Charts

Modular Curves as Riemann Surfaces

2.2 Charts

Note that Γ_τ also fixes the pointτ ∈ −H since Γ ⊂ SL₂(R). (“Elliptic point” is unrelated to “elliptic curve” from Chapter 1 other than each having a distant connection to actual ellipses.)

The next result will be established in the following section.

Proposition 2.2.2.Let Γ be a congruence subgroup ofSL₂(Z). For each el- liptic pointτ ofΓ the isotropy subgroupΓ_τ is finite cyclic.

Thus each pointτ∈ Hhas an associated positive integer, h_τ =|{±I}Γ_τ/{±I}|=

|Γτ|/2 if−I∈Γτ,

|Γ_τ| if−I /∈Γ_τ.

This hτ is called the periodof τ, with hτ >1 only for the elliptic points. If τ∈ Handγ∈SL2(Z) then the period ofγ(τ) underγΓ γ⁻¹is the same as the period of τ under Γ (Exercise 2.2.3). In particular, h_τ depends only onΓ τ, making the period well defined on Y(Γ), and ifΓ is normal in SL₂(Z) then all points ofY(Γ) over a point ofY(SL₂(Z)) have the same period. The space Y(Γ) depends onΓ as a group of transformations acting onH, and−I acts trivially, so defining the period as we did rather than simply takingh_τ =|Γ_τ| is natural. The definition correctly counts theτ-fixing transformations.

To put coordinates on Y(Γ) about a point π(τ), first use the map δτ = _1−τ

1−τ

∈ GL2(C) (the group of invertible 2-by-2 matrices with com- plex entries) to take τ to 0 and τ to ∞. The isotropy subgroup of 0 in the conjugated transformation group, (δτ{±I}Γ δ_τ⁻¹)0/{±I}, is the conjugate of the isotropy subgroup ofτ, δτ({±I}Γτ/{±I})δ⁻_τ¹, and therefore is cyclic of orderhτ as a group of transformations by Proposition 2.2.2. Since this group of fractional linear transformations fixes 0 and∞, it consists of maps of the formz→az, and since the group is finite cyclic these must be the rotations through angular multiples of 2π/hτ about the origin. Thusδτ is “straighten- ing” neighborhoods ofτ to neighborhoods of the origin in the sense that after the map, equivalent points are spaced apart by fixed angles. This suggests (as Figure 2.1 has illustrated withh_τ = 2) that a coordinate neighborhood ofπ(τ) inY(Γ) should be, roughly, theπ-image of a circular sector through angle 2π/h_τ aboutτ inH, and that the identifying action of πis essentially the wrapping action of theh_τth power map, taking the sector to a disk.

To write this down precisely and check that it works, start with another consequence of Proposition 2.1.1 (Exercise 2.2.4),

Corollary 2.2.3.LetΓ be a congruence subgroup ofSL₂(Z). Each pointτ ∈ Hhas a neighborhoodU in Hsuch that

for allγ∈Γ, if γ(U)∩U =∅ thenγ∈Γ_τ. Such a neighborhood has no elliptic points except possiblyτ.

U ∆ Ψ

Ρ

V

Figure 2.2.Local coordinates at an elliptic point

Now given any pointπ(τ)∈Y(Γ), take a neighborhoodU as in the corol- lary. Defineψ : U −→ C to beψ =ρ◦δ where δ =δτ and ρis the power functionρ(z) =z^h with h=hτ as above. Thusψ(τ) = (δ(τ))^h acts as the straightening mapδ followed by theh-fold wrappingρ. (See Figure 2.2.) Let V =ψ(U), an open subset ofCby the Open Mapping Theorem from complex analysis. Since the projectionπand the wrappingψ identify the same points of U, there should exist an equivalence between the images of U under the two mappings. To confirm this, consider the situation

U −→^π π(U)⊂Y(Γ), U −→^ψ V ⊂C and note that for any pointsτ1, τ2∈U,

π(τ1) =π(τ2) ⇐⇒ τ1∈Γ τ2 ⇐⇒ τ1∈Γττ2 by Corollary 2.2.3

⇐⇒ δ(τ1)∈(δΓτδ⁻¹)(δ(τ2)) ⇐⇒ δ(τ1) =µ^d_h(δ(τ2)) for somed, where µ_h =e^2πi/h, since δΓ_τδ⁻¹ is a cyclic transformation group of hrota- tions. So

π(τ1) =π(τ2) ⇐⇒ (δ(τ1))^h= (δ(τ2))^h ⇐⇒ ψ(τ1) =ψ(τ2),

as desired. Thus there exists an injection ϕ : π(U) −→ V such that the diagram

U

π

}}zzzzzzzzz ^ψ

>

>>

>>

>>

>

π(U) ^ϕ // V

commutes. Also, ϕsurjects because ψ surjects by definition of V. The map ϕis the local coordinate. The coordinate neighborhood aboutπ(τ) in Y(Γ) isπ(U) and the mapϕ:π(U)−→V is a homeomorphism (Exercise 2.2.5).

