Modular curves and moduli spaces

Modular Forms, Elliptic Curves, and Modular Curves

1.5 Modular curves and moduli spaces

℘(1/2) is real, as is ℘(i/2) = −℘(1/2). Compute some dominant terms of

℘(1/2) and ℘(i/2) to show that ℘(1/2) is the positive value. For what m does the complex torusC/mΛcorrespond to the elliptic curve with equation y²= 4x(x−1)(x+ 1)?

Reason similarly with Λ = Λµ3 to find the zeros of the corresponding Weierstrass function℘and to show that℘(1/2) is real. For whatmdoes the complex torus C/mΛ correspond to the elliptic curve with equation y² = 4(x−1)(x−µ3)(x−µ²₃)? (A hint for this exercise is at the end of the book.) 1.4.4.Forτ ∈ Hletp_τ(x) = 4x³−g₂(τ)x−g₃(τ). Show that the discriminant of p_τ equals ∆(τ) up to constant multiple, where ∆ is the cusp form from Section 1.1.

1.4.5.Show that when a₂ = 0 in Proposition 1.4.3 the desired lattice isΛ= mΛ_µ3 for a suitably chosen m. Prove the case a₃ = 0 in Proposition 1.4.3 similarly.

for 0 < n < N.) Two such pairs (E, Q) and (E, Q) are equivalent if some isomorphismE−→^∼ EtakesQtoQ. The set of equivalence classes is denoted

S1(N) ={enhanced elliptic curves forΓ1(N)}/∼. An element of S1(N) is denoted [E, Q].

An enhanced elliptic curve for Γ(N) is a pair (E,(P, Q)) where E is a complex elliptic curve and (P, Q) is a pair of points ofEthat generates theN- torsion subgroupE[N] with Weil pairingeN(P, Q) =e^2πi/N. From Section 1.3 eN(P, Q) is some primitive complex Nth root of unity, but this condition is more specific. Two such pairs (E,(P, Q)) and (E,(P, Q)) are equivalent if some isomorphismE−→^∼ EtakesP toPandQtoQ. The set of equivalence classes is denoted

S(N) ={enhanced elliptic curves forΓ(N)}/∼. An element of S(N) is denoted [E,(P, Q)].

Each of S0(N), S1(N), and S(N) is a space of moduli or moduli space of isomorphism classes of complex elliptic curves andN-torsion data. When N = 1 all three moduli spaces reduce to the isomorphism classes of complex elliptic curves as described at the beginning of the section.

For any congruence subgroup Γ of SL2(Z), acting on the upper half plane H from the left, the modular curve Y(Γ) is defined as the quotient space of orbits underΓ,

Y(Γ) =Γ\H={Γ τ :τ∈ H}.

The modular curves forΓ₀(N),Γ₁(N), andΓ(N) are denoted

Y₀(N) =Γ₀(N)\H, Y₁(N) =Γ₁(N)\H, Y(N) =Γ(N)\H. Chapter 2 will show that modular curves are Riemann surfaces and they can be compactified. Compact Riemann surfaces are described by polynomial equations. Thus modular curves have complex analytic and algebraic charac- terizations like complex elliptic curves. Chapter 7 will show how the moduli spaces arise from a single elliptic curve, and it will further show that the poly- nomials describingY0(N) and Y1(N) have rational coefficients. For now we continue to work complex analytically and show that the moduli spaces map bijectively to noncompactified modular curves. Recall the latticeΛτ =τZ⊕Z forτ ∈ H. Proving the following theorem amounts to checking that the torsion data defining the moduli spaces match the conditions defining the congruence subgroups.

Theorem 1.5.1.Let N be a positive integer.

(a)The moduli space forΓ₀(N) is

S₀(N) ={[E_τ,1/N+Λ_τ] :τ∈ H}.

Two points [E_τ,1/N+Λ_τ] and[E_τ,1/N+Λ_τ]are equal if and only if Γ0(N)τ=Γ0(N)τ. Thus there is a bijection

ψ0: S0(N)−→^∼ Y0(N), [C/Λτ,1/N+Λτ]→Γ0(N)τ.

