Meromorphic differentials

Dimension Formulas

3.3 Meromorphic differentials

nonzero elements for all even positive integers k. (A hint for this exercise is at the end of the book.)

3.2.4.Show that for any k ∈ Z, if f is a nonzero element of Ak(Γ) then Ak(Γ) =C(X(Γ))f.

3.2.5.Letα=₁₋₁

2−1

and letΓ =α⁻¹Γ₁(4)α. Show thatΓ_∞=−[^{1 1}_{0 1}]. (A hint for this exercise is at the end of the book.)

3.2.6.Verify the expression forg[S]24 in the proof of Proposition 3.2.2.

In other words, taking the pullback means changing variables in the meromor- phic differential, doing so compatibly with the change of variable formula for integrals. The pullback is obviously linear. The pullback is alsocontravariant, meaning that if ϕ1 =ϕ is as before and ϕ2 :V2 −→ V3 is also holomorphic then (ϕ2◦ϕ1)^∗=ϕ^∗₁◦ϕ^∗₂ (Exercise 3.3.1(a)). IfV1⊂V2 andι:V1 −→V2 is inclusion then its pullback is restriction,ι^∗(ω) =ω|V1 forω ∈Ω^⊗n(V2) (Ex- ercise 3.3.1(b)). It follows that ifϕis a bijection, making its inverse holomor- phic as well by complex analysis, then (ϕ^∗)⁻¹ = (ϕ⁻¹)^∗ (Exercise 3.3.1(c)).

Ifπ: V1−→ V2 is a holomorphic surjection of open sets inC then π^∗ is an injection (Exercise 3.3.1(d)).

Now we can piece together local differentials on a Riemann surfaceX. Let X have coordinate charts ϕ_j :U_j −→ V_j, wherej runs through some index set J, eachU_j is a neighborhood in X, and eachV_j is an open set in C. A meromorphic differential onX of degreenis a collection of local meromorphic differentials of degreen,

(ω_j)_j∈J∈

j∈J

Ω^⊗n(V_j),

that iscompatible. To define this, letV_j,k=ϕ_j(U_j∩U_k) andV_k,j =ϕ_k(U_j∩ U_k) for j, k ∈ J. Then the compatibility criterion is that pulling back any transition map

V_j,k ^ϕk,j //V_k,j

to get the corresponding map of local meromorphic differentials Ω^⊗n(Vj,k) Ω^⊗n(Vk,j)

ϕ^∗_k,j

oo

gives

ϕ^∗_k,j(ωk|V_k,j) =ωj|V_j,k.

The meromorphic differentials of degreenonX are denotedΩ^⊗n(X). Again this set forms a complex vector space and the sumΩ(X) =

n∈NΩ^⊗n(X) forms a ring.

For example, whenX is a complex elliptic curveC/Λthe coordinate maps ϕ_j : U_j −→ V_j are homeomorphic local inverses to natural projection π : C−→X. Letω= (dzj)j∈J. Thenω satisfies the compatibility criterion since the transition maps between coordinate patches take the formz →z+λfor λ∈Λ, and these pull dzk back to dzj. Thus the differentialdz makes sense globally on a complex torus even though a holomorphic variablezdoes not.

Returning to the congruence subgroup Γ of SL2(Z), we now map each meromorphic differentialω onX(Γ) to a meromorphic differentialf(τ)(dτ)ⁿ onH, theΓ-invariant object onHmentioned at the beginning of the section.

The map is the pullback of the natural projectionπ:H −→X(Γ),

π^∗:Ω^⊗n(X(Γ))−→Ω^⊗n(H).

To define this, recall from Chapter 2 that a collection of coordinate neigh- borhoods on X(Γ) is {π(Uj) : j ∈ J} where each Uj ⊂ H^∗ is a neighbor- hood of a point τj ∈ H or of a cusp sj ∈ Q∪ {∞}; the local coordinate map ϕj : π(Uj) −→ Vj is characterized by the relation ψj = ϕj ◦π where ψj : Uj −→ Vj mimics the identifying action of π but maps into C. Now letω = (ωj)j∈J be a meromorphic differential on X(Γ). For eachj ∈J set U_j =Uj∩ H and V_j =ψj(U_j) and ω_j =ωj|V_j. Then the pullback π^∗(ω) is defined locally onHas

π^∗(ω)|U_j =ψ_j^∗(ω_j) for allj. (3.4) To see that (3.4) gives a well defined global meromorphic differentialπ^∗(ω) = f(τ)(dτ)ⁿ onH, consider the commutative diagram

U_j∩U_k

ψj

yytttttttttt

π

ψ_k

%%K

KK KK KK KK K V_j,k

ϕ⁻_j¹

//π(Uj∩Uk) _ϕ

k //V_k,j.

