of James Milne’s online book on Abelian Varieties.
I will instead restrict myself to working over the complex numbers where we can consider a projective curve as a compact Riemann surface of genus and have more tools to play with. Consider be the space of holomorphic 1-forms on . We can make a subset of elements of the dual space called the periods of by sending a 1-form to where is a closed path on . If there are two closed paths where where is a closed subsurface of then on because closed 1-forms are exact and hence . Thus the periods of a Riemann surface depend only on the homology class of , so the periods are in correspondence with , so we call the set of periods the period lattice. We call the Jacobian of the curve.
But wait, why would this have a group structure isomorphic to ? Recall that on we have a distinguished rational point(well now that we are in the complex numbers, every point is rational) and so we can, for every point choose a path starting at and ending at . Thus we have a map called the Abel-Jacobi map. This map extends to divisors of and in particular to divisors of degree zero. It is a theorem of Jacobi that the map extending the Abel-Jacobi map is surjective and it is a theorem of (you guessed it) Abel that the kernel of is the set of principal divisors.
Next time I will move on to morphisms between curves, isogenies, dual isogenies and things of that sort.