In "A first course in modular forms" (Diamond-Shurman) the author defines something called an 'enhanced elliptic curve' for the congruence subgroups $\Gamma_0(N), \Gamma_1(N)$ and $\Gamma(N)$. For example an enhanced elliptic curve for $\Gamma_0(N)$ is an ordered pair $(E,C)$ where $E$ is a complex elliptic curve and $C$ is a cyclic subgroup of $E$ of order $N$. And later we prove the space of enhanced elliptic curves modulo the equivalence relation $(E_1,C_1) \sim (E_2, C_2)$ if $\exists$ an isomorphism of Elliptic curves $\phi: E_1 \rightarrow E_2$ such that $\phi(C_1)= C_2$ is in bijection with $\Gamma_0(N)\backslash\mathcal{H}$.
Now, I have a few questions :
a) If we accept that the set of elliptic curves upto isomorphism is in bijection with $G \backslash \mathcal{H}$, where $G$ is the full modular group; then in these particular cases what was the motivation behind considering a cyclic group of order $N$ (for $\Gamma_0(N)$, a point of order $N$ (for $\Gamma_1(N)$),etc., as the 'torsion data.'
And a related but slightly general question,
b)What is the intuition/technique behind deciding what extra torsion data should be added so that the bijection between elliptic curves with extra data (with an equivalence relation) and the modular curve works for a general congruence subgroup ?
Thanks in advance and I am sorry if the questions was not concisely worded.
Oh boy, this is a topic which one can (and one has) written entire books about, and which lies at the very beginning of an extremely rich field of math (the theory of Shimura varieties).
Let me just say some words that should hopefully make you happy.
a) So, we have a 'canonical' bijection
$$\{\text{elliptic curves over }\mathbb{C}\}/\text{iso.}\longleftrightarrow \mathbb{H}/\Gamma(1)\longleftrightarrow \mathbb{C}$$
(the second arrow actually being an isomorphism of open Riemann surface). The composite map taking an elliptic curve $E/\mathbb{C}$ to its $j$-invariant. This is great because what it allows for us to do is endow the set of elliptic curves with a geometry—the geometry coming from the bijection, provided by $j$, with $\mathbb{C}$ (an algebraic structure even!).
But, if we're going to give the set of elliptic curves over $\mathbb{C}$, a set with overwhelming mathematical importance, a geometry we better be damn sure that we're giving it the right geometry. And, what above guarantees that it's the right geometry? Who is to say that I can't create a 'natural bijection' between the elliptic curves over $\mathbb{C}$ and some quotient $\mathbb{C}^{75}\times \mathbb{P}^7_\mathbb{C}/G$ for some group $G$? Maybe I can. Why not give it the geometry coming from this 'natural bijection'?
The issue is that one cannot hope to give an object intrinsic geometry by understanding just its set of $\mathbb{C}$-points. Said more abstractly, if we want to think about $Y(1)$ as being the 'space of elliptic curves' as a complex manifold then, by Yoneda's lemma we know that to understand $Y(1)$ (as a complex manifold) is to understand the set of holomorphic maps $\mathrm{Hom}(X,Y(1))$ where $X$ ranges over ALL complex manifolds. Here one should interpret $\mathrm{Hom}(X,Y(1))$ (through the lens that $Y(1)$ is the 'space of elliptic curves') as being families of elliptic curves indexed by the points of $X$ (which vary 'holomorphically').
Now, by looking at just the set of elliptic curves over $\mathbb{C}$ we're just looking at the $\mathbb{C}$-points of $Y(1)$: the set $\mathrm{Hom}(\mathbb{C},Y(1))$. This is not sufficient to uniquely pin down $Y(1)$ again, as Yoneda suggets, we need to do this for all $X$. Ok fine, I'm just being pedantic, right? Let's just define $Y(1)$ as being the Riemann surface that satisfies precisely what I said it did above. It's the Riemann surface for which there is a canonical identification
$$\mathrm{Hom}(X,Y(1))\longleftrightarrow\{\text{holomorphically varying families of elliptic curves over }X\}$$
(canonical identification meaning identification of functors, but you can ignore that if you want). Then, whatever $Y(1)$ is, its geometry will be uniquely pinned down by this property, and we can stop all of this pedantry.
Unfortunately, no, we can't. There is NO complex manifold $Y(1)$ satisfying the above property. If we move over to the algebraic category this is because the functor 'elliptic curves over ____' is not a 'sheaf'. While this may not resonate with you, it certainly is not the case that the $j$-invariant provides an identification of $Y(1)$ (satisfying the property I said above!) with $\mathbb{A}^1=\mathbb{C}$ since this says any family whose fibers all have the same $j$-invariant is a trivial family.
