This post is a continuation of my previous post about elliptic functions.
We showed that the function which is -invariant satisfies the differential equation
where the constants are given in terms of
We did this by neutralizing the only pole of on , by adding to a suitable polynomial in .
Thus we can use the functions to parametrize the curve
in . In fact we’re really parametrizing the projective curve
in by using the map
What we’re doing is exactly analogous to the parametrization of a conic using trigonometric functions.
With a bit more work, we can see that the field of elliptic functions with respect to is precisely the abstract field , subject to the relation (i.e. the quotient field of ). This means that for each -elliptic function , we can construct a rational function of and which has the same poles and zeroes, and thus express as a rational function in and .
Now since is algebraically closed, we can factor our equation as
for suitable values of . It’s easy to see from this equation that , where runs over the zeroes of . Counted with multiplicities, there are points where vanishes, since of degree as a cover of . Using the fact that is an odd function and periodic with respect to , we can see that vanishes at the symmetry points of the fundamental paralellogram having coordinates
Thus we have
Moreover, these three values are distinct. Indeed, it’s easy to see that each one is taken with multiplicity two by construction (i.e. each is a double zero of ), and since takes each value exactly twice, no two of them can be equal. To see that these points are double points, notice that the derivative of vanishes at , so the point is a double point. This implies that the discriminant of does not vanish, which implies after a quick check that the curve , which is the locus of zeroes of in the projective plane, is actually a nonsingular curve. (From now on we’ll call both , curves.)
So we have two curves: is defined in an analytic way, because its function field is constructed as a subfield of the field of meromorphic functions on . On the other hand, the curve is an algebraic curve.
In fact, the curves and are exactly the same in every respect, as it turns out. This means that the curve can be made into a group, since the curve is naturally a group (it’s just the torus group). Of couse, the magical thing that happens is that the group law on has a beautiful geometric interpretation, and that it’s given by rational functions on .
Let’s compare again with trigonometric functions. Consider the locus of in . We know how to add points on the (usual) circle by adding angles. We can prove by elementary geometry, or using the series definitions of trigonometric functions, the formula
which shows that the group structure on is given by rational (polynomial!) functions. What is amazing is that the group structure is compatible with the parametrization of – in fact, the group law becomes a pair of “addition theorems”: one for and one for .
For the rest of this post, I will assume that the reader is familiar with the simple geometric interpretation of the group law on an elliptic curve. For an easy description, see the wikipedia page.
So it is easy to “discover” the addition theorem for if we take the group law on for granted. By a simple calculation, we obtain that, for ,
For example, to get the formula for , we find the -coordinate of the point
using the geometric law. To ease notation a bit, let
Now the line passing through and (assuming they are distinct, so and are distinct points) is , where . We substitute this value of in the equation and we get the cubic in
Which is, after dividing by and rearranging terms,
Now we already know two roots of this cubic; they are the -coordinates of and , by construction. Thus, by inspecting the coefficient of in this cubic, which is , we see that
and hence, by the definition of addition on the elliptic curve and by the (still unjustified) assumption that the group structure is compatible with the coordinates , that the function satisfies the addition theorem
As expected, this expression is symmetric in and . The doubling formula, i.e. the case , is obtained by taking the limit as in the addition theorem.
Of course, none of this is justified because we haven’t explained why the coordinates should be compatible with the group structure. In fact, it makes much more sense to think of the group structure on the elliptic curve as a consequence of the addition theorems. So, to understand why the group structure really is what it is, we have to understand where these addition theorems really come from.
Recall from complex analysis that, for a function meromorphic on a domain , the integral
equals , where and denote the number of zeroes and poles of on , each taken with appropriate multiplicity.
By multiplying the differential by a function holomorphic on , we obtain a weighted sum over the zeroes and poles of . More precisely,
where and run over the zeroes and poles of in , respectively. In particular, taking , we see that
where denotes the order of at , i.e. the greatest integer such that is holomorphic at .
Now if is an elliptic function, and we take for a fundamental parallelogram, we see immediately using the periodicity of , that equals a -linear combination of elements of . Thus we see that , for an elliptic function ,
In particular, if is an elliptic function of order , then we can determine the position of any zero of from knowledge of the position of the other two; the zeroes are always , , and (mod ).
Now apply this to the elliptic function
which is of order . By construction, it has zeroes at and at ; thus its third zero is at . This means precisely that the line passing through and also passes through . Of course, these three points lie on the cubic . This explains precisely where the group law comes from. It also shows why we must reflect the third point of intersection of the line through and across the -axis.
All of this discussion can be carried out in a quite abstract setting using the Riemann-Roch theorem, which allows us to endow any smooth, genus algebraic variety having at least one point with a group structure, as above. It follows from the general construction that the group structure on an elliptic curve is isomorphic to , the degree Picard group of .
In a future series of posts, I will discuss the similarities between the theory of number fields and the theory of elliptic curves, which lead to the Birch and Swinnerton-Dyer conjecture.