Cusps

Modular Curves as Riemann Surfaces

2.4 Cusps

2.3.6.In the proof of Corollary 2.3.5, need the 2d points in Y(Γ) listed in the description ofE_Γ be distinct? Need the points ofE_Γ all be elliptic points ofΓ?

2.3.7.Prove that there are no elliptic points for the following groups: (a) Γ(N) forN >1, (b)Γ₁(N) forN >3 (also, find the elliptic points forΓ₁(2) andΓ₁(3) given that each group has one), (c) Γ₀(N) for N divisible by any primep≡ −1 (mod 12). (A hint for this exercise is at the end of the book.) In the next chapter we will extend the technique of proving Proposition 2.3.3 to count the elliptic points ofΓ₀(N) for allN.

Also, show that if the congruence subgroupΓ does not contain the negative identity matrix−I thenΓ has no elliptic points of period 2.

2.3.8.Suppose SL2(Z) =

j{±I}Γ γj where Γ is a congruence subgroup.

Show that

jγ_jDsurjects toY(Γ). (A hint for this exercise is at the end of the book.)

lower left entry 0 then it takes∞to some rational numbers. Left multiplying by a suitableα⁻¹∈SL2(Z) takessback to∞, meaning the productγ =α⁻¹γ fixes∞and therefore has lower left entry 0 as desired.) Ifα∈SL2(Z) takes∞ to a rational number s then α transforms the fundamental domain D to a region that tapers to a cusp ats (cf. Figure 2.4), just asD itself tapers to a cusp at∞on the Riemann sphere. The isotropy subgroup of∞in SL2(Z) is the translations,

SL2(Z)_∞=

± 1m

0 1

:m∈Z .

Let Γ be a congruence subgroup of SL2(Z). To compactify the modular curveY(Γ) =Γ\H, defineH^∗=H∪Q∪{∞}and take the extended quotient

X(Γ) =Γ\H^∗=Y(Γ)∪Γ\(Q∪ {∞}).

The points Γ s in Γ\(Q∪ {∞}) are also called the cusps of X(Γ). For the congruence subgroups Γ₀(N), Γ₁(N), and Γ(N) we write X₀(N), X₁(N), andX(N).

Lemma 2.4.1.The modular curveX(1) = SL2(Z)\H^∗has one cusp. For any congruence subgroupΓ of SL2(Z)the modular curve X(Γ) has finitely many cusps.

Proof. Exercise 2.4.1.

The topology onH^∗consisting of its intersections with open complex disks (including disks{z:|z|> r} ∪ {∞}) contains too many points ofQ∪ {∞}in each neighborhood to make the quotientX(Γ) Hausdorff. Instead, to put an appropriate topology onX(Γ) start by defining for anyM >0 a neighborhood

NM ={τ∈ H: Im(τ)> M}.

Adjoin to the usual open sets in H more sets in H^∗ to serve as a base of neighborhoods of the cusps, the sets

α(NM ∪ {∞}) : M >0, α∈SL2(Z),

and take the resulting topology onH^∗. Since fractional linear transformations are conformal and take circles to circles, if α(∞) ∈ Q then α(NM ∪ {∞}) is a disk tangent to the real axis. (Figure 2.5 showsN1∪ {∞} and some of its SL₂(Z)-translates; note how this quantifies the discussion leading up to Definition 1.2.3.) Under this topology eachγ ∈SL₂(Z) is a homeomorphism ofH^∗. Finally, giveX(Γ) the quotient topology and extend natural projection toπ:H^∗−→X(Γ).

Proposition 2.4.2.The modular curve X(Γ) is Hausdorff, connected, and compact.

Figure 2.5. Neighborhoods of∞and of some rational points

Proof. The first statement requires distinct points x1, x2 ∈ X(Γ) to have disjoint neighborhoods. The case x1 = Γ τ1, x2 = Γ τ2 with τ1, τ2 ∈ H is already established as Corollary 2.1.2.

