Elliptic Curves and Jacobians

Dear Readers,

Today we begin by talking about elliptic curves. We define an elliptic curve to be an abstract nonsingular genus one curve over a field k 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 E as an abstract nonsingular complete curve of genus one with a distinguished rational point O and as the zero set of a homogeneous degree 3 polynomial Y^2Z + a_1XYZ + a_3YZ^2 -X^3 -a_2X^2Z -a_4XZ^2 -a_6Z^3 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 Pic^0(E).

Let O = [0:1:0](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 E. We make a map from E \to Pic^0(E) by P \mapsto P - O. This is in fact a group homomorphism. Let P,Q be rational points on E and l denote the line in \mathbf{P}^2 connecting P,Q. Let R denote the third point of intersection with E dictated by Bezout’s Theorem. Let l' be the line in \mathbf{P}^2 connecting R, O and call the similar third point of intersection P\oplus Q(the addition of the two points in the tangent-and-chord group law). It suffices then to show that (P -O) + (Q -O) and P\oplus Q -O are equivalent up to the divisor of a rational function and we give it explicitly.  Since l, l' are lines, they are the zero sets of linear forms f = \alpha X + \beta Y + \gamma Z and f' = \alpha'X +\beta'Y + \gamma'Z. Thus f/f' gives a well-defined rational function on E. The divisor of this function is (P +Q + R) - (R + P\oplus Q + O) = (P -O) + (Q -O) - (P\oplus Q -O). This shows that (P -O) + (Q -O) = P\oplus Q -O and thus our map is a group homomorphism.

This homomorphism is surjective. If we let D be a divisor of degree 0, then D + O is a divisor of degree 1. Since E has genus 1, by Riemann-Roch, the k-dimension of L(D + O) is 1, and so we can pick f so that \textrm{span}(f) = L(D+O). Thus we consider the divisor of f in the sense of this post, div(f) = \sum_P v_P(f)*P. Since div(f) \in L(D+O), div(f) + D + O is an effective divisor of degree 1 and is thus 1 times a point P. Hence D is equivalent to P - O.

Finally we check injectivity. Suppose there were two rational points on E, P and Q such that a rational function f with the property div(f) + P -O = Q -O. Then Q = P + div(f) is an effective divisor of degree 1, so div(f) is effective and thus f is a nonzero constant function and v_R(f) = 0 for all points R. Therefore P = Q, i.e. P and Q cannot be distinct.

This (as with most things with elliptic curves) is the base case of a general phenomenon with other varieties. Let C be a curve of genus g with a rational point O. The Jacobian variety of C is a g-dimensional variety with a group structure isomorphic to Pic^0(C). 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 g copies of the curve modded out by the symmetric group S_g be a nonsingular variety of dimension g?” 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 g and have more tools to play with. Consider \Omega_1(C) be the space of holomorphic 1-forms on C. We can make a subset of elements of the dual space called the periods of C by sending a 1-form \omega to \int_\gamma \omega where \gamma is a closed path on C. If there are two closed paths \gamma,\gamma' where \gamma' = \gamma + \partial S where S is a closed subsurface of C then \int_\gamma = \int_{\gamma'} on \Omega_1(C) because closed 1-forms are exact and hence \int_{\partial S} \omega=\int_S \textrm{d}\omega = \int_S 0 = 0. Thus the periods of a Riemann surface C depend only on the homology class of \gamma, so the periods are in correspondence with H_1(C;\mathbf{Z}) \cong \mathbf{Z}^{2g}, so we call the set of periods \Lambda the period lattice. We call \textrm{Hom}(\Omega_1(C), \mathbf{C})/\Lambda the Jacobian of the curve.

But wait, why would this have a group structure isomorphic to Pic^0(C)? Recall that on C we have a distinguished rational point(well now that we are in the complex numbers, every point is rational) O and so we can, for every point P choose a path \gamma_P starting at O and ending at P. Thus we have a map C \to Jac(C) P \mapsto \int_{\gamma_P}\bmod \Lambda called the Abel-Jacobi map. This map extends to divisors of C and in particular to divisors of degree zero. It is a theorem of Jacobi that the map \phi: Div^0(C) \to Jac(C) extending the Abel-Jacobi map is surjective and it is a theorem of (you guessed it) Abel that the kernel of \phi is the set of principal divisors.

Next time I will move on to morphisms between curves, isogenies, dual isogenies and things of that sort.

This entry was posted in Abelian Varieties, Curves. Bookmark the permalink.

3 Responses to Elliptic Curves and Jacobians

  1. Very interesting!

    I wonder where you heard the rumor that a genus one curve without a rational point is at least as interesting as a genus one curve with a rational point…but the following seems to me to be a pretty solid proof of it.

    If C is a genus one curve over k without a rational point, then Pic^0(C) is a genus one curve over k with a rational point, which is (among other things), isomorphic to the original curve C when C itself has a k-rational point.

    However this association is in general far from being one-to-one: the set of all genus one curves C with Pic^0(C) isomorphic (as a variety together with a Pic^0(C)-action) to a given elliptic curve is called the WC-group of Pic^0(C), which is in general a big (infinite) abelian group.

    A similar, but simpler, argument works nicely to show that arithmetic geometry — i.e., study of algebraic varieties over not necessarily algebraically closed fields — must be at least as interesting as algebraic geometry.

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