Local charts and the Riemann surface structure of a modular quotient

For compact sets $K,L\subset \mathfrak{H}$, only finitely many $\gamma \in \mathrm{\Gamma }$ satisfy $\gamma K\cap L\mathrm{\ne }\mathrm{\varnothing }$. Write $z\in K$, $w=\gamma z\in L$; if $y_K\le \mathrm{\Im }z\le Y_K$, $\mathrm{\Im }w\ge y_L>0$ and $\mathrm{\mid }\mathrm{\Re }z\mathrm{\mid }\le X_K$, the height formula gives $\mathrm{\mid }cz+d{\mathrm{\mid }}^2\le Y_K\mathrm{/}{y}_L$. Thus $\mathrm{\mid }c\mathrm{\mid }\le \sqrt{Y_K\mathrm{/}{y}_L}\mathrm{/}{y}_K$ and $\mathrm{\mid }d\mathrm{\mid }\le \sqrt{Y_K\mathrm{/}{y}_L}+\mathrm{\mid }c\mathrm{\mid }{X}_K$, leaving finitely many integer bottom rows. For any fixed bottom row, all determinant-one top rows are $(a_0,b_0)+k(c,d)$, so the corresponding maps are ${\gamma }_{0}z+k$; the real parts of ${\gamma }_0(K)$ and $L$ are bounded, leaving finitely many $k$. Apply this to a closed small disc about $\tau$. The stabiliser is finite cyclic: conjugation into $\bar{D}$ and [F1, F6] identify it with a subgroup of a cyclic group of order $1,2$ or $3$. For each of the finitely many non-stabilising maps meeting that disc, disjointness of its image of $\tau$ and $\tau$ permits shrinking the neighbourhood. Intersect its images under the finite stabiliser to obtain an invariant open $U$ with $\gamma U\cap U\mathrm{\ne }\mathrm{\varnothing }$ exactly for stabiliser elements.

If $q,q^{\mathrm{\prime }}\in \mathrm{\Gamma }\mathrm{\backslash }\mathfrak{H}$ are distinct, choose $\tau ,{\tau }^{\mathrm{\prime }}$ with $p(\tau )=q$, $p({\tau }^{\mathrm{\prime }})=q^{\mathrm{\prime }}$; the set $\{\gamma \in \mathrm{\Gamma }:\gamma B(\tau ,\delta )\cap B({\tau }^{\mathrm{\prime }},\delta )\mathrm{\ne }\mathrm{\varnothing }\}$ is finite for small $\delta$ by the same boundedness argument as in 1.1, and no element of it carries $\tau$ to ${\tau }^{\mathrm{\prime }}$ because $q\mathrm{\ne }{q}^{\mathrm{\prime }}$; shrinking $\delta$ therefore gives disjoint open neighbourhoods $U,U^{\mathrm{\prime }}$ of $\tau ,{\tau }^{\mathrm{\prime }}$ with $\gamma U\cap U^{\mathrm{\prime }}=\mathrm{\varnothing }$ for all $\gamma \in \mathrm{\Gamma }$. Then $\mathrm{\Gamma }U$ and $\mathrm{\Gamma }{U}^{\mathrm{\prime }}$ are disjoint $\mathrm{\Gamma }$-invariant open sets, so $q,q^{\mathrm{\prime }}$ have disjoint neighbourhoods and the quotient is Hausdorff. The map $p$ is open: for open $V\subseteq \mathfrak{H}$, the preimage $p^{-1}(p(V))={\bigcup }_{\gamma \in \mathrm{\Gamma }}\gamma V$ is open, so $p(V)$ is open by definition of the quotient topology.

Construction of charts. If ${\mathrm{Stab}}_{\mathrm{\Gamma }}(\tau )=\{1\}$, take $U$ as in 1.1; then $p{\mathrm{\mid }}_U$ is injective, $p(U)$ is open by 2.1, and $p{\mathrm{\mid }}_U^{-1}$ is a homeomorphism onto its image by [F5]; declare it a chart, and $p$ is the identity map in these coordinates. If ${\mathrm{Stab}}_{\mathrm{\Gamma }}(\tau )=⟨{\gamma }_0⟩$ has order $\nu >1$, then $\nu \in \{2,3\}$ by 1.1; by [F3] there is a biholomorphic coordinate $z$ on a disc $\mathrm{\Delta }$ centred at $\tau$, $z(\tau )=0$, in which ${\gamma }_0$ acts by $z\mapsto \zeta z$ with $\zeta$ a primitive $\nu$-th root of unity. Choose $\mathrm{\Delta }$ so that $\{\gamma :\gamma \mathrm{\Delta }\cap \mathrm{\Delta }\mathrm{\ne }\mathrm{\varnothing }\}=⟨{\gamma }_0⟩$; then $z_1^{\nu }=z_2^{\nu }$ for $z_1,z_2\in \mathrm{\Delta }$ exactly when $z_2={\zeta }^k{z}_1$ for some $k$ (both are $\nu$-th roots of the same number), so $\psi :=z^{\nu }$ induces a bijection $\mathrm{\Delta }\mathrm{/}⟨{\gamma }_0⟩\to {\mathrm{\Delta }}^{\mathrm{\prime }}$ onto a disc ${\mathrm{\Delta }}^{\mathrm{\prime }}$ and this bijection is a homeomorphism by [F5]; since $p(\mathrm{\Delta })=\mathrm{\Delta }\mathrm{/}⟨{\gamma }_0⟩$, the induced map $p(\mathrm{\Delta })\to \mathrm{\Delta }\mathrm{\prime }$ is a chart on the quotient. In these coordinates $p$ is the holomorphic map $z\mapsto z^{\nu }$, so $p$ is holomorphic for the atlas.

Transition maps are holomorphic away from elliptic centres by the local inverse theorem [F4]. At a centre of order $\nu$, a quotient-coordinate function pulled back to the uniformising coordinate has a holomorphic Taylor series $h(z)$ invariant under $z\mapsto \zeta z$; coefficient comparison gives $h(z)={\sum }_{m\ge 0}{a}_{m\nu }{z}^{m\nu }=g(z^{\nu })$, with $g$ holomorphic. Indeed its power series converges for $\mathrm{\mid }{z}^{\nu }\mathrm{\mid }<r^{\nu }$ whenever $h$ converges for $\mathrm{\mid }z\mathrm{\mid }<r$. This proves transition holomorphy also at the centre. Any other surface structure making $p$ holomorphic has the same property for the pullback of each of its charts, so its charts are holomorphic functions of our quotient charts. These functions are injective, since both charts are homeomorphisms; their derivatives are therefore nonzero and their inverses are holomorphic. The two maximal atlases agree, proving uniqueness. The quotient is second countable: images under the open map $p$ of a countable disc basis of $\mathfrak{H}$ form a basis; it is connected as the continuous image of $\mathfrak{H}$. Thus the atlas defines a Riemann surface with all the topological hypotheses.