What do elliptic functions look like?
I'm often asked about the connection between elliptic curves and ellipses. Elliptic curves don't look ellipses at all, so why are they called "elliptic curves?" The simple answer that they come from what you get when you do arc length integrals on an ellipse usually leads to a discussion of elliptic integrals and elliptic functions, which often ends up talking about what elliptic functions look like.
I'm basically very lazy, so if I get asked something more than once or twice I often put the answer here so that I can just point to it in the future instead of having to answer the question again. So here's some discussion of what elliptic functions look like that I hope will save me some work in the future.
Elliptic functions are functions of a complex variable that generalize the familiar periodic functions from trigonometry by having two periods instead of just one. We also want to make sure that both periods don't point the same way in the complex plane, and requiring that their ratio isn't a real number is a concise way to say that.
What would an elliptic function look like?
It turns out to not be too hard to create one. Working with a power series is an approach that's often useful for functions of a complex variable. What do we get if we try to find a power series for an elliptic function? Let's start with the easiest possible case first.
The simplest form of an elliptic function would be one that looks like
f(z) = a0 + a1 z + a2 z2 + …
This function has no poles. But Liouville's theorem tells us that an analytic function defined on the entire complex plane that has no poles has to be a constant, so we'd have to have just
f(z) = a0
That means that we can't have a non-trivial elliptic function that's that simple.
What about the next simplest possibility? Can we have an elliptic function with a pole of order 1?
If we could, we could write
f(z) = a-1 z-1 + a0 + a1 z + a2 z2 + …
Let's suppose that this function has periods ω1 and ω2 and look at what happens when we evaluate ∫C f(z) dz where C is the path show in this picture that encloses the point z = 0:
Because the opposite sides of C differ by the periods of f and have opposite orientations, they actually cancel each other out, leaving us with
∫C f(z) dz = 0
But we also know that
∫C f(z) dz = 2 π i Res(f,0)
Res(f,0) = 0
But Res(f,0) is just a-1, and if a-1 = 0 then we're back to the first case where f has to be constant. That means we can't get a non-trivial elliptic function that way either.
What about the case where we have poles of order 2?
If we have poles of order 2 at each point
ω = n1 ω1 + n2 ω2
then we might want to look at something that looks like
f(z) = Σ (z – ω)-2
where the sum is over all
ω = n1 ω1 + n2 ω2
That's not too far from things that we've seen before, like
π2 sin-2 πx = Σ-∞<n<∞ (x – n)-2
This f(z) clearly has the two periods that we're looking for because we can replace any z with z + n1 ω1 + n2 ω2 and get the same sum. That looks somewhat promising, but if we think about it for a while we find that this series doesn't actually converge. Comparing it to
Σ ω-2 = Σ (n1 ω1 + n2 ω2)-2
= Σn1 Σn2 (n1 ω1 + n2 ω2)-2
and remembering that
Σn≥1 n-2 = π2/6
is probably the easiest way to see that. This is when quoting Homer Simpson by saying "D'oh!" probably seems like a good idea. Having two periods instead of one is apparently enough to throw off what works for π2 sin-2 πx.
But if we think about this some more, we can find that it's not too hard to force this series to converge by tweaking it slightly. In particular, if we look at
f(z) = Σ (z – ω)-2 – ω-2
then that actually works. It actually converges uniformly on compact sets that don't include any of the points n1 ω1 + n2 ω2. To write this more carefully we need to handle the case where both n1 and n2 are zero, so we can write this as
f(z) = z-2 + Σω≠0 (z – ω)-2 – ω-2
That's just we're looking for – a function of a complex variable that has two periods. This particular function is very useful in understanding elliptic functions because any elliptic function can be constructed from it and its derivative, much like any periodic function can be constructed from the sine function and its derivative. And because it's so useful, it even has a special name: the Weierstrass ℘-function, where the ℘ that doesn't display properly in some browsers is an old German version of the letter P, reminding us of the fact that the mathematicians who did most of the work in understanding elliptic functions back in the 19th century were Germans.
Understanding the ℘-function is also a very useful tool in understanding elliptic curves. The way that we add points on an elliptic curve and the reason that it makes sense follows directly from the properties of the ℘-function, for example. The reason that we use the Weierstrass form of an elliptic curve also does.
In many cases it's possible to amaze and astound people by taking some complicated identity that some 19th century mathematician proved about the ℘-function and interpreting it in terms of elliptic curves. Just don't tell people where you got it and they'll probably think that you know way more about elliptic curves than you really do. People have written books like The Bluffer's Guide to Wine, The Bluffer's Guide to Philosophy and The Bluffer's Guide to Art. Maybe there's enough material there to fill The Bluffer's Guide to Elliptic Curves.