Last month I showed how to construct an elliptic function, a complex valued function with two periods. By the results proven in October, it must have a pole---otherwise it would be bounded and therefore constant---and since the sum of the residues at its poles must be zero, it must have at least two poles, which could occur as a double pole. But the periods of the elliptic function created last month are pretty boring. The periods are simply at right angles: $4K$ and $2K'i$. In fact, $\textit{all}$ parallelograms are possible. Choose any $\omega$ with positive imaginary part and consider the parallelograms with sides $1$ and $\omega$. By rescaling an elliptic function with these periods, rotating, and then adding a constant these can be transformed into any parallelogram you might want.

In 1850, Karl Weierstrass showed how to build an elliptic function with a double pole at the origin and periods 1 and $\omega$ for arbitrary $\omega$. Today it is known as the Weierstrass $\wp$-function. Let $\mathbb{L}$ be the set of points $\{ m + n\omega\:|\:m,n \in \mathbb{Z} \}$. These are the vertices of the paralellograms that we want to use to tile the plane. Then $$ \wp(z) := \frac{1}{z^2} + \sum_{\lambda \in \mathbb{L}\backslash 0} \left( \frac{1}{(z-\lambda)^2} - \frac{1}{\lambda^2} \right). $$

The Weierstrass $\wp$-function has many amazing properties, one of which is that it satisfies the differential equation in which the square of the derivative is a cubic polynomial, $$ \left(\frac{\wp'}{2}\right)^2 = \wp^3 + a \wp + b,$$ where the $a$ and $b$ can be computed from the values of $\wp$ at the half periods. This functional relationship, $y^2 = x^3 + ax + b$ has a graph that is known as the ``elliptic curve'' (see Figure 1). As we have seen, the connection to ellipses is not simple, making the name of this curve a bit misleading.

There is another way to construct elliptic functions, explained by Gauss in 1808. This involves complex-valued functions known as $\textit{modular functions}$. A simple example of a modular function is $$ \theta(z) = \sum_{n=-\infty}^{\infty} (-1)^n e^{\pi i n^2 z}. $$

These are only defined on the upper half-plane. They are periodic with a real period, but there also is a certain invariance when you replace the variable $z$ by the negative of its reciprocal, $-1/z$. Note that if you take a complex number with positive imaginary part, its reciprocal has negative imaginary part, and the negative of that has positive imaginary part, $$ a + bi \Longrightarrow \frac{1}{a+bi} = \frac{a-bi}{a^2 + b^2} \Longrightarrow \frac{-a+bi}{a^2 + b^2}. $$ Doing this does not leave the value of the modular function unchanged, but it changes the value in a simple and predictable way. The corresponding images are gorgeous (Figure 2). The values throughout the upper half-plane are uniquely determined by the values inside any pair of adjacent colored regions, known as the $\textit{fundamental domains}$.

Now we are into truly powerful modern mathematics. Modular functions have enabled the calculation of billions of digits of $\pi$. The proof of Fermat's Last Theorem rests on the connection between modular and elliptic functions. These functions and their generalizations lie at the foundations of statistical mechanics and our understandings of much of subatomic physics. They are intimately involved in some of most exciting collaborations between mathematicians and physicists today.