# On the group structure of elliptic curves

In last post we proved Abel’s theorem and as a corollary we saw that any compact Riemann surface of genus (a **smooth** **elliptic curve**) is biholomorphic to a complex torus, the biholomorphism being given by the Abel-Jacobi mapping:

which is given by

(mod )

where some , is a holomorphic 1-form and is the **period lattice** for the two **period vectors** for giving a basis for .

In particular, every such Riemann surface has a **group structure **and we will see that this description of the group structure on an elliptic curve is the same as the usual one given for elliptic curves in given by a cubic polynomial. By the usual group structure I mean the following: recall that by Bézout’s theorem, if you intersect a line in with the zero locus of a cubic polynomial, you get (counting multiplicities) exactly 3 points. Thus to add 2 points , you consider the line between them and declare that if is the third point of intersection of this line and the elliptic curve.

After a preliminary discussion of projective embeddings of Riemann surfaces, we will see that in fact any genus 1 compact Riemann surface can be embedded in as the zero locus of some cubic polynomial. In particular when the initial surface is a complex torus , this embedding is given by the so-called **Weierstrass -functions.** We will then see that this embedding is actually an inverse to the Abel-Jacobi mapping.

First the remarks on projective embeddings of compact Riemann surfaces. Suppose is a holomorphic line bundle over a compact complex manifold and are a basis for the global holomorphic sections of . Suppose that there is no point on such that every sections in vanish simultaneously. Then for every , we have a non-zero vector

.

By choosing a trivialisation, we can consider this vector as sitting in . Of course this vector will depend on the choice of trivialisation of around but different choices of trivialisations will only change the vector by a complex multiple. We thus get a map

defined by .

**Kodaira’s embedding theorem** states that when is a **positive** line bundle (i.e. admits a connection of positive curvature), there exists a such that for , the map

is well-defined (the global sections of are never all vanishing) and is an embedding of .

We can in fact give a sharper version of Kodaira’s embedding theorem for compact Riemann surfaces (you can find a detailed discussion of this on pages 214, 215 of Griffiths-Harris):

Theorem:Let be a compact Riemann surface and a holomorphic line bundle. If , then is well-defined and an embedding.

In this proposition, the degree of a line bundle is just under the identification or equivalently, since if then the Poincaré dual , the degree is . Also, is the **canonical bundle of ** and .

The case that will interest us is when the genus of is . What we show is that in fact we can take . Indeed, since because is trivial ( is topologically a torus), for any the above theorem tells us that for the bundle , the map gives an embedding of into for If , then are 3 linearly independent global holomorphic sections so . On the other hand, a global section of corresponds to a meromorphic function on having poles only at and at worst of degree . Such a function is completely determined by the first four coefficients in its Laurent expension around :

(the difference of two such functions having all of these 4 numbers equal would be a global holomorphic function vanishing at hence everywhere). Moreover, two such functions having the same and must have the same because otherwise their difference would be a meromorphic function having only one pole at and of order 1, which is impossible if the genus is not zero. So we need at most 3 parameters to determine such a function, hence . This shows .

We can write this embedding explicitly as follows: by Kodaira’s vanishing theorem, we have so considering the long exact sequence associated to the short exact sequence

where is the skyscraper sheaf at , we find so there exists a meromorphic function on having a double pole at and no other poles. Since , there is also on a nonzero holomorphic 1-form . But as remarked earlier, since the canonical bundle is trivial, the divisor has zero Poincaré dual so is itself zero, i.e. is nowhere vanishing. We consider . It is a meromorphic form with only one pole at , of order 2, and since the sum of the residues of a meromorphic 1-form must be zero, .

So if is a local holomorphic coordinate near , we can write

near

after possibly multiplying by a constant and adding a constant. Considering also the meromorphic function , which is holomorphic except at where it has a triple pole, we get a global meromorphic function which is

near

for the right choice of ‘s. Since and are three linearly independant global sections of , they form a basis and thus the embedding is given by

.

We can use this to explicitly describe as the zero set of a polynomial in : around , we have

and

.

Thus the meromorphic function has at most a simple pole at and no other poles, so is constant. This tells us that is included in the zero locus of the polynomial which can be rewritten by making suitable linear changes in the coordinates by

for . Note that here is a set of affine coordinates in on the open set . Since there are both topological torus (the second by the degree-genus formula), they must be equal, i.e. the zero locus of this polynomial in is exactly .

We will show that in fact the above construction gives an inverse for the Abel-Jacobi map which realises such a non-singular planar curve as a complex torus.

Consider and as above, with their pole at . Take as a basis for and the coordinate on such that . In this context, the function is the so-called **Weierstrass -function**. Its derivative is denoted . If the Laurent expansions of around contained a term of odd degree, then killing the order 2 pole by we would obtain a noncontant holomorphic function on . So it has no term of odd degree and we can write

.

We find the relation

,

where and . The corresponding embedding is then

where ,

and is the zero locus of the polynomial

,

written in suitable affine coordinates. On the other hand, the Abel-Jacobi map from to is given by

(mod )

with let’s say . Indeed, so is a non-zero holomorphic 1-form on . Moreover, is actually inverse to the embedding because

.

We have thus found **an inverse for the Abel-Jacobi mapping of a smooth elliptic curve in **.

We are now in a position to discuss the **group structure** from different point of views. Any Riemann surface of genus inherits a group structure simply by letting

for

There is also a group structure induced by an embedding as above : if are colinear in . These are in fact the same. To see this, consider as embedded in , take 3 points and denote the corresponding points in the complex torus. Then by Abel’s theorem,

iff with .

But suppose for , i.e. with the line joining (in affine coordinates), we also have . Then

is a meromorphic function on the torus having divisor so the function has divisor . Thus if are collinear, we have and conversely, if there is a meromorphic function on such that , then is a meromorphic function with and this forces because otherwise we would have a meromorphic function having a simple pole at and no other poles.