Modular Forms (MA4H9)

Marc Masdeu Copyright c 2014 Marc Masdeu

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1 Modular Forms for SL2(Z) ...... 5 1.1 What can modular forms do for us5 1.1.1 Class numbers...... 5 1.1.2 An example: a modular form of level 11 ...... 5 1.2 Ode to the upper half-plane6 1.3 The modular group7 1.4 Fourier expansions of Eisenstein series9 1.5 Valence formula 10 1.6 A product formula for ∆(z) 12 1.6.1 The Dedekind-eta function...... 12

2 Modular Forms on Congruence Groups ...... 15 2.1 Congruence subgroups 15 2.2 Fourier expansion at inﬁnity 16 2.3 Expansions at cusps 16

3 Complex tori ...... 19 3.1 Lattices and tori 19 3.2 Tori and elliptic curves 21 3.2.1 Meromorphic functions on C/Λ ...... 21 3.2.2 The Weierstrass ℘-function...... 21

Bibliography ...... 25

Index ...... 27

What can modular forms do for us Class numbers An example: a modular form of level 11 Ode to the upper half-plane The modular group Fourier expansions of Eisenstein series Valence formula A product formula for ∆(z) The Dedekind-eta function

1. Modular Forms for SL2(Z)

1.1 What can modular forms do for us In order to motivate the study of modular forms, we give two example results whose proof relies on the existence of these objects.

1.1.1 Class numbers Let D < 0 be the discriminant of an imaginary quadratic field, and let h(D) the class number (the size of the class group). This number is interesting, for a famous problem studied by Gauss is to know when h(D) = 1 (there are only finitely many such), or in general, list all D for which h(D) = h0 for a fixed h0. Here is a simple modular form:

∞ 2 θ(z) = ∑ qn = 1 + 2q + 2q4 + 2q9 + , q = e2πiz. n= ∞ ··· − where z H = z C: ℑ(z) > 0 . Note that q < 1 so the above makes sense. Take its cube: ∈ { ∈ } | | ∞ H(z) = θ(z)3 = 1 + 6q + 12q2 + 24q163 + 168q251 + = ∑ r(n)qn. ··· ··· ··· n=0

Lemma 1.1.1 If D ∼= 3 (mod 8), then 24h(D) = r(D). There are similar results to the above for other D, the point here is that this gives an easy way to calculate h(D)!

1.1.2 An example: a modular form of level 11 Consider another modular form: ∞ ∞ f (z) = q ∏(1 qn)2(1 q11n)2 = q 2q2 q3 + 2q4 + q5 + 2q6 + = ∑ a(n)qn. n=1 − − − − ··· n=1 6 Chapter 1. Modular Forms for SL2(Z)

Theorem 1.1.2 1. a(nm) = a(n)a(m) whenever (n,m) = 1. 2. a(p) 2√p for all prime p. | | ≤ Consider the equation:

Y 2 +Y = X3 X2 10X 20, − − − and let N(p) be the number of solutions in Fp. An easy heuristic reasoning leads us to think that N(p) p. ' Theorem 1.1.3 — Hasse. p N(p) 2√p. | − | ≤ It turs out that a(p) = p N(p) for all p, so again we can easily calculate (from f ) what is − N(p) for all p. We say in this case that E “is modular”. The book [DS05] (but not this course) explains how to attach an elliptic curve to a modular form (this is called “Eichler–Shimura”). It is much harder to reverse this process, and this is what A.Wiles did in order to prove Fermat’s Last Theorem.

1.2 Ode to the upper half-plane This section introduces the seemingly innoccuous upper half-plane H. Deﬁnition 1.2.1 The upper half-plane H is the set of complex numbers with positive imaginary part:

H = z = x + iy ℑ(z) > 0 . { | } The upper half-plane appears in the classiﬁcation of Riemann surfaces: there are only three of them which are simply connected which are the complex plane, the complex sphere, and H. The general linear group GL2(R) consists of all 2 2 invertible matrices with entries in R. It + × constains the subgroup GL2 (R) of matrices with positive determinant. The special linear group + a b + SL2(R) GL (R) consists of those matrices with determinant 1. For γ = GL (R) and ⊂ 2 c d ∈ 2 z H, deﬁne γz as: ∈ a b az + b γz = z = . (1.1) c d cz + d

+ Lemma 1.2.1 Let γ = a b GL (R). Then: c d ∈ 2 det(γ)ℑ(τ) ℑ(γτ) = . cτ + d 2 | | Proof. Exercise (direct computation).