(b)The moduli space forΓ₁(N)is

S1(N) ={[Eτ,1/N+Λτ] :τ∈ H}.

Two points [E_τ,1/N+Λ_τ] and [E_τ,1/N+Λ_τ] are equal if and only if Γ₁(N)τ =Γ₁(N)τ. Thus there is a bijection

ψ1: S1(N)−→^∼ Y1(N), [C/Λτ,1/N+Λτ]→Γ1(N)τ.

(c)The moduli space forΓ(N)is

S(N) ={[C/Λ_τ,(τ /N+Λ_τ,1/N+Λ_τ)] :τ ∈ H}.

Two points[C/Λτ,(τ /N+Λτ,1/N+Λτ)],[C/Λτ,(τ/N+Λτ,1/N+Λτ)]

are equal if and only if Γ(N)τ =Γ(N)τ. Thus there is a bijection ψ: S(N)−→^∼ Y(N), [C/Λ_τ,(τ /N+Λ_τ,1/N+Λ_τ)]→Γ(N)τ.

Proof. Parts (a) and (c) are left as Exercise 1.5.1. For (b), take any point [E, Q] of S1(N). SinceEis isomorphic toC/Λτ for someτ ∈ Has discussed in Section 1.3, we may take E = C/Λτ. Thus Q = (cτ +d)/N +Λτ for somec, d∈Z. Then gcd(c, d, N) = 1 because the order ofQis exactlyN, i.e., ad−bc−kN = 1 for some a, b, and k, and the matrixγ =_{a b}

c d

∈M2(Z) reduces modulo N into SL2(Z/NZ). Modifying the entries of γ modulo N doesn’t affect Q, so since SL₂(Z) surjects to SL₂(Z/NZ) we may take γ = _{a b}

c d

∈SL₂(Z). Letτ=γ(τ) and letm=cτ+d. Thenmτ =aτ+b, so mΛτ=m(τZ⊕Z) = (aτ+b)Z⊕(cτ+d)Z=τZ⊕Z=Λτ

(using Lemma 1.3.1 for the third equality), and m

1 N +Λ_τ

= cτ+d

N +Λ_τ =Q.

This shows that [E, Q] = [C/Λτ,1/N+Λτ] whereτ ∈ H.

Suppose two pointsτ, τ ∈ HsatisfyΓ1(N)τ =Γ1(N)τ. Thusτ =γ(τ) whereγ=_{a b}

c d

∈Γ1(N). Again letm=cτ+d. Then as just shown,

mΛτ=Λτ, m 1

N +Λτ

=cτ+d N +Λτ.

But since (c, d)≡(0,1) (modN) the second equality is nowm(1/N+Λ_τ) = 1/N+Λ_τ. Thus [C/Λ_τ,1/N+Λ_τ] = [C/Λ_τ,1/N+Λ_τ].

Conversely, suppose [C/Λ_τ,1/N+Λ_τ] = [C/Λ_τ,1/N+Λ_τ] withτ, τ ∈ H. Then for somem∈C, mΛ_τ =Λ_τ (by Corollary 1.3.3) and m(1/N+Λ_τ) = 1/N+Λτ. By Lemma 1.3.1 the first of these conditions means that

mτ m

=γ τ

1

for someγ= a b

c d

∈SL2(Z), (1.10) so in particularm=cτ+d. Now the second condition becomes

cτ+d

N +Λ_τ= 1 N +Λ_τ,

showing that (c, d) ≡ (0,1) (modN) and γ ∈ Γ₁(N). Since τ = γ(τ) by (1.10), it follows thatΓ1(N)τ =Γ1(N)τ. Specializing to N = 1, the theorem shows that the space of isomor- phism classes of complex elliptic curves parameterizes the modular curve Y₀(1) = Y₁(1) = Y(1) = SL₂(Z)\H as mentioned in Section 1.3 and at the beginning of this section. This lets us associate a complex number to each isomorphism class. Recall the modular invariantj from Section 1.1, an SL2(Z)-invariant function on H. Each isomorphism class of complex elliptic curves has an associated orbit SL2(Z)τ ∈ SL2(Z)\H and thus has a well defined invariant j(SL2(Z)τ). This value is also associated to any complex elliptic curveE in the isomorphism class and correspondingly denotedj(E).

