Definition 18.4 (Complex structure): a ComplexStructure on a topological space
X is a ChartedSpace ℂ X together with the property that X is a real-analytic
manifold modelled on ℂ (i.e., the transition maps between charts are holomorphic).
- atlas : Set (OpenPartialHomeomorph X ℂ)
- chartAt : X → OpenPartialHomeomorph X ℂ
- isManifold : IsManifold (modelWithCornersSelf ℂ ℂ) ⊤ X
Instances
Definition 18.5 (Riemann surface): a Riemann surface is a connected Hausdorff
topological space equipped with a one-dimensional ComplexStructure.
- atlas : Set (OpenPartialHomeomorph X ℂ)
- chartAt : X → OpenPartialHomeomorph X ℂ
- isManifold : IsManifold (modelWithCornersSelf ℂ ℂ) ⊤ X
- t2 : T2Space X
- connected : ConnectedSpace X
Instances
The reduction-mod-N group homomorphism SL₂(ℤ) → SL₂(ℤ/Nℤ).
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Definition 18.11: the principal congruence subgroup Γ(N) ⊆ SL₂(ℤ), consisting of
matrices congruent to the identity modulo N.
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Membership criterion for Γ(N): a matrix γ ∈ SL₂(ℤ) lies in Γ(N) iff its
entries reduce mod N to the identity matrix.
The Hecke-type congruence subgroup Γ₀(N) ⊆ SL₂(ℤ) of matrices that are upper
triangular modulo N.
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The Hecke-type congruence subgroup Γ₁(N) ⊆ SL₂(ℤ) of matrices that are
upper-unitriangular modulo N.
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Membership criterion for Γ₁(N): a matrix A lies in Γ₁(N) iff its diagonal
entries are ≡ 1 (mod N) and its lower-left entry is ≡ 0 (mod N).
Inclusion Γ(N) ⊆ Γ₁(N): every matrix congruent to the identity mod N is in
particular upper-unitriangular mod N.
Inclusion Γ₁(N) ⊆ Γ₀(N).
Definition 18.11: a subgroup H ⊆ SL₂(ℤ) is a congruence subgroup if it
contains some principal congruence subgroup Γ(N).
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Γ(N) is itself a congruence subgroup (for N ≠ 0).
Γ₁(N) is a congruence subgroup (for N ≠ 0).
Γ₀(N) is a congruence subgroup (for N ≠ 0).
The open modular curve Y_Γ := Γ \ ℍ, the quotient of the upper half-plane by a
congruence subgroup Γ.
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The extended upper half-plane ℍ* = ℍ ∪ ℚ ∪ {∞} (cusps adjoined), modelled here
as an opaque type pending a formal definition.
Axiom: there is a natural SL₂(ℤ)-action on ℍ*.
Instances For
The SL₂(ℤ)-action on the extended upper half-plane, registered as an instance.
Axiom: ℍ* has a natural topology (extending the topology on ℍ with neighborhoods
of cusps).
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The topology on ℍ*, registered as an instance.
Axiom: ℍ* is compact (adjoining cusps to ℍ produces a compactification).
Compactness of ℍ*, registered as an instance.
Axiom: ℍ* is connected.
Connectedness of ℍ*, registered as an instance.
Axiom: ℍ* is Hausdorff.
The Hausdorff property of ℍ*, registered as an instance.
Axiom: ℍ* is locally compact.
Local compactness of ℍ*, registered as an instance.
Axiom: the SL₂(ℤ)-action on ℍ* is continuous.
Continuity of the SL₂(ℤ)-action on ℍ*, registered as an instance.
Axiom: the SL₂(ℤ)-action on ℍ* is properly discontinuous (the key topological
input for the modular curve to be Hausdorff).
Proper discontinuity of the SL₂(ℤ)-action on ℍ*, registered as an instance.
Definition 18.11: the (compactified) modular curve X_Γ := Γ \ ℍ* for a congruence
subgroup Γ.
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Bundled assertion that Γ(N), Γ₁(N), and Γ₀(N) are all congruence subgroups.
The topology on X(1) inherited as a quotient of ℍ*.
X(1) is compact, since it is a quotient of the compact space ℍ*.
X(1) is Hausdorff, since the SL₂(ℤ)-action on ℍ* is properly discontinuous
and ℍ* is T2.
X(1) is connected, since it is a quotient of the connected space ℍ*.
Theorem 18.3: X(1) is a connected compact Hausdorff space.