Suppose x1 = Γ s1, x2 = Γ τ2 with s1 ∈ Q∪ {∞} and τ2 ∈ H. Then s₁ = α(∞) for some α ∈ SL₂(Z). Let U₂ be any neighborhood of τ₂ with compact closureK. Then the formula

Im(γ(τ))≤max{Im(τ),1/Im(τ)} forτ∈ Handγ∈SL₂(Z) (Exercise 2.4.2(a)) shows that forM large enough, SL₂(Z)K∩ NM =∅. Let U₁=α(NM ∪ {∞}). Thenπ(U₁) andπ(U₂) are disjoint (Exercise 2.4.2(b)).

Supposex₁ =Γ s₁, x₂ =Γ s₂ with s₁, s₂ ∈Q∪ {∞}. Then s₁ =α₁(∞) and s2 = α2(∞) for some α1, α2 ∈ SL2(Z). Let U1 = α1(N2∪ {∞}) and U2 = α2(N2∪ {∞}). Then π(U1) and π(U2) are disjoint, for if γα1(τ1) = α2(τ2) for some γ ∈ Γ and τ1, τ2 ∈ N2 then α⁻₂¹γα1 takes τ1 to τ2 and (sinceN2 is tessellated by the integer translates ofDand contains no elliptic points) therefore must be ±[¹_{0 1}^m] for some m ∈ Z. Thus α⁻₂¹γα1 fixes ∞ and consequentlyγ(s1) =s2, contradicting thatx1 and x2 are distinct. This completes the proof thatX(Γ) is Hausdorff.

SupposeH^∗=O1∪O2 is a disjoint union of open subsets. Intersect with the connected setH to conclude that O₁ ⊃ H and so O₂ ⊂Q∪ {∞}. But then O₂ is not open after all unless it is empty. Thus H^∗ is connected and hence so is its continuous imageX(Γ).

For compactness, first note that the setD^∗=D∪{∞}is compact in theH^∗ topology (Exercise 2.4.3). SinceH^∗= SL₂(Z)D^∗=

jΓ γ_j(D^∗) where theγ_j are coset representatives, X(Γ) =

jπ(γ_j(D^∗)). Since each γ_j is continuous andπis continuous and [SL2(Z) :Γ] is finite, the result follows.

MakingX(Γ) a compact Riemann surface requires giving it charts. Retain the coordinate patchesπ(U) and maps ϕ:π(U)−→V from Section 2.2 for neighborhoodsU ⊂ H. For each cusp s ∈ Q∪ {∞} some δ = δ_s ∈ SL₂(Z) takessto∞. Define thewidthofsto be

h_s=h_s,Γ =|SL₂(Z)_∞/(δ{±I}Γ δ⁻¹)_∞|.

Figure 2.6. Local coordinates at a cusp

(The Γ subscript is suppressed when context makes the group clear.) This notion is dual to the period of an elliptic point, being inversely proportional to the size of an isotropy subgroup. Recall that the period of an elliptic point is the number of sectors of the disk at the point that are identified under isotropy.

At a cusp, infinitely many sectors come together, sectors most easily seen after translating to∞as unit vertical strips, and the width of the cusp is the number of such strips are that are distinct under isotropy (see Figure 2.6). The group SL2(Z)_∞={±I}[^{1 1}_{0 1}]is infinite cyclic as a group of transformations, and so the width is characterized by the conditions{±I}(δΓ δ⁻¹)_∞ ={±I}[¹_{0 1}^h], h >0. The width is finite (Exercise 2.4.4(a)) and independent ofδsince in fact h_s=|SL₂(Z)_s/{±I}Γ_s| (Exercise 2.4.4(b)). Ifs∈Q∪ {∞} andγ ∈SL₂(Z) then the width of γ(s) under γΓ γ⁻¹ is the same as the width of sunder Γ (Exercise 2.4.4(c)). In particular, h_s depends only on Γ s, making the width well defined on X(Γ), and if Γ is normal in SL₂(Z) then all cusps ofX(Γ) have the same width. Now define U = U_s = δ⁻¹(N²∪ {∞}) and as before defineψas a compositeψ=ρ◦δwhere this timeρis theh-periodic wrapping mapρ(z) =e^2πiz/h. Note thatψ is exactly the change of variable embedded in condition (3) of Definition 1.2.3. Also as before, letV = im(ψ), an open subset ofC, to get

ψ:U −→V, ψ(τ) =e^2πiδ(τ)/h.