The Modularity Theorem states that the elliptic curves with rationalj-values arise from modular forms, as discussed back in the preface.

It follows from Theorem 1.5.1 that maps of the modular curves Y0(N), Y1(N), andY(N) give rise to maps of moduli spaces. For example, the natural map fromY₁(N) toY₀(N) taking orbitsΓ₁(N)τtoΓ₀(N)τ becomes the map from S₁(N) to S₀(N) taking equivalence classes [E, Q] to [E,Q], forgetting the generator but keeping the group it generates. For another example, since Γ₁(N) is a normal subgroup ofΓ₀(N) the quotient group acts onY₁(N) and therefore on S₁(N). The action works out to

Γ₁(N)γ: [E, Q]→[E, dQ] whereγ≡_{a b}

c d

(mod N).

This self-map will recur in Chapter 5 and thereafter as a Hecke operator of the sort mentioned in the preface. For details of these examples and others see Exercises 1.5.2 through 1.5.6. The hybrid maps from Exercise 1.5.6,

[E, C, Q]→[E, Q], [E, C, Q]→[E/C, Q+C], (1.11) where C is a cyclic subgroup of prime order pand Q is a point of order N and C∩ Q = {0_E}, combine to describe the other Hecke operator to be introduced in Chapter 5,

[E, Q]→

C

[E/C, Q+C].

The bijections between moduli spaces and modular curves give more ex- amples of modular forms. The idea is that a class of functions of enhanced elliptic curves corresponds to the weight-k invariant functions on the upper half plane. Letk be an integer and letΓ be one ofΓ0(N), Γ1(N), orΓ(N).

A complex-valued functionF of the enhanced elliptic curves forΓ isdegree-k homogeneous with respect toΓ if for every nonzero complex numberm,

F(C/mΛ, mC) F(C/mΛ, mQ) F(C/mΛ,(mP, mQ))

⎫⎬

⎭=

⎧⎪

⎨

⎪⎩

m^−kF(C/Λ, C) ifΓ =Γ₀(N), m^−kF(C/Λ, Q) ifΓ =Γ1(N), m^−kF(C/Λ,(P, Q)) ifΓ =Γ(N).

(1.12)

Given such a function F, define the corresponding dehomogenized function f :H −→C by the rule

f(τ) =

⎧⎪

⎨

⎪⎩

F(C/Λτ,1/N+Λτ) ifΓ =Γ0(N), F(C/Λ_τ,1/N+Λ_τ) ifΓ =Γ₁(N), F(C/Λτ,(τ /N+Λτ,1/N+Λτ)) ifΓ =Γ(N).

(1.13)

Thenf is weight-kinvariant with respect toΓ. To see this, letγ=_{a b}

c d

∈Γ and for anyτ ∈ Hletm= (cτ+d)⁻¹. Then, e.g., forΓ =Γ1(N), using the condition (c, d)≡(0,1) (modN) at the third step,

f(γ(τ)) =F(C/Λ_γ(τ),1/N+Λ_γ(τ)) =F(C/mΛτ, m(cτ+d)/N+mΛτ)

=m^−kF(C/Λ_τ,1/N+Λ_τ) = (cτ+d)^kf(τ).

For instance, the lattice Eisenstein series from Section 1.4 are degree-khomo- geneous with respect to SL2(Z), dehomogenizing to Eisenstein series on the upper half plane.

Conversely, letf be weight-kinvariant with respect toΓ whereΓ is one of Γ₀(N),Γ₁(N), orΓ(N). Then formula (1.13) turns around to define a func- tionF on enhanced elliptic curves of the special type (C/Λ_τ,(torsion data)) in terms off (Exercise 1.5.7). If two such enhanced elliptic curves are equiv- alent, e.g., (C/Λ_τ,1/N+Λ_τ) = (C/mΛ_τ, m/N+mΛ_τ), thenτ =γ(τ) and m=cτ+dfor someγ=_{a b}

c d

∈Γ as in the proof of Theorem 1.5.1. Thus f(τ) =m^kf(τ) and soF obeys formula (1.12) for those two points,

F(C/Λ_τ,1/N+Λ_τ) =f(τ) =m^−kf(τ) =m^−kF(C/Λ_τ,1/N+Λ_τ).

