Why are elliptic curves projective?

Posted on July 14, 2012 by charlesflorian

Before we go on to discuss what makes a complex torus algebraic, let us perhaps return to the case of elliptic curves. The claim is that all elliptic curves are algebraic; other than the one case explicitly provided in the first post, this has by no means been shown, so let us dwell on this a little further.

There are a few ways that we can see this fact. We will go over the most obvious one first.

Fix an element \tau in the upper half-plane (and so in effect, fix a lattice \Lambda_\tau and hence an elliptic curve). Consider the complex function

\displaystyle \wp(z) = \frac{1}{z^2} + \sum_{w \in \Lambda_\tau}\!\!{}'\ \Big(\frac{1}{(z + w)^2} - \frac{1}{w^2}\Big)

where by convention the primed summation means that we sum over all non-zero elements of the lattice. This function is called the Weirstrass \wp-function. Note that this function satisfies

\wp(z + w) = \wp(z)

for all z \in \mathbb{C} and w \in \Lambda_\tau. While it is not holomorphic (it has a pole of order 2 at every lattice point), it is meromorphic, and it is translation invariant for every element in the lattice. It thus yields a well-defined meromorphic function on the elliptic curve E_\tau = \mathbb{C}/\Lambda_\tau.

We claim the following relationship holds:

\displaystyle \big(\wp(z)'\big)^2 = 4\big(\wp(z)\big)^3 - g_2\wp(z) - g_3

where g_2, g_3 are given by the expressions

\displaystyle g_2 = 60 \sum_{w\in \Lambda_\tau}\!\!{}'\ w^-4

\displaystyle g_3 = 140 \sum_{w\in \Lambda_\tau}\!\!{}'\ w^-6

It is interesting to remark that these expressions g_2, g_3 are in fact the Eisenstein series of weights 4 and 6, respectively. This can be checked by verifying how they transform as functions of \tau under the two transformations

\tau \mapsto \tau + 1 \qquad \tau \mapsto -\frac{1}{\tau}

and noting that they are thus modular forms of weights 4 and 6, respectively; if you happen to know that these spaces are one-dimensional, then you are done. If not, then the following exercise is somewhat instructive.

Exercise Show that the function g_2(\tau) (where we now mention its explicit \tau-dependence) can also be written as

\displaystyle g_2(\tau) = \frac{4\pi^4}{3}\Big(1 + 240\sum_{k=1}^\infty \sigma_3(k)e^{2\pi i k \tau}\Big)

It is helpful to start with the identity

\pi \cot \pi x = \sum_{k \in \mathbb{Z}} \frac{1}{x + k}

which can be checked by looking at the zeros and poles of these two functions.

Now, the shown identity can be verified by comparing the poles on the left- and right-hand sides of the expression; as they match up, the two expressions must be equal (why?)

The point of all this, of course, is that the expression above provides us an explicit description of our elliptic curve as the affine plane curve

y^2 = 4x^3 - g_2x - g_3

and in particular, we see that every elliptic curve can be written as the curve cut out by a cubic polynomial.

Now, while this is all true, it is not particularly enlightening. If, after reading this, someone were to ask you “Why are elliptic curves projective?”, all you could answer would be “Because the Wierstrass \wp-function exists and satisfies a certain differential equation”. It would not answer, in any way, the question as to why complex tori are not necessarily projective. So if our goal was to answer that question, then this approach, while interesting, fails.

For us to move on, we need to perhaps consider more what exactly it means for a variety to be projective.

About charlesflorian

Mathematician. Climber. Living in Copenhagen.
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3 Responses to Why are elliptic curves projective?

  1. Atsushi says:
    July 15, 2012 at 5:21 pm

    There should be a function like Weirstrass p-function in higher dimensional case, although it may not provide us with projective embeddings. I am also interested in Ableian varieties associated to interesting lattices such as E8. They should have special properties.

    Reply
    • charlesflorian says:
      July 16, 2012 at 6:19 pm

      I’m not so sure that there is an obvious generalization of the p-function to higher dimensional cases; certainly, the naive approach of summing over the larger lattice and replacing the z with (z_1, \ldots, z_n) doesn’t really make sense.

      As for using E8, I’m not exactly sure what that would mean, really. A lattice in this sense is simply a co-compact subgroup of the Lie group \mathbb{C}^n, as opposed to a free abelian group together with a symmetric bilinear form. This makes me realize that the word ‘lattice’ in mathematics has at least three distinct meanings, two of which are somewhat confusingly close to each other…

      Reply
      • Atsushi says:
        August 1, 2012 at 5:02 pm

        You are right. A lattice in your sense is more general than “a free abelian group together with a symmetric bilinear form on \Z”. I just mean that the standard lattice with \tau=i and the hexagon lattice with \tau=e^{2*\pi*i/6} (the latter is also “lattice” if multiplied by \sqrt{2}) certainly yield more symmetric elliptic curves than general ones. In fact, they are useful to produce very interesting higher-dimensional varieties, such as rigid Calabo-Yau threefolds. Probably the standard lattice is not very interesting but the hexagon lattice is very special in the sense that it provide us with the best sphere packing in dimension 2. So I think an analogue of hexagon lattice can be E8 and the abelian 4-fold associate to E8 should have some special properties.

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