The cube root of unity ρ = -1/2 + i√3/2 ∈ ℍ, one of the two elliptic points
(along with i) in the standard fundamental domain 𝓕.
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Part of Lemma 18.7: for a generic point z in the fundamental domain (not i,
ρ, or ρ'), its SL₂(ℤ)-stabilizer is the center {±I}, isomorphic to ℤ/2ℤ.
Lemma 18.7 at z = i: the stabilizer of i in SL₂(ℤ) is the cyclic group of
order 4 generated by S = ((0,-1),(1,0)).
Lemma 18.7 at z = ρ: the stabilizer of ρ in SL₂(ℤ) is the cyclic group of
order 6 generated by S * T.
Lemma 18.7 at the cusp ∞: the stabilizer of ∞ in SL₂(ℤ) consists of matrices
with lower-left entry 0, i.e., powers of ±T.
The matrix S = ((0,-1),(1,0)) ∈ SL₂(ℤ) has order 4.
The product S*T ∈ SL₂(ℤ) has order 6.
The classical identity |cz + d|² = Im(z) / Im(γz) for γ = ((a,b),(c,d)) ∈ SL₂(ℤ)
acting on the upper half-plane.
Expansion of |cz + d|² = (c·Re(z) + d)² + (c·Im(z))² as a sum of two squares of
real numbers, used to extract bounds on the matrix entries c, d.
Lemma 18.1: for any compact sets A, B ⊆ ℍ, the set of γ ∈ SL₂(ℤ) such that
γA ∩ B ≠ ∅ is finite. Proved by bounding the four entries a, b, c, d of γ
using |cτ + d|² = Im(τ)/Im(γτ) and the compactness of A, B.
If f maps ball 0 R into itself, then so does its k-th iterate.
If f(z) = ζ·z on ball 0 R and f maps the ball to itself, then
f^k(z) = ζ^k · z for every iterate.
Schwarz lemma / disk-automorphism rigidity: any holomorphic self-map of an open
disk fixing the origin that admits a holomorphic inverse (also fixing 0) must be a
rotation z ↦ ζz for some unimodular ζ.
Strengthening of the rotation lemma: if additionally f^n = id on the ball with
n minimal, then the rotation constant ζ is a primitive n-th root of unity.
This is the analytic core used in proving Lemma 18.8.
The Cayley-type uniformizer δ_x(τ) = (τ - τ_x)/(τ - conj(τ_x)) mapping the upper
half-plane biholomorphically to the open unit disk, sending τ_x ↦ 0. Used in
Lemma 18.8 to transfer disk automorphism rigidity to the upper half-plane.
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Lemma 18.8: let τ_x ∈ ℍ and φ : ℍ → ℍ be holomorphic with φ(τ_x) = τ_x and
φ^n = id with n minimal. Then there exists a primitive n-th root of unity ζ
such that δ_x(φ(τ)) = ζ · δ_x(τ) for every τ ∈ ℍ. Proved by conjugating with the
Cayley-type map deltaMap and applying disk_aut_primitive_root.
The action map τ ↦ γ • τ on the upper half-plane is continuous for every
γ ∈ SL₂(ℤ).
Lemma 18.2: for any τ₁, τ₂ ∈ ℍ*, there exist open neighborhoods U₁, U₂ of
τ₁, τ₂ such that for every γ ∈ SL₂(ℤ), some z ∈ U₁ has γ•z ∈ U₂ iff
γ•τ₁ = τ₂. In particular each τ has a neighborhood in which it is the unique
representative of its Γ-orbit.
Axiom: the open cover {U_x} and atlas {ψ_x} of X(1) (from §18.3) give a
charted-space structure modelled on ℂ.
Instances For
The charted-space structure on X(1) modelled on ℂ, registered as an instance.
Axiom: the atlas on X(1) is real-analytic, making it a complex manifold.
X(1) is a real-analytic (hence holomorphic) complex 1-manifold.
Assemble the complex structure on X(1) from the charted space and manifold
instances.
X(1) is a Riemann surface: it carries a complex structure and is Hausdorff and
connected (from Theorem 18.3).
Theorem 18.9: the cover and atlas {ψ_x} define a complex structure on X(1),
i.e., X(1) is a compact complex manifold of dimension 1.
The chosen triangulation of X(1) has 3 vertices.
Theorem 18.10: X(1) is a compact Riemann surface of genus 0. The genus claim is
extracted from the explicit triangulation.