+ Corollary 1.2.2 GL2 (R) acts on the left on H. Note that the determinant gives a decomposition

+ GL (R) = SL2(R) R, 2 × λ 0 and since the scalar matrices (those of the form 0 λ ) act trivially on H, from now on we will 1 0 restrict our attention to SL2(R). In fact, since the scalar matrix −0 1 belongs to SL2(R), the 1.3 The modular group 7

above action on H factors through PSL2(R) = SL2(R)/ 1 , which is called the projective special linear group. {± } From this action we can deduce a right action on functions on H, by precomposing:

( f γ)(z) = f (γz). | We will need slightly more general actions on functions, but before we introduce a piece of notation that will later prove useful. Deﬁnition 1.2.2 The automorphy factor is the function

j : SL2(R) H C × → given by:

a b j(γ,z) = cz + d, γ = . c d

The following lemma gives a very interesting property of the automorphy factor, sometimes referred to as the cocycle relation:

Lemma 1.2.3 For every γ1, γ2 in SL2(Z) and for every z H we have: ∈

j(γ1γ2,z) = j(γ1,γ2z) j(γ2,z).

Finally, we define an action of SL2(R) on functions f : H C, for each k Z. → ∈ Definition 1.2.3 The weight-k slash operator is defined as

k ( f kγ)(z) = j(γ,z)− f (γz). (1.2) | The cocycle property implies that if f is a function, then:

f k(γ1γ2) = ( f kγ1) kγ2, γ1,γ2 SL2(R). | | | ∀ ∈

That is, for each k the weight-k slash operator deﬁnes an action of SL2(R) on functions on the upper-half plane.

1.3 The modular group

Let Γ = SL2(Z)SL2(R) be the subgroup of matrices with entries in Z (and determinant 1), which of course still acts on functions as we have seen. Deﬁnition 1.3.1 A holomorphic function f : H C is called weakly-modular of weight → k Z for Γ if: ∈ f (γ z) = j(γ,z)k f (z), γ Γ. (1.3) · ∀ ∈ We will need an extra analytic property to deﬁne modular forms for Γ. For now, note that:

d(γ z) · = j(γ,z) 2, dz −

so we can rewrite the weakly-modular property by asking that the differential f (z)(dz)k/2 is invariant under Γ. It also shows that if (1.3) holds for γ1 and γ2, then it also holds for γ1γ2. 8 Chapter 1. Modular Forms for SL2(Z)

Lemma 1.3.1 Γ is generated by:

1 1 0 1 T = , S = . 0 1 1− 0

The lemma and the observation before it imply that for f to be weakly-modular it is enough to check (1.3) for T and S:

f (z + 1) = f (z), f ( 1/z) = zk f (z). − Example 1.1 — Eisenstein Series. For k 3, deﬁne ≥ k Gk(z) = ∑ 0 (mz + n)− . (m,n) Z2 ∈

Lemma 1.3.2 Gk converges absolutely for all z, and uniformly on compact subsets.

Therefore we may rearrange the series, and deduce that Gk is weakly-modular.

R If k is odd then Gk(z) is identically zero. For even k, we have: 1 lim Gk(z) = = 2ζ(k). z i∞ ∑ nk → k Z ∈

The transformation property (rather, the fact that f (z+1) = f (z)) implies that f has a Fourier expansion. Another way to think about it is that there is a holomorphic map:

2πiz exp: H 0 < q < 1 , z q = e . → { | | } 7→ If f is holomorphic and 1-periodic, then we can deﬁne g(q) = f (z). That is, we may deﬁne:

logq g(q) = f , 2πi where we may choose any branch of the logarithm because of the periodicity of f . The function g is holomorphic on D0, and thus it has a Laurent expansion

∞ g(q) = ∑ a(n)qn. n= ∞ − Therefore f has an expansion

∞ f (z) = ∑ a(n)e2πinz. n= ∞ −

Deﬁnition 1.3.2 We say that f is meromorphic at ∞ (respectively holomorphic at ∞) if n f (z) = ∑n n0 a(n)q (respectively if in addition n0 = 0). ≥ Note that checking that f is holomorphic at inﬁnity is the same as checking that f (z) is bounded as z approaches i∞. 1.4 Fourier expansions of Eisenstein series 9

Deﬁnition 1.3.3 We say that f vanishes at ∞ if n0 = 1.

Similarly, note that f vanishing at ∞ is equivalent to f 0 as z i∞. → → Example 1.2 For all even k 4, the Eisenstein series Gk(z) 2ζ(k) as z i∞, so it is ≥ → → holomorphic at ∞ (but does not vanish).