(3.5)

The transition mapϕ_k,j =ϕ_kϕ⁻_j¹|V_j,k satisfiesϕ_k,j◦ψ_j=ψ_k onU_j∩U_k, and pulling back givesψ^∗_k=ψ^∗_j◦ϕ^∗_k,j onV_k,j. Therefore, lettingV_j,k =ψ_j(U_j∩U_k) andV_k,j =ψ_k(U_j∩U_k),

ψ^∗_k(ω_k|V_k,j ) =ψ^∗_j(ϕ^∗_k,j(ω_k|V_k,j )) =ψ^∗_j(ω_j|V_j,k ) by compatibility, so the overlapping local pullbacks have a common valuef(τ)(dτ)ⁿ|U_j∩U_k and the global pullbackπ^∗(ω) =f(τ)(dτ)ⁿ is well defined.

Since the pullback comes from an object on the quotientX(Γ) it must be Γ-invariant. That is, for anyγ∈Γ,

f(τ)(dτ)ⁿ =γ^∗(f(τ)(dτ)ⁿ) = (f(γ(τ)))(γ(τ))ⁿ(dτ)ⁿ

=j(γ, τ)⁻²ⁿf(γ(τ))(dτ)ⁿ= (f[γ]2n)(τ)(dτ)ⁿ. Thus the meromorphic functionf defining the pullback is weakly modular of weight 2n. This is the motivation promised back in Chapter 1 for the definition of weak modularity, at least for even weights.

The function f also satisfies condition (3) in Definition 3.2.1, thatf[α]_2n is meromorphic at∞ for any α ∈ SL₂(Z). To see this, let s = α(∞). The local mapψ :U −→V abouts takes the formψ =ρ◦δ :τ →z →q with δ=α⁻¹andρ(z) =e^2πiz/hwherehis the width ofs. Sinceωis meromorphic at the cusps of X(Γ), the local differential ω|V takes the form g(q)(dq)ⁿ whereg is meromorphic at 0. The functionf onU− {s}comes from pulling

backω|V−{0} under ψ to f(τ)(dτ)ⁿ. Computing this gives f = ˜f[δ]_2n where (Exercise 3.3.2(a))

f˜(z) = (2πi/h)ⁿqⁿg(q), q=e^2πiz/h.

Thus (f[α]2n)(z) = ˜f(z) is meromorphic at ∞ as claimed. In sum, every meromorphic differential ω of degree n on X(Γ) pulls back to a meromor- phic differentialπ^∗(ω) =f(τ)(dτ)ⁿ onHwhere f is an automorphic form of weight 2nwith respect to Γ.

The calculation that π^∗(ω) is well defined can be turned around (Exer- cise 3.3.2(b))—given a collection (ω_j)∈

j∈JΩ^⊗n(V_j) of local meromorphic differentials, exclude the cusps by settingU_j =U_j∩ H andV_j=ψ_j(U_j) and ω_j =ω_j|V_j for allj ∈J. If theω_j pull back underψ_j^∗ to restrictions of some meromorphic differentialf(τ)(dτ)ⁿ onHthen the original local differentials ω_j are compatible and so (ω_j)∈Ω^⊗n(X(Γ)). Thus, a given collection (ω_j) of local meromorphic differentials is compatible, giving a meromorphic differen- tial onX(Γ), if and only if its local elements pull back to restrictions of some f(τ)(dτ)ⁿ∈Ω^⊗n(H) withf ∈ A2n(Γ).

Conversely, given an automorphic form f ∈ A2n(Γ) we will construct a meromorphic differentialω(f) ∈ Ω^⊗n(X(Γ)) such that π^∗(ω) = f(τ)(dτ)ⁿ. By the previous paragraph it suffices to construct local differentials that pull back to restrictions of f(τ)(dτ)ⁿ. Thus the idea is to express f(τ)(dτ)ⁿ in local coordinates.

