elliptic curves over the complex number field

The most fruitful field over which the theory of elliptic curves is developed is perhaps \mathbb{C}.

First we shall explain the origin of the name, ‘elliptic curve’. This comes from elliptic integrals. Suppose that (x/a)^2+(y/b)^2=1 is an ellipse, then the length function with respect to x is just F(x)=\int_0^x\sqrt{1+(dy/dx)^2}dx. After some easy calculations, we see that F(x)=a\int_0^{x/a}\sqrt{(1-k^2x^2)/(1-x^2)}dx with k^2=1-b^2/a^2(we suppose that b<a). This leads to the study of the following kind of integrals

G(x)=\int_0^x\frac{1}{\sqrt{(1-k^2x^2)(1-x^2)}}dx

which is an elliptic integral(of the first kind, to be more precise). It is not this integral which interests us, it is the inverse function of the elliptic integral that leads to the study of elliptic curves.

In fact, when we allow x to take complex values, we can show that the inverse G^{-1} is double-periodic. This is rather interesting. This, in some sense, means that, the above integral is not well defined if we allow x to take complex values. Just like the logarithm function, log(z)=log|z|+i\theta+2ki\pi(k\in\mathbb{Z}), the elliptic integral G has similar property: the value of G(x) is unique up to an additive factor in \mathbb{Z}\omega_1+\mathbb{Z}\omega_2. This is the essence of the elliptic integrals.

Apart from constructing elliptic functions from elliptic integrals, do we have other methods? Indeed, we do have. One famous example is the Weierstrass \wp function. For a lattice \Gamma=\mathbb{Z}\omega_1+\mathbb{Z}\omega_2 in \mathbb{C}, we define

\wp(z,\Gamma)=1/z^2+\sum_{w\in \Gamma-0}(1/(z-w)^2-1/w^2)(z\in\mathbb{C})

It is easily shown that \wp(z,\Gamma)(meromorphic function) is indeed \omega_1,\omega_2-periodic. What is more, it has a double pole on 0\in \mathbb{C}/\Gamma, it is an even function. Its derivative, \wp(z,\Gamma), has a pole on 0 of order 3, it is an odd function. We can show that, the field of meromorphic functions \mathfrak{M}(\mathbb{C}/\Gamma)(that is, the elliptic functions with periods in \Gamma) can be expressed using only these two functions, that is

\mathfrak{M}(\mathbb{C}/\Gamma)=\mathbb{C}(\wp(z,\Gamma),\wp'(z,\Gamma))

In fact, we can show that, if f\in\mathfrak{M}(\mathbb{C}/\Gamma), then it is a rational function of \wp, \wp'. Suppose that f has a pole at P of order n. If P=0 on the torus, then we can find m,p\in\mathbb{Z} such that \wp^m\wp'^p has also a pole on P of order n. Then choosing some constant c, we have that f-c\wp^m\wp'^p has a pole on P of order less than n. In this way, we eliminate the pole of f on P=0 of order n(we can continue this procedure to eliminate all the poles on P=0). If f does not have a pole on 0, we suppose that f has n poles altogether(on the torus, of course, not on the complex plane). Then consider f(z)-f(0), this function has also n poles altogether ans still lies in \mathfrak{M}(\mathbb{C}/\Gamma). So, the function 1/(f(z)-f(0)) has a pole on 0 of order at least 1. So, we can use the above procedure to eliminate its pole on 0, and so get an elliptic function having less than n poles altogether. So, with this method, we can reduce the number of poles of these rational functions of f. At last, we will surely get an elliptic function having no poles, which means that it must be a constant, thus proving the statement.

Another surprising result is that, \wp,\wp' are not independent. In fact, we have that

\wp'^2=4\wp^3-g_2\wp-g_3

where g_2=g_2(\Gamma)=60G_4(\Gamma),g_3=g_3(\Gamma)=140G_6(\Gamma) with G_k(\Gamma)=\sum_{w\in \Gamma-0}1/w^k(k>2).

This formula ressembles much to the polynomial defining an elliptic curve. In fact, we can show that

For any elliptic curve E(\mathbb{C}), we can find some lattice \Gamma such that up tp isomorphism E is defined by a polynomial y^2=4x^3-g_2(\Gamma)x-g_3(\Gamma).

In this way, we establish a one-to-one correspondance between the set of elliptic curves and the set of lattice up to scalar factor. Here the ‘scalar factor’ needs to be explained. In fact, this comes from an easy fact, \Gamma should be the same as a\Gamma(\mathbb{C}/\Gamma should be the same as \mathbb{C}/a\Gamma) where a is a non-zero complex number. In this way, we can always assume that \Gamma=\mathbb{Z}+\mathbb{Z}\tau with Im(\tau)>0. The most important result about this correspondance is perhaps the following:

Suppose an elliptic curve E: y^2=4x^3-g_2(\Gamma)x-g_3(\Gamma), then the map \phi:\mathbb{C}/\Gamma\rightarrow E, z\mapsto (\wp(z),\wp'(z))(0\mapsto e_E) is a bi-holomorphism and isomorphism of groups.

In this way, we use the natural group structure on \mathbb{C}/\Gamma to induce a group structure on the elliptic curve, and surprisingly, this induced group structure coincides with the more well-known group structure on the elliptic curve defined using intersection of straight lines with the elliptic curve.

In summary, we have three objets: an elliptic curve E, a lattice \Gamma(and the associated torus \mathbb{C}/\Gamma), and the function field \mathbb{C}(x,y)/(y^2-(4x^3-g_2x-g_3)).

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