As with elliptic points, ψ mimics the identifying action of π. This time the fractional linear transformationδ straightens neighborhoods of s by making identified points differ by a horizontal offset, and then the exponential mapρ wraps the upper half plane into a cylinder which, held to one’s eye like a telescope, becomes in perspective a disk with ∞ at its center. (Again see Figure 2.6, noting how the shape of the shaded sector in U motivates the term “cusp.”)

To confirm thatψcarries out the same identification asπabouts, compute that forτ₁, τ₂∈U,

h 1

U

∆ Ρ

Ψ

V

π(τ₁) =π(τ₂) ⇐⇒ τ₁=γ(τ₂) ⇐⇒ δ(τ₁) = (δγδ⁻¹)(δ(τ₂))

for someγ∈Γ. But this makesδγδ⁻¹a translation sinceδ(τ1) andδ(τ2) both lie inN2∪ {∞}. Soδγδ⁻¹∈δΓ δ⁻¹∩SL2(Z)_∞= (δΓ δ⁻¹)_∞⊂ ±[¹_{0 1}^h]and

π(τ1) =π(τ2) ⇐⇒ δ(τ1) =δ(τ2) +mh for somem∈Z

⇐⇒ ψ(τ₁) =ψ(τ₂).

As in Section 2.2 there exists a bijectionϕ:π(U)−→V such that the diagram U

π

}}zzzzzzzzz ^ψ

>

>>

>>

>>

>

π(U) ^ϕ // V

commutes. As before, the coordinate neighborhood aboutπ(τ) inY(Γ) isπ(U) and the coordinate map isϕ:π(U)−→V, a homeomorphism (Exercise 2.4.5).

Againψ : U −→ V is the composition ψ = ρ◦δ, and the local coordinate ϕ:π(U)−→V is defined by the conditionϕ◦π=ψ.

Checking that the transition maps are holomorphic is similar to the process carried out for coordinate patchesU₁, U₂⊂ Hin Section 2.2, but now at least one patch is a cusp neighborhood.

SupposeU₁⊂ Hhas correspondingδ₁=δ_τ1 ∈GL₂(C) whereτ₁has width h₁, and supposeU₂=δ⁻₂¹(N2∪ {∞}). As before, for eachx∈π(U₁)∩π(U₂) write x = π(˜τ₁) = π(τ₂) with ˜τ₁ ∈ U₁, τ₂ ∈ U₂, and τ₂ = γ(˜τ₁) for some γ∈Γ. LetU1,2=U1∩γ⁻¹(U2), a neighborhood of ˜τ1inH. Thenϕ1(π(U1,2)) is a neighborhood ofϕ1(x) inV1,2 =ϕ1(π(U1)∩π(U2)). Note that ifh1>1 then τ1 ∈/ U1,2, else the point δ2(γ(τ1))∈ N2 is an elliptic point forΓ, but N2 contains no elliptic points for SL2(Z) (Exercise 2.4.6). So ifh1 >1 then 0 ∈/ ϕ1(π(U1,2)). As before, an input point ϕ1(x) to ϕ2,1 in V1,2 takes the formq= (δ1(τ))^h1. This time the corresponding output is

ϕ2(x) =ϕ2(π(γ(τ))) =ψ2(γ(τ)) = exp(2πiδ2γ(τ)/h2)

= exp(2πiδ2γδ⁻₁¹(q^1/h1)/h2).

So the only case where the transition map might not be holomorphic is when h1 >1 and 0∈ϕ1(π(U1,2)), but we have seen that this cannot happen. The discussion here also covers the case with the roles of U1 and U2 exchanged since the inverse of a holomorphic bijection is again holomorphic.

