[proofplan] We work over $\mathbb{C}$ and interpret the moduli assertion as a coarse moduli statement, so points classify isomorphism classes rather than supplying a universal family in every case. The open curve $Y_0(N)$ is the coarse moduli curve for elliptic curves equipped with a cyclic subgroup of order $N$, and its complex points are analytically uniformized by $\Gamma_0(N)\backslash\mathfrak{H}$. The compact curve $X_0(N)$ is obtained by adjoining the rational boundary classes $\Gamma_0(N)\backslash\mathbb{P}^1(\mathbb{Q})$, and the generalized elliptic curve compactification identifies these added points with Tate-curve degenerations carrying compatible $\Gamma_0(N)$-level structure. [/proofplan] [step:Declare the moduli functor represented on the open curve] Fix $N \in \mathbb{N}$ and work over $\operatorname{Spec}\mathbb{C}$. Define the $\Gamma_0(N)$ moduli functor $\mathcal{F}_0(N)$ on complex schemes by sending a complex scheme $S$ to the set of isomorphism classes of pairs $(\pi:E \to S, C)$, where $\pi:E \to S$ is an elliptic curve over $S$ and $C \le E[N]$ is a finite flat subgroup scheme which is locally for the fppf topology generated by one section and has constant rank $N$. This is the family version of a cyclic subgroup of order $N$; over $\operatorname{Spec}\mathbb{C}$ it is exactly an ordinary cyclic subgroup $C \le E[N](\mathbb{C})$ of order $N$. The standard Deligne--Rapoport and Katz--Mazur construction of modular curves with $\Gamma_0(N)$-level structure applies over $\mathbb{C}$ and gives $Y_0(N)$ as the coarse moduli scheme for $\mathcal{F}_0(N)$. Therefore the non-cuspidal complex points of $Y_0(N)$ classify precisely isomorphism classes of elliptic curves over $\mathbb{C}$ equipped with a cyclic subgroup of order $N$. [guided] The object being classified must be stated before compactifying it. For a scheme $S$, define $\mathcal{F}_0(N)(S)$ to be the set of isomorphism classes of pairs $(\pi:E \to S, C)$, where $\pi:E \to S$ is an elliptic curve over $S$ and $C \le E[N]$ is a cyclic finite flat subgroup scheme of rank $N$. The finite flat condition is the scheme-theoretic version of having order $N$ in families, and over an algebraically closed field it reduces to an ordinary cyclic subgroup of the finite group $E[N]$. The standard construction theorem of Deligne--Rapoport and Katz--Mazur for modular curves with $\Gamma_0(N)$-level structure applies to this functor over $\mathbb{C}$: its hypotheses are exactly that the object is an elliptic curve over the base together with a finite flat cyclic subgroup scheme of rank $N$. The theorem gives a coarse, not necessarily fine, moduli curve $Y_0(N)$ for $\mathcal{F}_0(N)$. Hence a complex point of $Y_0(N)$ represents exactly an isomorphism class of an elliptic curve over $\mathbb{C}$ together with a cyclic subgroup of order $N$. This proves the asserted open moduli interpretation before the boundary points are added. [/guided] [/step] [step:Identify the complex points with the quotient by $\Gamma_0(N)$] Let $\mathfrak{H} := \{\tau \in \mathbb{C} : \operatorname{Im}(\tau) > 0\}$ be the upper half-plane, and let $\Gamma_0(N) := \{\begin{pmatrix} a & b \\ c & d \end{pmatrix} \in SL_2(\mathbb{Z}) : c \equiv 0 \pmod N\}$. For $\tau \in \mathfrak{H}$, define the lattice $\Lambda_\tau := \mathbb{Z}\tau + \mathbb{Z} \subset \mathbb{C}$ and the elliptic curve \begin{align*} E_\tau := \mathbb{C}/\Lambda_\tau. \end{align*} Define $C_\tau \le E_\tau[N]$ to be the cyclic subgroup generated by the class of $1/N \in \mathbb{C}$. The analytic [uniformization theorem](/theorems/3376) for $\Gamma_0(N)$-level modular curves applies because every complex elliptic curve is analytically isomorphic to $\mathbb{C}/\Lambda$ for a lattice $\Lambda \subset \mathbb{C}$, and changing a basis of $\Lambda$ acts on $\mathfrak{H}$ through $SL_2(\mathbb{Z})$. It identifies \begin{align*} Y_0(N)(\mathbb{C}) \cong \Gamma_0(N)\backslash\mathfrak{H}, \end{align*} and under this identification the orbit of $\tau$ corresponds to the isomorphism class of $(E_\tau,C_\tau)$. Thus the analytic quotient and the coarse moduli interpretation describe the same open curve. [guided] The analytic model explains why the group $\Gamma_0(N)$ is the correct congruence subgroup. Let \begin{align*} \mathfrak{H} := \{\tau \in \mathbb{C} : \operatorname{Im}(\tau) > 0\} \end{align*} be the upper half-plane, and define \begin{align*} \Gamma_0(N) := \{\begin{pmatrix} a & b \\ c & d \end{pmatrix} \in SL_2(\mathbb{Z}) : c \equiv 