Since every enhanced elliptic curve is equivalent to an enhanced elliptic curve of the special type, formula (1.12) in its full generality extendsFto a degree-k homogeneous function of enhanced elliptic curves forΓ.

As an example of the correspondence, letN >1 and letv= (c_v, d_v)∈Z² be a vector whose reduction v modulo N is nonzero. Define a function of enhanced elliptic curves

F₂^v(C/Λ,(P, Q)) = 1

N²℘_Λ(c_vP+d_vQ).

(The superscript v is just a label, not an exponent.) Then F₂^v is degree-2 homogeneous with respect toΓ(N) (Exercise 1.5.8(a)). Chapter 4 will use the corresponding function

f₂^v(τ) = 1 N²℘_τ

cvτ+dv

N

to construct Eisenstein series of weight 2. Similarly, recall the lattice constants g2(Λ) = 60G4(Λ) and g3(Λ) = 140G6(Λ), the integer multiples of lattice Eisenstein series from Section 1.4, and define functions of enhanced elliptic curves forΓ(N),Γ1(N), andΓ0(N),

F₀^v(C/Λ,(P, Q)) = g₂(Λ)

g3(Λ)℘_Λ(c_vP+d_vQ), F₀^d(C/Λ, Q) = g₂(Λ)

g3(Λ)℘_Λ(dQ), d∈Z, d≡0 (modN)

=F₀^(0,d)(C/Λ,(P, Q)) for suitableP , F₀(C/Λ, C) = g₂(Λ)

g3(Λ)

Q∈C−{0}

℘_Λ(Q)

=

N−1 d=1

F₀^d(C/Λ, dQ) for any generatorQofC.

Each of these functions is degree-0 homogeneous with respect to its group (Exercise 1.5.8(b)). The corresponding weight-0 invariant functions are

f₀^v(τ) =g₂(τ) g3(τ)℘_τ

c_vτ+d_v N

, f₀^d(τ) =g₂(τ)

g₃(τ)℘_τ d

N

=f₀^(0,d)(τ), f0(τ) =g2(τ)

g3(τ)

N−1 d=1

℘_τ d

N

=

N−1 d=1

f₀^d(τ).

Chapter 7 will use these functions to show how polynomial equations describe modular curves. We could have defined the functions withg2(τ)g3(τ)/∆(τ) where we haveg2(τ)/g3(τ), to avoid poles inH, but the meromorphic functions here are normalized more suitably for our later purposes.

Exercises

1.5.1.Prove the other two parts of Theorem 1.5.1. (A hint for this exercise is at the end of the book.)

1.5.2.The containments Γ(N) ⊂ Γ₁(N) ⊂ Γ₀(N) give rise to surjections Y(N) −→ Y1(N) and Y1(N) −→ Y0(N) given by Γ(N)τ → Γ1(N)τ and Γ1(N)τ→Γ0(N)τ. Describe the corresponding maps between moduli spaces.

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

1.5.3.Since the group containments mentioned in the previous exercises are normal, it follows that the quotientΓ1(N)/Γ(N) ∼=Z/NZ (cf. Section 1.2) acts on the modular curveY(N) by multiplication from the left, and similarly forΓ0(N)/Γ1(N)∼= (Z/NZ)^∗ andY1(N). Describe the corresponding action ofZ/NZon the moduli space S(N), and similarly for (Z/NZ)^∗ and S1(N).

(A hint for this exercise is at the end of the book.) 1.5.4.Let w_N = _{0 −1}

N 0

∈ GL⁺₂(Q). Show that w_N normalizes the group Γ₀(N) and so gives an automorphism Γ₀(N)τ → Γ₀(N)w_N(τ) of the mod- ular curve Y₀(N). Show that this automorphism is an involution (meaning it has order 2) and describe the corresponding automorphism of the moduli space S₀(N). (A hint for this exercise is at the end of the book.)