Each local map ψj : Uj −→ Vj is a compositeψj =ρj ◦δj : τ →z → q with δj ∈ GL2(C). Since δj is invertible it is easy to transform f(τ)(dτ)ⁿ intoz-coordinates locally. First extend the weight-koperator [γ]k to matrices γ∈GL2(C) by definingj(γ, τ) =cτ+das before and

(f[γ]_k)(τ) = (detγ)^k/2j(γ, τ)^−kf(γ(τ))

when this makes sense. The exponentk/2 of detγ is a normalization conve- nient for the next calculation because of the formula γ(τ) = detγ/j(γ, τ)² (Exercise 3.3.3); in other contexts the factor (detγ)^k−1 is more convenient.

Now letU_j=U_j∩ H. Thenf(τ)(dτ)ⁿ|U_j is a pullbackδ^∗_j(λ_j) whereλ_j is the differential obtained by pulling f(τ)(dτ)ⁿ|U_j forward to z-space under δ_j⁻¹. Lettingα=δ_j⁻¹,

λj=α^∗(f(τ)(dτ)ⁿ|U_j) =f(α(z))(d(α(z)))ⁿ

= (detα)ⁿj(α, z)⁻²ⁿf(α(z))(dz)ⁿ= (f[α]2n)(z)(dz)ⁿ.

Thusλj isδjΓ δ_j⁻¹-invariant sincef(τ)(dτ)ⁿ is Γ-invariant. Further pushing λj forward fromz-space toq-space isn’t quite as easy sinceρj is not invertible in general, but it isn’t hard to find local forms ωj that pull back under ρj

toλj as desired.

Specifically, ifU_j⊂ H, i.e., if we are not working at a cusp, then δ_j takes τ_j to 0 and the quotient {±I}(δ_jΓ δ_j⁻¹)0/{±I} is cyclic of order h, gener- ated by the rotationrh :z →µhz whereµh =e^2πi/h. By δjΓ δ⁻_j¹-invariance the pullback r_h^∗(λj) = (f[α]2n)(µhz)µⁿ_h(dz)ⁿ must equal λj, or equivalently µⁿ_h(f[α]2n)(µhz) = (f[α]2n)(z), or (µhz)ⁿ(f[α]2n)(µhz) = zⁿ(f[α]2n)(z). In other words, the functionzⁿ(f[α]2n)(z) takes the formgj(z^h) for some mero- morphic function gj. Note for later reference in proving formula (3.8) that hν0(gj) =n+ντ_j(f) and thusν0(gj) =ν_π(τj)(f) +n/h. Define a local mero- morphic differential inq-coordinates,

ω_j= g_j(q)

(hq)ⁿ(dq)ⁿ onV_j. (3.6)

Sinceρ_j(z) =z^h, thisω_j pulls back underρ_j toλ_j (Exercise 3.3.4(a)), which in turn pulls back underδ_jto the originalf(τ)(dτ)ⁿ|Uj. Thus, for eachU_j⊂ H the differential ω_j onV_j pulls back underψ_j to a suitable restriction of the global differentialf(τ)(dτ)ⁿ onH.

On the other hand, ifUj contains a cuspsj thenδj takessj to∞and the function (f[α]2n)(z) takes the formgj(e^2πiz/h) wheregj is meromorphic at 0 andhis the width ofs. The relevant local differential is now

ω_j = g_j(q)

(2πiq/h)ⁿ(dq)ⁿ onV_j, (3.7) which is meromorphic atq= 0. Sinceq=ρ_j(z) =e^2πiz/h, againω_j pulls back under ρj to λj (Exercise 3.3.4(b)), and as before it follows that ψ^∗_j(ωj) = f(τ)(dτ)ⁿ|U_j. Putting all of this together gives

Theorem 3.3.1.Let k ∈ N be even and let Γ be a congruence subgroup ofSL₂(Z). The map

ω:Ak(Γ)−→Ω^⊗k/2(X(Γ))

f →(ω_j)_j∈J where(ω_j)pulls back to f(τ)(dτ)^k/2∈Ω^⊗k/2(H) is an isomorphism of complex vector spaces.

Proof. The mapωis defined since we have just constructedω(f). Clearlyωis C-linear and injective. Andωis surjective because every (ω_j)∈Ω^⊗k/2(X(Γ)) pulls back to somef(τ)(dτ)^k/2∈Ω^k/2(H) withf ∈ Ak(Γ).