0 \pmod N\}. \end{align*} For each $\tau \in \mathfrak{H}$, define the lattice $\Lambda_\tau := \mathbb{Z}\tau + \mathbb{Z} \subset \mathbb{C}$ and the elliptic curve $E_\tau := \mathbb{C}/\Lambda_\tau$. The element $1/N \in \mathbb{C}$ has order $N$ modulo $\Lambda_\tau$, so its class generates a cyclic subgroup $C_\tau \le E_\tau[N]$ of order $N$. The analytic uniformization theorem for modular curves with $\Gamma_0(N)$-level structure gives a biholomorphic identification \begin{align*} Y_0(N)(\mathbb{C}) \cong \Gamma_0(N)\backslash\mathfrak{H}. \end{align*} Under this identification, the orbit of $\tau$ corresponds to the isomorphism class of $(E_\tau,C_\tau)$. The congruence condition $c \equiv 0 \pmod N$ is exactly the condition ensuring that the change of lattice basis preserves the distinguished cyclic subgroup. Therefore the analytic quotient realizes the same moduli problem as the algebraic open curve $Y_0(N)$. [/guided] [/step] [step:Add the boundary points and interpret them as cusps] The compact modular curve $X_0(N)$ is the smooth compactification of $Y_0(N)$. Let \begin{align*} \mathbb{P}^1(\mathbb{Q}) := \mathbb{Q} \cup \{\infty\} \end{align*} denote the rational projective line, equipped with the fractional linear action of $\Gamma_0(N)$. Analytically the compactification is obtained from $\Gamma_0(N)\backslash\mathfrak{H}$ by adjoining the finite set \begin{align*} \Gamma_0(N)\backslash\mathbb{P}^1(\mathbb{Q}). \end{align*} These added points are, by definition, the cusps of $X_0(N)$. The Tate-curve boundary theorem applies on a punctured formal neighbourhood $\operatorname{Spec}\mathbb{C}((q))$ of each cusp and extends over $\operatorname{Spec}\mathbb{C}[[q]]$ to a generalized elliptic curve whose generic fiber is a smooth elliptic curve and whose special fiber is a degenerate Neron polygon. In this compactified moduli problem, the $\Gamma_0(N)$-level structure is required to lie in the smooth locus and to be a cyclic subgroup scheme of rank $N$ meeting the relevant components compatibly; the Tate-curve construction supplies exactly such an extension. Hence every added point of $X_0(N) \setminus Y_0(N)$ corresponds to a degenerate generalized elliptic curve with compatible $\Gamma_0(N)$-level structure. [guided] Compactification adds precisely the limiting objects that are missing from the open moduli problem. By construction, $X_0(N)$ is the smooth compactification of $Y_0(N)$. Define \begin{align*} \mathbb{P}^1(\mathbb{Q}) := \mathbb{Q} \cup \{\infty\}, \end{align*} the rational projective line, and let $\Gamma_0(N)$ act on it by fractional linear transformations. In the analytic model, compactification is described by adjoining this rational boundary of the upper half-plane: \begin{align*} X_0(N)(\mathbb{C}) = \Gamma_0(N)\backslash\bigl(\mathfrak{H} \cup \mathbb{P}^1(\mathbb{Q})\bigr). \end{align*} Thus the added analytic points are exactly the finite quotient set $\Gamma_0(N)\backslash\mathbb{P}^1(\mathbb{Q})$, and these points are called cusps. It remains to connect the word cusp with the moduli interpretation. The Tate-curve boundary theorem is applied on the local base $\operatorname{Spec}\mathbb{C}[[q]]$, with punctured generic part $\operatorname{Spec}\mathbb{C}((q))$. It says that the elliptic curve over the punctured base extends over $q = 0$ to a generalized elliptic curve: the nearby fibers are smooth elliptic curves, while the limiting fiber is a degenerate Neron polygon. The $\Gamma_0(N)$ datum also has a limiting meaning in the compactified moduli problem: it is a cyclic finite flat subgroup scheme of rank $N$ contained in the smooth locus and compatible with the component structure of the generalized elliptic curve. Therefore the added points are not arbitrary compactification points; they parameterize degenerate generalized elliptic curves equipped with compatible $\Gamma_0(N)$-level structure. [/guided] [/step] [step:Combine the open and boundary descriptions] From the first step, $Y_0(N)$ is the coarse moduli space over $\mathbb{C}$ of pairs $(E,C)$ with $E$ an elliptic curve and $C \le E[N]$ cyclic of order $N$. From the compactification step, $X_0(N)$ is obtained from $Y_0(N)$ by adding exactly the cusps, and those cusps correspond in the generalized elliptic curve compactification to degenerate fibers with compatible $\Gamma_0(N)$-level structure. Therefore $X_0(N)$ is the compactification of the stated coarse moduli space, with added points precisely the cusps described in the theorem. [/step]