= Solution
Adjoin $\mathbb P^1(\mathbb Q)$ to the <complex upper half-plane>. Ordinary points retain their usual neighborhoods. At a rational boundary point $a=\sigma\infty$, a neighborhood basis consists of $a$ together with $\sigma\{z\in\mathbb H:\operatorname{Im}z>Y\}$, with $Y$ arbitrarily large. Different choices of $\sigma\in SL_2(\mathbb Z)$ change it by a translation and sign, so give the same neighborhood system. Give $X(1)$ the <quotient topology>. The <modular group> acts properly discontinuously on $\mathbb H$; its <modular cusp> neighborhoods descend to punctured neighborhoods with their <modular cusp> point filled in. This quotient is Hausdorff: interior orbits can be separated by proper discontinuity, and a sufficiently high <modular cusp> neighborhood avoids any prescribed compact interior neighborhood.
Away from fixed points of the effective group $PSL_2(\mathbb Z)$, a small disk is disjoint from its nonidentity translates, so its coordinate gives a <holomorphic> quotient coordinate. At an elliptic fixed point $\tau_0$, use
$$
u=\frac{\tau-\tau_0}{\tau-\overline{\tau_0}}.
$$
Its effective <stabilizer> rotates $u$ by an $e$th root of unity, where $e=2$ at $i$ and $e=3$ at $\rho$. The quotient coordinate is $w=u^e$, not $u$. All rational boundary points lie in one <modular cusp> class. At infinity the <stabilizer> acts by $\tau\mapsto\tau+1$, and
$$
q=e^{2\pi i\tau}
$$
is a punctured-disk coordinate; adjoining $q=0$ fills in the <modular cusp>. These charts and their <holomorphic> transition maps define the complex structure of the <compactified modular curve>.
For <compactness> use the standard region
$$
\mathcal D=\{\tau\in\mathbb H:|\operatorname{Re}\tau|\leq1/2,\ |\tau|\geq1\}.
$$
Its part with $\operatorname{Im}\tau\leq Y$, $Y>1$, is compact: the imaginary part is at least $\sqrt3/2$. The quotient of its upper tail, with its two vertical sides identified and infinity added, is the closed disk $|q|\leq e^{-2\pi Y}$. These two compact images cover $X(1)$, including the boundary-edge identifications. Thus \b[$X(1)$ is a <compact Riemann surface>].
The $j$ function also extends over the <modular cusp> with one <simple pole>, since $j=q^{-1}+O(1)$. Its local expressions at <elliptic points> descend to <holomorphic functions> of $u^e$. By Question 1 it maps the remaining quotient bijectively onto $\mathbb C$, so its extension is a degree-one <holomorphic map> $X(1)\to\mathbb P^1$. A degree-one map of compact Riemann surfaces is <biholomorphic>: any local ramification would already give degree at least two. Therefore $X(1)$ is the <Riemann sphere>; this will also identify the base <genus> in Question 5.
Back to article page