Exercises 3.2.3 and 3.2.4 showed that forkpositive and even,Ak(Γ) takes the form C(X(Γ))f where C(X(Γ)) is the field of meromorphic functions onX(Γ) andf is any nonzero element ofAk(Γ). Thus, Theorem 3.3.1 shows thatΩ^⊗k/2(X(Γ)) =C(X(Γ))ω(f) for such k.

The aim of this chapter is to compute the dimensions of the subspaces Mk(Γ) andSk(Γ) ofAk(Γ). Now that we know thatAk(Γ) andΩ^⊗k/2(X(Γ))

are isomorphic, the final business of this section is to describe the images ω(Mk(Γ)) andω(Sk(Γ)) in Ω^⊗k/2(X(Γ)). Some Riemann surface theory to be presented in the next section will then find the desired dimensions by computing the dimensions of these image subspaces instead in Sections 3.5 and 3.6.

So take any automorphic formf ∈ Ak(Γ) and let ω(f) = (ωj)j∈J. For a point τj ∈ H, the local differential (3.6) with n= k/2 vanishes at q = 0 to (integral) order (Exercise 3.3.5)

ν₀(ω_j)^def=ν₀

g_j(q) (hq)^k/2

=ν_π(τj)(f)−k 2

1−1

h

. (3.8)

In particular, at a nonelliptic point, when h = 1, the order of vanishing is ν₀(ω_j) =ν_π(τj)(f), the order of the original function. For a cusps_j the local differential (3.7) with n = k/2 vanishes at q = 0 to order (Exercise 3.3.5 again)

ν₀(ω_j)^def=ν₀

g_j(q) (2πiq/h)^k/2

=ν_π(sj)(f)−k

2. (3.9)

When k ∈ N is even, formulas (3.8) and (3.9) translate the conditions ν_π(τj)(f)≥0 andν_π(sj)(f)≥0 characterizingMk(Γ) as a subspace ofAk(Γ) into conditions characterizingω(Mk(Γ)) as a subspace ofΩ^⊗k/2(X(Γ)), and similarly forSk(Γ) andω(Sk(Γ)). In particular, the weight 2 cusp formsS2(Γ) are isomorphic as a complex vector space to the degree 1holomorphicdiffer- entials onX(Γ), denotedΩ_hol¹ (X(Γ)) (Exercise 3.3.6). This special case will figure prominently in the later chapters of the book.

Exercises

3.3.1.(a) Show that the pullback is contravariant.

(b) Show that ifι:V₁−→V₂is inclusion then its pullback is the restriction ι^∗(ω) =ω|V1 forω∈Ω^⊗n(V₂).

(c) Show that if ϕ is a holomorphic bijection of open sets in C then (ϕ⁻¹)^∗= (ϕ^∗)⁻¹.

(d) Show that if π : V₁ −→ V₂ is a holomorphic surjection of open sets inCthenπ^∗ is an injection. Ifi:V₁−→V₂ is an injection of open sets inC, needi^∗ be a surjection?

3.3.2.(a) Let ψ = ρ ◦ δ where δ ∈ SL₂(Z) and ρ(z) = e^2πiz/h. Let ω =g(q)(dq)ⁿ. Show that ψ^∗(ω) = f(τ)(dτ)ⁿ with f = ˜f[δ]_2n and ˜f(z) = (2πi/h)ⁿqⁿg(q) whereq=e^2πiz/h.

(b) Consider the situation π : H −→ X(Γ) as described in the small in diagram (3.5). Show that if the local meromorphic differentials ω_j pull back underψ_j^∗to restrictions of some common global meromorphic differential f(τ)dτ onHthen they are compatible. Exercise 3.3.1(d) is relevant.

3.3.3.For γ ∈ GL₂(C), show thatγ(τ) = detγ/j(γ, τ)². (This generalizes Lemma 1.2.2(e).)

3.3.4.(a) For a neighborhood Uj ⊂ H, show that the local differential ωj

in (3.6) pulls back underρj toλj. How wasωj found in the first place?

(b) Same question forωj in (3.7) whenUj contains a cusp.

3.3.5.Prove formulas (3.8) and (3.9).

3.3.6.Prove that S²(Γ) and Ω_hol¹ (X(Γ)) are isomorphic as complex vector spaces. Your proof will incidentally show that the elements ofS2(Γ) vanish at the elliptic points ofX(Γ). Argue this directly as well by examining the text leading up to (3.6).