I fell a bit behind on the continuation of the class field theory series because I was setting up a new laptop. Before I resume that, I want to talk about something very weird that I learned today.

Let ${U \subset \mathbb{C}}$ be a set that omits at least two points. If ${f: U \rightarrow U}$ holomorphic and is such that $f(w)=w, {f'(w)=1}$ at one ${w \in U}$, then ${f}$ is the identity.

This is a striking rigidity phenomenon!

But how do we prove it? The idea is to consider the sequence of iterates ${f, f \circ f, \dots}$. Suppose for simplicity ${P=0}$. Then in a neighborhood of ${0}$, we can write ${f = z + cz^m + \dots }$, where the ${\dots}$ are omitted higher terms. If ${f}$ is not identically the identity, then ${c \neq 0}$.

So, similarly, by direct computation, in some neighborhood of ${P}$, we have ${f \circ f = z + 2c z^m + \dots}$. Similarly, if we define ${g_1 = f, g_2 = f \circ f, }$ for notational convenience, we have

$\displaystyle g_k = z + kc z^m + \dots.$

But the ${g_k}$ are all holomorphic maps into ${U}$. Since ${U}$ omits at least two points, the family ${g_k}$ is normal by Montel’s theorem and consequently has a subsequence ${g_{k_i}}$ that converges uniformly on compact sets.

Thus the derivatives ${g^{(m)}_{k_i}(0) = m! k_i c}$ converge, which is impossible unless ${c=0}$.

Huh? I didn’t exactly see that coming. If ${U}$ is the unit disk, then at least it looks familiar. A holomorphic map ${f}$ of the unit disk into itself sending zero to zero must satisfy ${|f'(0)| \leq 1}$, and if equality holds ${f}$ is a rotation. So perhaps this result should be thought of as a generalization of Schwarz’s lemma? (Nevertheless, the use of Montel’s theorem is quite a sledgehammer to prove something as elementary as Schwarz.)

I should say where I got this from: Krant’z Complex Analysis: The Geometric Viewpoint. Krantz didn’t prove exactly this, but the argument is the same.  Either this is standard fare that I missed when learning basic complex analysis, or I’m turning Climbing Mount Bourbaki into a comedy routine.