June 9th: Elliptic Functions

Jun 9—2025

Given an elliptic curve $\mathbb{C}/{\Lambda}$ , how do we define holomorphic functions on it?

A mistake I made: Take a holomorphic function on $\mathbb{C}$ and then make it periodic in terms of the basis ${\xi_1, \xi_2}$ , i.e. $f(x + a\xi_1 + b\xi_2) = f(x)$ where a, b are integers. In my mind, I had been thinking about really nice functions, example: $f(z) = z$ .

I forgot about Liouville’s theorem…

If I take a nice function defined on the parallelogram spanned by $\xi_1, \xi_2$ and then make it periodic, it’s going to be bounded on the entire complex plane. Then Liouville’s theorem tells me that this will be a constant.

So to make life interesting, we need poles.

Now, the question comes: how do we get a holomorphic function with poles to be defined on $\mathbb{C}/{\Lambda}$ ? i.e. how do we get a function to be invariant under the symmetries of the lattice?

Key: given a space V and a group G acting on it, we can construct an element v in V invariant under G by taking the sum of its orbit. Recall the construction of the spherically invariant harmonic function by averaging the sum of the orbit of a harmonic function under the action of SO(n).

So here, we consider:

$\Sigma_{g \in G} g(v)$ .

In the scenario we are dealing with, V = set of meromorphic functions, and G = symmetries of the lattice (i.e. translations by multiples of $\xi_1, \xi_2$ ).

$f(z) = \Sigma_{\lambda \in L} f(z + \lambda)$ (where L is the group of translations that we are interested)

Obviously, there is a big problem. L is an infinite set, so this sum might not converge… and so f may not be well-defined.

In order for things to converge, we need $f(z) \rightarrow 0$ as $|z| \rightarrow \infty$ . So something like $f(z) = \frac{1}{z^n}$ might work. But for what n?

When n=1 (our encounter with this function in the real numbers already foreshadows the fact that this might be divergent), we have:

$f(z) = \Sigma_{\lambda \in L} \frac{1}{z-\lambda}$ .

In his first video on the elliptic functions, Borcherds gives a very nice argument for why this diverges. Suppose the lattice is $\mathbb{Z} \bigoplus \mathbb{Z}i$ . We divide our complex planes into rings of radius n (where $n \in \mathbb{N}$ ) centered at the origin. For n large, there are approximately n values (or a multiple of n?) of $\frac{1}{z - \lambda}$ such that it falls into the region between the circle of radius n and that of radius n-1. Why? My reasoning: The area between the two circles is $\pi(n^2 - (n-1)^2) = \pi (2n - 1) \approx 6n$ (intuitive explanation: we can fit approx. 6n unit squares into the region. For n large, we can recover around n ‘whole’ unit squares, so we can find around n such points). Within each of the region bounded by the two circles, $|\frac{1}{z - \lambda}| \approx \frac{1}{n}$ , and so, the absolute value of the sum of the points in the region is approximately 1. Thus, the series diverges.

For $\frac{1}{z^2}$ , applying the same argument as above, we see that the absolute value of the sum of the region bounded between two concentric circles is approximately $\frac{1}{n}$ , and so, the series diverges as well.