Modular curve $X(N)$ (reviewed)

There are three variants of the modular curve $Y(N)$:

1. There is a functor sending each $\Z[1/N]$-algebra $R$ to the set of (isomorphism classes of) pairs $(E,\phi)$ such that $E$ is an elliptic curve over $R$ and $\phi \colon (\Z/N\Z)^2 \to E[N]$ is an isomorphism of group schemes. Suppose that $N \ge 3$; then this functor is represented by a smooth affine $\mathbb{Z}[1/N]$-scheme $Y(N)^{\text{big}}$, called the big modular curve of level $N$. (If $N<3$, it is representable only by an algebraic stack, and one must take the coarse moduli space to get a scheme.) For any field $k$ with $\operatorname{char} k \nmid N$, the set $Y(N)^{\text{big}}(k)$ is the set of isomorphism classes of triples $(E,P,Q)$ , where $E$ is an elliptic curve over $k$ and $P,Q \in E(k)$ form a $(\Z/N\Z)$-basis of $E[N]$. The curve $Y(N)^{\text{big}}_{\Q}$ is integral but typically has several geometric components.

2. Fix a primitive $N$th root of unity $\zeta$ in $\overline{\Q}$. There is a functor sending each $\Z[1/N,\zeta]$-algebra $R$ to the set of pairs $(E,\phi)$ such that $E$ is an elliptic curve over $R$ and $\phi \colon (\Z/N\Z)^2 \to E[N]$ is an isomorphism of group schemes such that the resulting elements $P,Q \in E[N](R)$ satisfy $e_N(P,Q)=\zeta$. For $N \ge 3$, this functor is represented by a smooth affine $\Z[1/N,\zeta]$-scheme $Y(N)$, called the classical modular curve of level $N$. Over any $\Z[1/N,\zeta]$-field $k$, the curve $Y(N)_k$ is geometrically integral.

3. There is a functor sending each $\Z[1/N]$-algebra $R$ to the set of pairs $(E,\phi)$ consisting of an elliptic curve $E$ over $R$ and a symplectic isomorphism $\phi \colon \Z/N\Z \times \mu_N \to E[N]$. For $N \ge 3$, this functor is represented by a smooth affine $\mathbb{Z}[1/N]$-scheme $Y(N)^{\text{arith}}$, called the arithmetic modular curve of level $N$. Over any field $k$ with $\operatorname{char} k \nmid N$, the curve $Y(N)^{\text{arith}}_k$ is geometrically integral.

Relationships: Over any $\Z[1/N,\zeta]$-field $k$, the curve $Y(N)^{\text{arith}}_k$ is isomorphic to $Y(N)_k$ and to a connected component of $Y(N)^{\text{big}}_k$.

Complex points: The group $\Gamma(N)$ acts on the upper half-plane $\mathfrak{h}$, and the quotient $\Gamma(N) \backslash \mathfrak{h}$ is biholomorphic to $Y(N)(\mathbb{C})$.

Compactifications: For each variant, there is a corresponding smooth projective model, denoted $X(N)^{\text{big}}$, $X(N)$, or $X(N)^{\text{arith}}$.

Quotients: For each open subgroup $H \le \GL_2(\widehat\Z)$, there is a quotient $X_H$ of $X(N)^{\text{big}}$.