Let, $H = \{\text{complex numbers whose imaginary part is positive}\}$. First, you take the fundamental domain $\Omega$ (a subset of $H$) from figure 12.1.
Any modular form is determined by its values on $\Omega$. Now $\Omega$ is much smaller than the whole upper half-plane $H$. It’s a bit lopsided though.
We have included the left-hand portion of its boundary but not its right-hand portion.
It is much more equitable to work with $\Omega$, where we include both boundaries. (This is called the closure of $\Omega$.) But $\Omega$ is a little too big to be a fundamental domain. If $z$ is a point on its right-hand border, then the point $z − 1$ is on its left-hand border, and the two points are in the same orbit of $SL_2(\mathbb Z)$. Also, a point $z$ on the right half of the semicircle is in the same orbit as a certain point on the left half, namely $−1/z$.
So the fair thing to do is to work with all of $\Omega$ but to “identify’’ or “sew together’’ (à la topology) the right and left vertical borders and the right and left semicircles by attaching each $z$ in the boundary of $\Omega$ to the other point in its orbit in the boundary. When we do this sewing, we get something that looks like a stocking with a very pointy toe at $ρ$. (The point ρ is the sixth root of unity on the right.) There is also a less pointy place in the heel at $i$ (the square root of −1). Other than these two “singular’’ points, the rest of the stocking is nice and smooth. Because we built this shape out of a piece of the complex plane, the stocking still is a “complex space,’’ meaning we can do complex analysis on it. Let’s call this stocking $Y$.
There is a way to smooth out the two singular points $ρ$ and $i$ to make all of $Y$ into what is called a Riemann surface.
Above excerpts are taken from the book "Summing It Up" by Avner Ash andRobert Gross, 2016 (see Page 176).
PROBLEMS:
I don't understand why "We have included the left-hand portion of its boundary but not its right-hand portion".
I cannot visualize the stocking $Y$.
I don't understand how smoothing out the two singular points make all of $Y$ into what is called a Riemann surface. To be specific, what is a Riemann surface in this context?
REQUEST:
Can anyone provide a 2-D or 3-D picture of stocking $Y$ with descriptions that resolves above 3 problems?
A fundamental domain for a group action is defined to consist of a single point from every orbit. The goal here is that you should be able to specify a function which is stable under the group action by picking a function on the fundamental domain, and this choice shouldn't have any restrictions or redundancy. This last principle is why the left boundary and the right boundary can't both be in the fundamental domain: if I want to construct a periodic function $f$ and I pick a value for $f(1/2+i)$, this determines the value at $f(-1/2+i)$ since they're in the same orbit.
Are you familiar with how we construct a torus from a square by identifying opposite edges? Something similar is going on here. The left and right edges of $\Omega$ are identified, which means we can sort of pick them up out of the plane and bend $\Omega$ in to something which looks like a boba straw. But we're not done yet: we need to identify the two sides of the bottom and sew those together. If you could stretch the straw a little bit, this wouldn't be too hard, but you'd still have a sharp point at the corners of the straw opening.
Here a Riemann surface means what it usually does: a 1-dimensional complex manifold. (We mean 1 complex dimension here.) The idea is that $Y$ is already very close to being a 1-dimensional complex manifold: every point except for the corner points has a little ball which is isomorphic to a little ball in $\Bbb C^1$, so all we need to do is to fix those corner points and we're good. Intuitively, fixing means smoothing them out. For a precise statement, you would want to resolve the singularities of $Y$.