Suppose U₁ = δ⁻₁¹(N2∪ {∞}) with δ₁ : s₁ → ∞ and U₂ = δ₂⁻¹(N2∪ {∞}) withδ₂ : s₂ → ∞. If π(U₁)∩π(U₂)=∅ then γδ⁻₁¹(N2∪ {∞}) meets δ₂⁻¹(N2∪ {∞}) for someγ∈Γ, i.e.,δ₂γδ₁⁻¹moves some point inN2∪ {∞}to another and therefore must be a translation±[¹_{0 1}^m]. Soγ(s₁) =γδ₁⁻¹(∞) =

±δ⁻₂¹[¹_{0 1}^m] (∞) =s₂. It follows thath₁=h₂and the transition map takes an input point inϕ₁(π(U_1,2)),

q=ψ₁(τ) = exp(2πiδ₁(τ)/h), to the output point

ψ₂(γ(τ)) = exp(2πiδ₂γδ⁻₁¹(δ₁(τ))/h) = exp(2πi(δ₁(τ) +m)/h)

=e^2πim/hq.

This is clearly holomorphic.

For any congruence subgroupΓ of SL₂(Z) the extended quotientX(Γ) is now a compact Riemann surface. Figure 2.7 summarizes the local coordinate structure for future reference.

Topologically every compact Riemann surface is ag-holed torus for some nonnegative integer g called its genus. In particular, complex elliptic curves have genus 1. The next chapter will compute the genus ofX(Γ) and study the meromorphic functions and differentials onX(Γ). This will give counting results on modular forms as well as being interesting in its own right.

π:H^∗−→X(Γ) is natural projection.

U ⊂ H^∗ is a neighborhood containing at most one elliptic point or cusp.

The local coordinateϕ:π(U)−→^∼ V satisfiesϕ◦π=ψ whereψ:U −→V is a compositionψ=ρ◦δ.

Aboutτ₀∈ H: Abouts∈Q∪ {∞}:

The straightening map isz=δ(τ) The straightening map is z=δ(τ) whereδ=

1 −τ₀ 1−τ₀

,δ(τ₀) = 0. whereδ∈SL₂(Z),δ(s) =∞. δ(U) is a neighborhood of 0. δ(U) is a neighborhood of∞. The wrapping map isq=ρ(z) The wrapping map isq=ρ(z)

whereρ(z) =z^h,ρ(0) = 0 whereρ(z) =e^2πiz/h,ρ(∞) = 0 with periodh=|{±I}Γτ0/{±I}|. with width h=|SL2(Z)s/{±I}Γs|. V =ρ(δ(U)) is a neighborhood of 0. V =ρ(δ(U)) is a neighborhood of 0.

Figure 2.7.Local coordinates onX(Γ)

Exercises

2.4.1.Prove Lemma 2.4.1.

2.4.2.(a) Justify the formula Im(γ(τ)) ≤ max{Im(τ),1/Im(τ)} for τ ∈ H andγ∈SL₂(Z) used in the proof of Proposition 2.4.2.

(b) In the same proof, verify thatπ(U₁) andπ(U₂) are disjoint.

2.4.3.Show that the setD^∗=D ∪ {∞}is compact in theH^∗ topology.

2.4.4.(a) Show that the widthh_sis finite. (Hints for this exercise are at the end of the book.)

(b) Show thath_s=|SL₂(Z)_s/{±I}Γ_s|.

(c) Show that if s ∈ Q∪ {∞} and γ ∈ SL₂(Z) then the width of γ(s) underγΓ γ⁻¹is the same as the width ofsunderΓ.

2.4.5.Explain why ϕ:π(U)−→V as defined in the section is a homeomor- phism.

2.4.6.Show that N2 contains no elliptic points for SL2(Z). (A hint for this exercise is at the end of the book.)

2.4.7.Suppose thatΓ₁ andΓ₂are congruence subgroups of SL₂(Z) and that γis an element of GL⁺₂(Q) (see Exercise 1.2.11) such thatγΓ₁γ⁻¹⊂Γ₂. Show that the formulaΓ₁τ→Γ₂γ(τ) defines a holomorphic mapX(Γ₁)−→X(Γ₂).