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The j-invariant uniformizes X(1)

Statement

The function $j$ descends to a holomorphic map $\overset{ˉ}{j} : X (1) \to \hat{C}$ of compact Riemann surfaces, with $\overset{ˉ}{j} ([\infty]) = \infty$ and a simple pole at the cusp, and $\overset{ˉ}{j}$ is a biholomorphism. Consequently $X (1)$ is biholomorphic to the Riemann sphere and $j$ is a Hauptmodul. The quotient map $π : H \to Y (1) \subset X (1)$ has local degree $2$ at $i$ and $3$ at $ω$ and $ω + 1$ , representatives of the classes of $i$ and $ω$ ; more generally its local degree is $2$ throughout $P S L_{2} (Z) \cdot i$ , $3$ throughout $P S L_{2} (Z) \cdot ω$ , and $1$ elsewhere. These ramification indices belong to $π$ , while $\overset{ˉ}{j}$ itself is unramified, so the composed map $j = \overset{ˉ}{j} \circ π : H \to \hat{C}$ has local degree $2$ at $i$ and $3$ at $ω$ and $ω + 1$ .

Facts & Assumptions

Given: The compact Riemann surface $X (1) = P S L_{2} (Z) \ H^{*}$ with its cusp $[\infty]$ , the open part $Y (1)$ , and the quotient charts at elliptic points and at the cusp (The compactified level-one modular curve X(1), The cusp chart and compactness of X(1), Local charts and the Riemann surface structure of a modular quotient); the modular function $j$ with $j (γ τ) = j (τ)$ , holomorphic on $H$ , and with $j = q^{- 1} + 744 + O (q)$ at the cusp (The modular discriminant and the j-invariant).

[F1]

$j$ is injective on $P S L_{2} (Z)$ -classes: if $j (τ) = j (τ^{'})$ then the lattices $Λ_{τ}, Λ_{τ^{'}}$ are homothetic, hence $τ^{'} = γ τ$ for some $γ \in S L_{2} (Z)$ and $[τ] = [τ^{'}]$ in $Y (1)$ (The j-invariant classifies complex tori).

[F2]

For $λ \in C$ the form $E_{4}^{3} - λ Δ \in M_{12}$ is nonzero with value $1$ at the cusp, and its valence sum is $12 / 12 = 1 > 0$ , so it has a zero in $H$ ; hence $j$ takes the value $λ$ (The level-one valence formula, The modular discriminant and the j-invariant, The zeros of E4 and E6 at the elliptic points, The graded ring of level-one modular forms).

[F3]

Local normal form: a nonconstant holomorphic map of Riemann surfaces has, near a point, the form $z \mapsto z^{ν}$ in suitable coordinates; an injective holomorphic map of domains has nowhere-vanishing derivative and is biholomorphic onto its image (Local power-map normal form on Riemann surfaces, Ramification index, ramification order and branch value, An injective holomorphic map has no critical point and is biholomorphic onto its image, Biholomorphic maps between complex domains).

[F4]

In the quotient chart of Local charts and the Riemann surface structure of a modular quotient the map $π$ near $i$ is $z \mapsto z^{2}$ and near $ω$ or $ω + 1$ is $z \mapsto z^{3}$ , these being the local degrees $2$ and $3$ ; $\overset{ˉ}{j}$ will be unramified once it is shown biholomorphic, since biholomorphic maps have local degree one (Ramification index, ramification order and branch value).

Proof

1.1F4givenalgebra

On $Y (1)$ the invariance and holomorphy of $j$ produce a holomorphic function $\overset{ˉ}{j} : Y (1) \to C$ : at ordinary points use a local inverse of $π$ ; at a point of stabiliser order $ν$ , the invariant Taylor series in a uniformising coordinate $z$ has only powers $z^{m ν}$ , so it is holomorphic in the quotient coordinate $z^{ν}$ . Near the cusp, in the $q$ -chart of The cusp chart and compactness of X(1), $j$ is $q^{- 1} (1 + 744 q + O (q^{2}))$ , a meromorphic function of $q$ with a simple pole at $q = 0$ ; since the cusp chart identifies the cusp with $q = 0$ , the formula $q \mapsto q^{- 1} (1 + 744 q + \dots)$ defines a holomorphic map of a punctured disc into $\hat{C}$ extending to $0$ with value $\infty$ . Hence $\overset{ˉ}{j}$ extends to a holomorphic map $X (1) \to \hat{C}$ with $\overset{ˉ}{j} ([\infty]) = \infty$ and a simple pole in the $q$ -coordinate.

2.1F1step 1.1givenalgebra

$\overset{ˉ}{j}$ is injective. If $\overset{ˉ}{j} ([τ]) = \overset{ˉ}{j} ([τ^{'}])$ then $j (τ) = j (τ^{'})$ (for $τ, τ^{'} \in H$ ) and [F1] gives $[τ] = [τ^{'}]$ ; on $Y (1)$ this is the claim, and the cusp is the only remaining point, with $\overset{ˉ}{j} ([\infty]) = \infty$ not attained on $Y (1)$ because $j$ is holomorphic on $H$ .

3.1F2step 2.1givenalgebra

$\overset{ˉ}{j}$ is surjective: given $λ \in C$ , [F2] provides $τ \in H$ with $j (τ) = λ$ , so $λ$ is in the image, while $\infty = \overset{ˉ}{j} ([\infty])$ ; hence $\overset{ˉ}{j} (X (1)) = \hat{C}$ .

4.1F3F4step 3.1givenalgebra∎

A bijective holomorphic map of compact Riemann surfaces is a biholomorphism: at every point the local normal form is $z \mapsto z^{ν}$ , and injectivity forces $ν = 1$ , so the derivative is never zero and the map is locally biholomorphic by [F3]; a locally biholomorphic bijection has holomorphic inverse, so $\overset{ˉ}{j}$ is biholomorphic and $X (1) ≅ \hat{C}$ with $j$ a Hauptmodul. The local degrees of $π$ at $i$ and at $ω, ω + 1$ are $2$ and $3$ by [F4], the same indices hold at all their modular translates because the group acts by biholomorphisms and $π \circ γ = π$ ; outside these two orbits the stabilisers are trivial and the local degree is $1$ ; since $\overset{ˉ}{j}$ is unramified, the composition $j = \overset{ˉ}{j} \circ π$ has exactly these local degrees at the corresponding points, in particular $2$ at $i$ and $3$ at $ω$ and $ω + 1$ .

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Sources

J. S. Milne, Modular Functions and Modular Forms (v1.31, 2017) (standard reference, not scraped)
C. T. McMullen, Advanced Complex Analysis, Math 213a course notes (Harvard, 2010) (standard reference, not scraped)
D. Zagier, Elliptic Modular Forms and Their Applications, in The 1-2-3 of Modular Forms (Universitext, Springer, 2008) (standard reference, not scraped)