Tommy was talking about Riemann mappings. I can give you an idea of the Riemann mapping in question. The one he is searching for.
\[
\begin{align}
f(x) &= \sum_{n=0}^\infty \frac{x^n}{1+x^n}\frac{1}{2^n}\\
f(1) &= 1\\
f'(1) &= 1/2 \\
f(1+t) &= 1 + \frac{t}{2} + O(t^2)\\
\end{align}
\]
This is a standard quadratic Riemann surface branch. It is the only one on the unit circle. So we can think of this as a "quadratic" Riemann surface, with a single branch point.
Think of this point as the panama canal on the 3d topology of the earth. Where the western hemisphere is the unit disk \(|x| < 1\) and the eastern hemisphere is the unit disk \(|x|>1\).
We can now think of the panama canal as the value \(1 \in \mathbb{C}\) topologically. Where 1, is the sole point through, but we can't encircle this point without passing through a branch; traveling from water to land...
But ignoring this stupid metaphor....
This is what we do when we deal with Abel functions. But the study of Abel functions is explicit. We want to calculate them. We want to break it down to it's atoms. But Milnor, in his book, starts with basic Riemann mapping, Riemann surface. And consistently uses this as a foundation for existence of these things. In similar fashion, I am arguing for this Riemann mapping.
Since the topological mapping is quadratic. It looks like \(\sqrt{x}\), which has a single branch point and is quadratic... It's a second order Abel graph, in many ways.
This means the function \(f(1+t)\) is mappable to:
\[
\sqrt{t}\\
\]
For \(t \approx 0\). So that:
\[
h(f(1+h^{-1}(t))) = \sqrt{t}\\
\]
The function:
\[
\sqrt{t} : \mathbb{C}/(-\infty,0) \to \mathbb{C}/(-\infty,0)\\
\]
Thereby we can find a Riemann mapping of:
\[
h:\mathbb{C}/(U/1) \to \mathbb{C}/(-\infty,0]\\
\]
Where then, (Again, this is just implicit existence. Not construction.), we get:
\[
\sqrt{t} = h(f(h^{-1}(t)+1))\\
\]
If anyone has any questions, I will point out that they are called Riemann surfaces, because they relate to multidimensional surfaces of the Cauchy-Riemann equations. This stuff gets super advanced. So I can point you in the direction, I can't teach it to you. And I am not proving a solution. I am simply narrating my memory on the existence of these transforms.
The main point and the most important part, is that:
\[
h(1+t) = 1 + \frac{t}{2} + O(t^2)\\
\]
Let's take \(q \in U/1\)... then:
\[
h(q+ t) = \infty\,\,\,\text{or, at best}\,\, h(q) + O(t^2)\\
\]
This is also a common Riemann surface. It relates to the border being the Real Line. But at infinity, both functions merge...
Maybe, because you're interested in Modular functions Caleb, you'd be interested in this...
Nothing but love, Tommy. But this is how I remember these arguments going.
\[
\begin{align}
f(x) &= \sum_{n=0}^\infty \frac{x^n}{1+x^n}\frac{1}{2^n}\\
f(1) &= 1\\
f'(1) &= 1/2 \\
f(1+t) &= 1 + \frac{t}{2} + O(t^2)\\
\end{align}
\]
This is a standard quadratic Riemann surface branch. It is the only one on the unit circle. So we can think of this as a "quadratic" Riemann surface, with a single branch point.
Think of this point as the panama canal on the 3d topology of the earth. Where the western hemisphere is the unit disk \(|x| < 1\) and the eastern hemisphere is the unit disk \(|x|>1\).
We can now think of the panama canal as the value \(1 \in \mathbb{C}\) topologically. Where 1, is the sole point through, but we can't encircle this point without passing through a branch; traveling from water to land...
But ignoring this stupid metaphor....
This is what we do when we deal with Abel functions. But the study of Abel functions is explicit. We want to calculate them. We want to break it down to it's atoms. But Milnor, in his book, starts with basic Riemann mapping, Riemann surface. And consistently uses this as a foundation for existence of these things. In similar fashion, I am arguing for this Riemann mapping.
Since the topological mapping is quadratic. It looks like \(\sqrt{x}\), which has a single branch point and is quadratic... It's a second order Abel graph, in many ways.
This means the function \(f(1+t)\) is mappable to:
\[
\sqrt{t}\\
\]
For \(t \approx 0\). So that:
\[
h(f(1+h^{-1}(t))) = \sqrt{t}\\
\]
The function:
\[
\sqrt{t} : \mathbb{C}/(-\infty,0) \to \mathbb{C}/(-\infty,0)\\
\]
Thereby we can find a Riemann mapping of:
\[
h:\mathbb{C}/(U/1) \to \mathbb{C}/(-\infty,0]\\
\]
Where then, (Again, this is just implicit existence. Not construction.), we get:
\[
\sqrt{t} = h(f(h^{-1}(t)+1))\\
\]
If anyone has any questions, I will point out that they are called Riemann surfaces, because they relate to multidimensional surfaces of the Cauchy-Riemann equations. This stuff gets super advanced. So I can point you in the direction, I can't teach it to you. And I am not proving a solution. I am simply narrating my memory on the existence of these transforms.
The main point and the most important part, is that:
\[
h(1+t) = 1 + \frac{t}{2} + O(t^2)\\
\]
Let's take \(q \in U/1\)... then:
\[
h(q+ t) = \infty\,\,\,\text{or, at best}\,\, h(q) + O(t^2)\\
\]
This is also a common Riemann surface. It relates to the border being the Real Line. But at infinity, both functions merge...
Maybe, because you're interested in Modular functions Caleb, you'd be interested in this...
Nothing but love, Tommy. But this is how I remember these arguments going.
