$SL_2(\mathbf Z)$ has a fundamental domain $\Omega$ for which it “acts’’ on the upper half-plane $\mathbf H$. Suppose that $f (z)$ is a modular form for a fixed congruence group $\Gamma$. We only know the transformation law for $γ$ in $\Gamma$, which may well be considerably smaller than $SL_2(\mathbf Z)$. From the values of $f$ on $\Omega$, we don’t have enough information to determine the values of $f$ everywhere. So enlarge $\Omega$, so that it becomes a fundamental domain for $\Gamma$.
It turns out that there is a finite set of matrices $g_1, . . . , g_t$ in $SL_2(\mathbf Z)$ with the following property: If we define $\Omega_i$ = $g_i\Omega = \{z \in \mathbf H| z = g_i(w)$ for some $w ∈ \Omega \}$,
and then we form the union $\Psi$ = $\Omega_1 \cup \Omega_2 \cdots \Omega_t$, then $\Psi$ is a fundamental domain for $\Gamma$ in exactly the same sense that $\Omega$ is a fundamental domain for $SL_2(\mathbf Z)$.
3 (almost similar) questions pertain to the following quote from chapter 14 section 3 of "Summing it Up: From One Plus One to Modern Number Theory" by Avner Ash and Robert Gross (page 183):
"There will be a finite number of points on $\mathbf R \cup \{i\infty\}$ to which $\Psi$ will “go out’’ in the same way that $\Omega$ “goes out’’ to $i\infty$. These points are the cusps of $\Psi$."
I'm pretty new to the subject....so, I want to know/understand following:
Question 1: What does “go out’’ mean here?
Question 2: What does " a finite number of points on $\mathbf R \cup \{i\infty\}$ to which $\Psi$ will 'go out' " mean?
Question 3: How does "$\Omega$ “go out’’ to $i\infty$"?
From a Professor:
First of all, R you recall is the set of real numbers and we are viewing them as the points on the number line which is the x-axis, that is the points (x,0). Then there is i∞ which is harder to explain, since it is not a point in the complex plane. It is a point in the "extended complex plane", called an "ideal point". You do need some elementary topology to understand this rigorously, but I don't have a good reference for you. Any elementary book on complex analysis should contain all you need to know, but much more as well. ..., you could read pages 42-49 of our book Elliptic Tales. In the diagram on p. 49, on the right, you can see i∞ at the top, where it is labeled ∞.
The way you should think of i∞ is that it is an "extra point" that we stick on the upper half plane in the vertical direction "at infinity" i.e. infinitely far up. We could denote it by the coordinates (a,∞), where it is doesn't matter what a is. There is just one point "at infinity" in the vertical direction. When we say Omega "goes out" to i∞, we mean that if you try to go on an infinitely long walk in Omega, steadily increasing your y-coordinate, you may meander back and forth but "eventually" (after an infinite amount of time) you will "reach" i∞.
Now you can see this much more easily at any other cusp, i.e. a rational point in R (the x-axis.) For example, look at the diagram on page 184. This shaded fundamental domain has two cusps. One is i∞ and the other is 0. Do you see how the bottom part of the domain "funnells" into 0? That's what we mean by "going out" to the cusp: if you take a walk in the fundamental domain steadily going downwards towards 0, you will get closer and closer to 0 and eventually (after an infinite amount of time) you will "reach" 0.
Warning: for the last paragraph to make sense you have to measure distances using the "hyperbolic metric" in the upper halfplane. You could try looking at this wikipedia article: https://en.wikipedia.org/wiki/Poincar%C3%A9_half-plane_model.
The rigorous answer requires some knowledge of basic topology.
The compactified hyperbolic plane $\overline{\mathbb{H}}$ is the one-point compactification of the set $\{z \in \mathbb C \mid \text{Im}(z) \ge 0\}$, obtained by adding the one point denoted $i \infty$ (I think the notation $\infty$ is better for this point, but I'll stick to the notation $i\infty$ for this answer). Under the standard correspondence between the upper half plane and the Poincare disc, the point $i\infty$ of $\overline{\mathbb H}$ corresponds to the point $i$ of the closed Poincare disc.
So, with that in mind, to say that "$\Omega$ goes out to $i\infty$" means that $i\infty$ is the unique point in $\overline{\mathbb R} = \mathbb R \cup \{i\infty\}$ which is a limit point of the subset $\Omega$ with respect to the topology on $\overline{\mathbb{H}}$.
Similarly, each individual $\Omega_i$ will have a unique limit point in $\overline{\mathbb R}$; that's the point that $\Omega_i$ "goes out to".
One last important point: if you consider the collection of all translated fundamental domains $g \Omega$, as $g \in SL_2(\mathbb Z)$ varies, each of those translates has a unique limit point in $\overline{\mathbb R} = \mathbb R \cup \{i\infty\}$, and the set of all such limit points (as $g$ varies) is exactly $\mathbb Q \cup \{i\infty\}$. So the set of cusps of $\Psi$ becomes a finite subset of $\mathbb Q \cup \{i\infty\}$.