One can get modular curves by the following procedures: first take the uper half plane and the rationan numbers on the x-axis, then we consider the quotient by a congruence subgroup. Now we get a compact Riemann surface, and by Chow's results, it is algebraic–an algebraic variety. Here we start with an analytic object and finally we get an algebraic one. But can we use algebraic methords only (e.g. by quotients of group schemes) to get modular curves? Or, can we find a meanful moduli problem solved by a modular curve?
[Math] schemetical construction for modular curves over the rationals
ag.algebraic-geometrynt.number-theory
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For the groups $\Gamma$ in Conway-Norton, there is always a moduli problem of $\Gamma$-structures, but since the groups always contain $\Gamma_0(N)$ for some $N$, you won't be able to construct a universal family (because there is a $-1$ automorphism in the way). However, you will sometimes get a ``relatively representable'' problem in the sense of Katz-Mazur.
The upper half-plane quotients will be coarse spaces parametrizing objects of the following general form: You have a diagram of elliptic curves, with some isogenies of specified degrees between them, together with some data that tell you how much symmetry in the diagram you should remember. Since all of the groups normalize $\Gamma_0(N)$ for some $N$, the diagrams will typically involve cyclic isogenies of degree $N$ in some way, and the symmetrization will involve a subgroup of the finite quotient $N_{SL_2(\mathbb{R})}(\Gamma_0(N))/\Gamma_0(N)$.
The standard example is $\Gamma_0(p)^+$ for a prime $p$, which is generated by $\Gamma_0(p)$ as an index two subgroup, together with the Fricke involution $\tau \mapsto \frac{-1}{p\tau}$. The $\Gamma_0(p)$ quotient parametrizes diagrams $E \to E'$ of elliptic curves equipped with a degree $p$ isogeny between them. Taking the quotient of the moduli problem by the Fricke involution amounts to symmetrizing the diagram, so the $\Gamma_0(p)^+$ quotient parametrizes tuples $( \{ E_1, E_2 \}, E_1 \leftrightarrows E_2)$ of unordered pairs of elliptic curves, with dual degree $p$ isogenies between them. Equivalently, you can ask for a set of diagrams $\{E_1 \to E_2, E_2 \to E_1 \}$ where the maps are dual isogenies.
A less well-known example is the 3C group, which is an index 3 subgroup of $\Gamma_0(3|3)$, with Hauptmodul $\sqrt[3]{j(3\tau)} = q^{-1} + 248q^2 + 4124q^5 + \dots$. This group is labeled $\Gamma_0(3|3)$ in the Conway-Norton paper, because $\Gamma_0(3|3)$ is the eigengroup, namely the group that takes the Hauptmodul to constant multiples of itself. The 3C group contains $\Gamma_0(9)$ as a normal subgroup, with quotient $\mathbb{Z}/2\mathbb{Z} \times \mathbb{Z}/2\mathbb{Z}$. You can view the upper half-plane quotient as a parameter space of quadruples of elliptic curves, with a rather complicated system of cyclic 9-isogenies and correspondences that get symmetrized (more on this in the last paragraph). A more succinct expression follows from using the matrix $\binom{30}{01}$ to conjugate $\Gamma_0(3|3)$ to $\Gamma(1)$ and $\Gamma_0(9)$ to $\pm \Gamma(3)$. Then you're basically looking at a moduli problem that parametrizes elliptic curves $E$ equipped with an unordered octuple of symplectic isomorphisms $E[3] \cong (\mathbb{Z}/3\mathbb{Z})^2$ that form a torsor under the characteristic 2-Sylow subgroup $Q_8 \subset Sp_2(\mathbb{F}_3) \cong SL_2(\mathbb{Z})/\Gamma(3)$.
In general, you can encode moduli problems attached to arithmetic groups using the fact that congruence groups like $\Gamma(N)$ and $\Gamma_0(N)$ stabilize distinguished finite subcomplexes of the product of all $p$-adic Bruhat-Tits trees. Conway gives a explanation (that doesn't use the word "moduli") with pictures in his paper Understanding groups like $\Gamma_0(N)$. For example, when $N$ is a product of $k$ distinct primes, $\Gamma_0(N)$ stabilizes a $k$-cube. Given a finite stable subcomplex, there is a standard way to make a moduli problem out of it by assigning elliptic curves to the vertices, isogenies to the edges, such that the induced transformations on the Tate module behave as you would expect from traversing the product of buildings. To symmetrize, just enumerate orbits of the transformations you want, and demand a torsor structure.
In the case of the 3C group in the above paragraph, $\Gamma_0(9)$ pointwise stabilizes a subgraph of the 3-adic tree that is an X-shaped configuration spanned by 5 vertices. The edges coming out of the central vertex are in noncanonical bijection with points in $\mathbb{P}^1(\mathbb{F}_3)$, and to symmetrize, you can make an unordered 4-tuple of diagrams of 5 elliptic curves, related by the action of the subgroup $V_4 \subset PSL_2(\mathbb{F}_3)$ that preserves the cross-ratio.
In the model you describe, the cusp $\infty$ of $X_1(N)$ is not defined over ${\bf Q}$ (but the cusp $0$ is). A way to see this is that the marked elliptic curve $({\bf C}/({\bf Z}+\tau{\bf Z}),1/N)$ is isomorphic to the marked Tate curve $E_q=({\bf C}^\times/q^{\bf Z},e^{2\pi i/N})$ with $q=e^{2\pi i\tau}$. When you let $\tau \to \infty$, you get $q \to 0$ so that $E_q \to ({\bf G}_m,e^{2i\pi/N})$, which is not defined over ${\bf Q}$. This fact is explained in Diamond-Im, Modular forms and modular curves, see 9.3.5 and 9.3.6.
There is an alternative model $Y_\mu(N)$ classifying elliptic curves $E$ together with a closed immersion $\mu_N \hookrightarrow E$ (see loc. cit. 8.2.2). In this model the cusp $\infty$ is defined over ${\bf Q}$, so it gives an affirmative answer to your second question.
You can switch from one model to another with the Atkin-Lehner involution $W_N$, which becomes an isomorphism defined over ${\bf Q}$ — it is only defined over ${\bf Q}(\mu_N)$ when considered as an involution of either $X_1(N)$ or $X_{\mu}(N)$. But I don't see a nice way to characterize those functions which are rational for the canonical model in terms of the $q$-expansion at $\infty$.
Best Answer
Modular curves are moduli spaces of elliptic curves with additional (torsion) structure and can be constructed purely algebraically. If you start with an elliptic curve $E$ with transcendental $j$-invariant $j$, over an arbitrary algebraically closed field $k$, and adjoin to $k(j)$ the $x$-coordinates of the $N$-torsion points of $E$ you get a function field over $k$ which is the function field of $X(N)$. See Rohrlich's paper in the Cornell-Silverman-Stevens volume on Fermat's Last Theorem or Katz-Mazur.
Edit: Note, however, Brian's warning in the comments.