1.5.5.Let N be a positive integer and letp be prime. Define maps π1, π2 : Y0(N p) −→ Y0(N) to be π1(Γ0(N p)τ) = Γ0(N)τ and π2(Γ0(N p)τ) = Γ0(N)(pτ). How do the corresponding maps ˆπ1,πˆ2 : S0(N p) −→ S0(N) act on equivalence classes [E, C]? Same question, but forΓ1(N p) andΓ1(N) and S1(N p) and S1(N), and then again forΓ(N p), etc.

1.5.6.Let N be a positive integer and let p be prime. Define congruence subgroups of SL2(Z),

Γ⁰(p) = a b

c d

∈SL2(Z) : a b

c d

≡ ∗0

∗ ∗

(mod p) and

Γ₁⁰(N, p) =Γ₁(N)∩Γ⁰(p), and define a corresponding modular curve,

Y₁⁰(N, p) =Y(Γ₁⁰(N, p)).

An enhanced elliptic curve forΓ₁⁰(N, p) is an ordered triple (E, C, Q) where Eis a complex elliptic curve,C is a cyclic subgroup ofE of orderp, andQis a point ofE of orderN such thatC∩ Q={0E}. Two such triples (E, C, Q) and (E, C, Q) are equivalent if some isomorphism E −→^∼ E takes C to C andQtoQ. The moduli space forΓ₁⁰(N, p) is the set of equivalence classes,

S⁰₁(N, p) ={enhanced elliptic curves forΓ₁⁰(N, p)}/∼. An element of S⁰₁(N, p) is denoted [E, C, Q].

(a) By the proof of Theorem 1.5.1(b), every element of S⁰₁(N, p) takes the form [C/Λ_τ, C,1/N +Λ_τ]. Show that C has a generator of the form

(τ+j)/p+Λ_τ or 1/p+Λ_τ, this last possibility arising only ifpN. (This idea will be elaborated in Section 5.2.) Show that thereforeC and Qcan be assumed to take the form

C=

%aτ+b p +Λ_τ

&

, Q= cτ+d N +Λ_τ,

a b c d

∈SL₂(Z).

Let γ = _{a b}

c d

and let τ = γ(τ). Show that [C/Λ_τ, C,1/N +Λ_τ] = [C/Λ_τ,τ /p+Λ_τ,1/N+Λ_τ]. (A hint for this exercise is at the end of the book.)

(b) Part (a) shows that the moduli space forΓ₁⁰(N, p) is S⁰₁(N, p) ={[C/Λ_τ,τ /p+Λ_τ,1/N+Λ_τ] :τ∈ H}.

Show that two points of S⁰₁(N, p) corresponding toτ, τ ∈ Hare equal if and only ifΓ₁⁰(N, p)τ=Γ₁⁰(N, p)τ, and thus there is a bijection

ψ₁⁰: S⁰₁(N, p)−→Y₁⁰(N, p), [C/Λτ,τ /p+Λτ,1/N+Λτ]→Γ₁⁰(N, p)τ.

(c) Show that the mapsΓ₁⁰(N, p)τ→Γ1(N)τ andΓ₁⁰(N, p)τ→Γ1(N)τ /p fromY₁⁰(N, p) toY1(N) are well defined. Show that the corresponding maps from S⁰₁(N, p) to S1(N) are as described in (1.11).

1.5.7.Show that if two enhanced elliptic curves for Γ of the special type (C/Λτ,(torsion data)) are equal thenτ =γ(τ) for someγ= [¹_{0 1}^b]∈Γ. Show that consequently iff :H −→Cis weight-kinvariant with respect toΓ then formula (1.13) gives a well defined functionF on enhanced elliptic curves of the special type.

1.5.8.(a) Show that the functionF₂^vfrom the section is degree-2 homogeneous with respect toΓ(N).

(b) Similarly, show that the functionsF₀^v,F₀^d, andF₀are degree-0 homo- geneous with respect toΓ(N),Γ₁(N), andΓ₀(N) respectively.