The issue with this 'moduli problem', the thing that makes it not a 'sheaf', is that elliptic curves have automorphisms. This turns out to affect the non-sheafyness and so affect the inability to find a $Y(1)$ satisfying the above property.
There are two ways of proceeding from here. We could a) go to the nuclear option and not ask for a complex manifold (or scheme) which satisfies he desired properties of $Y(1)$ but ask for an orbifold (DM stack) which does. Or, we could b) rigidify the problem (rigidify the objects under consideraiton) to eliminate automorphisms. I don't think you want to do a) (it also has intrinsic downsides), and so we can do b). And, one of the most fruitful ways to rigidify elliptic curves is by adding in level data—torsion data about the elliptic curves we are considering.
For example, let's fix an integer $N$ and define an elliptic curve with 'level $\Gamma(N)$' data as being a pair $(E,\alpha)$ where $E$ is an elliptic curve and $\alpha$ is an isomorphism $E[N]\xrightarrow{\approx}(\mathbb{Z}/N\mathbb{Z})^2$. Said differently, we're choosing a basis for the $N$-torsion. Note that it's only true that abstractly is $E[N]$ equal to $(\mathbb{Z}/N\mathbb{Z})^2$, there is no canonical isomorphism—no canonical basis.
One can show then that for $N\geqslant 3$ there is no non-trivial automorphism of an elliptic curve $E$ preserving a level $N$-data. In other words, for a pair $(E,\alpha)$ there is not automorphism of $E$ taking $\alpha$ to $\alpha$ (it will always take one level $\Gamma(N)$ data to another, but we want to demand it takes it to the same one). Thus, we have a fighthing chance of representing the functor 'families of elliptic curves with level $\Gamma(N)$ data'. And, lo and behold, the functor $Y(N)$ taking a complex manifold $X$ to the families of elliptic curves over $X$ with level $\Gamma(N)$ data is representable by a smooth affine curve (which we also call $Y(N)$. One can show that as a complex manifold one has that $Y(N)\approx \mathbb{H}/\Gamma(N)$ (this is a bit of a lie, but one of not particular importance for you--it's true that the LHS has connected components isomorphic to the RHS).
Essentially the same story goes through if we replace $Y(N)$ with $Y_1(N)$. The functor taking a complex manifold $X$ to holomorphically varying families of elliptic curves over $X$ with a points of order $N$ is representable for $N\geqslant 3$.
Unfortunately, $Y_0(N)$ with the obvious 'family interpretation' is never representable. That said, one can still ask for a complex manifold which 'best approximates' the moduli problem (perhaps the moduli problem for $Y_0(N)$). This best approximation doesn't exist for arbitrarily moduli problems (=functors) but it does for the $Y_0(N)$ and for $Y_1(1)=Y(1),Y_1(2),Y(2)$. We call these best approximations, these coarse moduli spaces, also $Y_i(N)$.
For examples, there is a coarse moduli space for $Y(1)$ and it is, as you might have guessed it, is just $\mathbb{A}^1=\mathbb{C}$. But this is already an indication that the coarse moduli space is undesirable in some situations. There is no way that the geometry of the space of families of elliptic curves is dictate by the geometry of $\mathbb{C}$ even if they agree on $\mathbb{C}$-points (as sets).
This is one reason to add in level data, to rigidify.
Another obvious one is that we just care about torsion data, thus it's a natural choice of rigidification. To understand an elliptic curve over $\mathbb{Q}$ is, in some very coarse sense, to understand its Mordell-Weil group $E(\mathbb{Q})$. This has two components, the free part and torsion part. So a natural question we want to understand for elliptic curves is what their torsion parts look like. For example. do all torsion groups show up in the torsion of some Mordell-Weil group over $\mathbb{Q}$? Said differently, is it true that $Y_1(N)$ has a $\mathbb{Q}$-point for all $N$?
b) This also has a satisfactory answer, but one which is more difficult to explain. It is a good question. Why not consider elliptic curves with "blah blah blah torsion data"? Why these moduli problems? Well, besides the fact that they are somewhat 'natural choices', they are (in the $\Gamma(N)$ level case) somewhat 'cofinal' in all possible level data. Namely, there is a nice theory of generalized level data which feeds in some complicated objects (having to do with subgroups of $\mathrm{GL}_2(\mathbb{Z})$, or, more precisely, $\mathrm{GL}_2(\mathbb{A}_f$), the adeles). But for the level data we can deal with on a rigorous level (the compact open subroups) they all are 'congruence', they contain a $\Gamma(N)$ (or, rather, its completion) which makes this a sort of exemplifying case.
In particular, one usually cares about the 'tower' of these level data (the so-called Shimura variety for $\mathrm{GL}_2$ or its inverse limit, and this inverse limit doesn't care if one passes over just a cofinal system—in this case provided by the $\Gamma(N)$ level data.
I would be more than happy to clarify on any of the above points.