In sum, U is a neighborhood with no elliptic point except possibly τ; ψ:U −→V is the compositionψ=ρ◦δthat matches the identifying action ofπby wrappingU onto itself as prescribed by the period ofτ; and the local coordinateϕ:π(U)−→V, defined by the condition ϕ◦π=ψ, transfers the identified image ofU in the modular curve back to the wrapped image of U in the complex plane.

We need to check that the transition maps between coordinate charts are holomorphic. That is, given overlapping π(U1) and π(U2), we need to check ϕ2,1, the restriction ofϕ2◦ϕ⁻₁¹ toϕ1(π(U1)∩π(U2)). Let V1,2=ϕ1(π(U1)∩ π(U2)) andV2,1=ϕ2(π(U1)∩π(U2)) and write the commutative diagram

π(U₁)∩π(U₂)

ϕ₂

&&

MM MM MM MM MM MM V1,2

ϕq⁻₁q¹qqqqqq88 qq

q ϕ2,1 // V2,1.

For eachx ∈ π(U1)∩π(U2) it suffices to check holomorphy in some neigh- borhood ofϕ1(x) in V1,2. Write x= π(τ1) = π(τ2) with τ1 ∈ U1, τ2 ∈ U2, andτ2 =γ(τ1) for someγ ∈Γ. LetU1,2 =U1∩γ⁻¹(U2). Then the projec- tionπ(U_1,2) is a neighborhood ofxin π(U₁)∩π(U₂) and soϕ₁(π(U_1,2)) is a corresponding neighborhood ofϕ₁(x) in V_1,2.

Assume first thatϕ₁(x) = 0, i.e., the first straightening map isδ₁=δ_τ1. Then an input pointq=ϕ₁(x) to ϕ_2,1in this neighborhood takes the form

q=ϕ₁(π(τ)) =ψ₁(τ) = (δ₁(τ))^h1 for someτ∈U_1,2

whereh1is the period ofτ1. Letting ˜τ2∈U2be the point such thatψ2(˜τ2) = 0 and lettingh2be its period, the corresponding output is

ϕ2(x) =ϕ2(π(γ(τ))) =ψ2(γ(τ)) which is defined sinceγ(τ)∈U2

= (δ2(γ(τ)))^h2 = ((δ2γδ⁻₁¹)(δ1(τ)))^h2

= ((δ2γδ₁⁻¹)(q^1/h1))^h2.

This calculation shows that the only case where the transition map might not be holomorphic is when h₁ > 1, meaning τ₁ is elliptic and hence so is τ2 = γ(τ1) with the same period. Recall that U2 contains at most one elliptic point by construction and then the local coordinate takes it to 0. So whenh1 >1 the point τ2 is the point ˜τ2 ∈U2 mentioned above, the second straightening map isδ2=δτ2, andh2=h1. Thus

0 ^δ1−1 //τ₁ ^γ //τ₂ ^δ2 //0, ∞ ^δ−11 //τ₁ ^γ //τ₂ ^δ2 //∞, showing that δ₂γδ₁⁻¹ = _α0

0β

for some nonzero α, β ∈ C. The formula for ϕ_2,1 becomes

q→_α0

0β

(q^1/h) h

= (α/β)^hq and the map is clearly holomorphic.

So far the argument assumes that ϕ₁(x) = 0. But it also covers the case ϕ₂(x) = 0 since the inverse of a holomorphic bijection is again holomorphic.

And in general,ϕ_2,1is a compositeϕ_2,3◦ϕ_3,1whereϕ₃:π(U₃)−→V₃takesx to 0, so in fact the argument suffices for all cases.

Exercises

2.2.1.Let τ ∈ H be fixed only by the identity transformation in Γ. Use Proposition 2.1.1 to show that some neighborhood U of τ is homeomorphic to its image inY(Γ).

2.2.2.Compute _1−i

1 i 0−1 1 0

1−i 1 i

₋1

and explain how this shows that the mapγ(τ) =−1/τ acts in the small as 180-degree rotation abouti.

2.2.3.Show that if τ ∈ H and γ ∈ SL₂(Z) then the period of γ(τ) un- derγΓ γ⁻¹ is the same as the period ofτ underΓ.

2.2.4.Prove Corollary 2.2.3.

2.2.5.Explain why ϕ:π(U)−→V as defined in the section is a homeomor- phism. (A hint for this exercise is at the end of the book.)