The following is from the book Modular Forms by W Stein:
My questions:
$1-$ The very same book defines a cusp form as a modular form when $f(\infty)=a_0=0$. Is the set of cusps a different concept irrelevant to a cusp form? Otherwise how $\mathbb{P^1(Q)}$ is a cusp form?
$2-$ By the book Algebra by P Aluffi, the orbit of $a ∈ A$ under an action of a group $G$ is the set $O_G(a):={\{g.a | g∈G }\}$. So $$C(\Gamma)={\{\gamma.q \ | \ \gamma∈\Gamma, \forall q ∈\mathbb{Q}\cup{\infty} }\} = {\{\ \dfrac{az+b}{cz+d} \ | \ \text{matrix [a,b,c,d]} ∈\Gamma, z ∈\mathbb{Q}\cup{\infty} }\}.$$ Is that right? (Sorry I couldn't right matrix 2*2 in latex)
$3-$ With definition on $Q.2$ (if it is right) :
$a)$ how $C(SL_2(\mathbb{Z}))={\{ [\infty] }\}$?! and,
$b)$ does ${\{ [\infty] }\}={\{ \infty }\}$ mean?
$4-$ How the lemma is related to the set/equality in $Q.3$? That is, how the (proof of the) lemma implies $C(SL_2(\mathbb{Z}))={\{ [\infty] }\}$?
Cusp forms are distinguished by their behavior at cusps. $\infty$ is a cusp. Moreover, because modular forms transform in a particular way under the action of $SL_2(\mathbb{Z})$, describing the behavior of a modular form at $\infty$ implies things about its behavior at every other cusp as well.
No. What your notation describes is just all of $\mathbb{P}^1(\mathbb{Q})$. The definition refers to the set of orbits.
$SL_2(\mathbb{Z})$ acts transitively on $\mathbb{P}^1(\mathbb{Q})$, so the action consists of a single orbit. The parenthetical says: "we will often identify elements of $C(\Gamma)$ with a representative element from the orbit." So the notation $\{ [\infty] \}$ means: there is one orbit, which is the orbit represented by (containing) $\infty$.
The lemma says precisely that the action is transitive.