Deﬁnition 1.3.4 Let k Z and let f : H C. We say that f is a modular form of weight k ∈ → for SL2(Z) if: 1. f is holomorphic, k a b 2. f (γz) = (cz + d) f (z) for all γ = SL2(Z), and c d ∈ 3. f is holomorphic at ∞. A cusp form is a modular form which vanishes at ∞. The space of modular forms of weight k is written Mk = Mk(SL2(Z)), and it contains the space of cusp forms of weight k, which in turn is written Sk = Sk(SL2(Z)).

R If we replace “holomorphic” with “meromorphic” above, we obtain what can be called meromorphic modular forms. Other authors call them modular functions, but this name is used in different contexts and we will avoid it.

Note that both Mk and Sk are C-vector spaces. Also, multiplication of functions gives L M = k Z Mk the structure of a graded ring. That is, MrMs Mr+s. Finally, for all odd k one ∈ ⊆ has Mk = 0 . { }

1.4 Fourier expansions of Eisenstein series Recall that we have deﬁned, for k 4 even, ≥ k Gk(z) = ∑ (mz + n)− . (m,n) Z2 ∈

Deﬁnition 1.4.1 The Bernoulli numbers are deﬁneda by:

x ∞ xk 1 1 x2 1 x4 = Bk = 1 x + + . ex 1 ∑ k! − 2 6 2 − 30 24 ··· − k=0 aThese are called “ﬁrst Bernoulli numbers”, and differ by a sign from those deﬁned originally by Bernoulli. The Bernoulli numbers appear also naturally in the formula:

k (2πi) Bk ζ(k) = − , k 2. 2 k! ∀ ≥

Bn ζ(1 n) = , n 1. − − n ∀ ≥ An odd prime p is called regular if p does not divide the numerator of B2, B4,. . . Bp 3. This is p − equivalent to p not dividing the class number of Q(√1). Under this assumption, Fermat’s Last Theorem was proved by Kummer around 1850, and probably by Fermat himself. It is not know whether inﬁnite regular primes exist, although Siegel conjectured that about 60% of primes are regular! 10 Chapter 1. Modular Forms for SL2(Z)

Theorem 1.4.1 Let k 4 be even. Then ≥ ∞ 2k n Gk(z) = 2ζ(k)Ek(z), where Ek(z) = 1 ∑ σk 1(n)q Q[[q]]. − Bk n=1 − ∈ Here we write:

k 1 σk 1(n) = ∑d − . − d n | Proof. Consider the Fourier expansion of π cot(πz):

∞ π cot(πz) = πi 2πi ∑ qm. − m=0 Note that the left-hand side is the logarithmic derivative of sin(πz):

∞ z2 1 ∞ 1 1 sin(πz) = 1 = + + . ∏ − n2 z ∑ z d z + d n=1 d=1 − Differentiating both sides k 1 times we get the formula. −

Example 1.3 ∞ n E4 = 1 + 240 ∑ σ3(n)q M4 n=1 ∈

∞ n E6 = 1 504 ∑ σ5(n)q M6. − n=1 ∈ 3 2 Since both E4 and E6 are both in M12, its difference is also there. Computing we see that: 3 2 E E = (1 + 720q + ) (1 1008q + ) = 1728q + S12. 4 − 6 ··· − − ··· ··· ∈ Thus we may deﬁne

3 2 E4 E6 2 3 ∆(z) = − = q 24q + 252q + S12. 1728 − ··· ∈

1.5 Valence formula

Write vP( f ) be the order of vanishing f at P H ∞ . of f at ∞. ∈ ∪ { } Theorem 1.5.1 — Valence formula. Let f be a non-zero weakly modular form of weight k on SL2(Z). Then: 1 1 k v ( f ) + v ( f ) + v ( f ) + v ( f ) = . ∞ 2 i 3 ρ ∑ τ 12 τ Γ H ∈ \ 0 Here the sum runs through the orbits in Γ H other than those of i and ρ. \ Proof. TODO. 1.5 Valence formula 11

i = √ 1 −

2πi ρ¯ ρ = e 3 −

Figure 1.1: Fundamental domain for SL2(Z)

Corollary 1.5.2 1. M0 = C. 2. Mk = CEk and Sk = 0 for 4 k 10 and k = 14. { } ≤ ≤ 3. Mk = CEk Sk for k 4. ⊕ ≥ 4. S12 = C∆, and Sk = ∆Mk 12 for k 12. 5. In general, − ≥

( k 1 + 12 k 2 (mod 12), dim(Mk) = b c ≡ k else. b 12 c

Example 1.4 This allows to write down all spaces of modular and cusp forms for SL2(Z). For example,

M18 = CE18 S18 = CE18 ∆M6 = CE18 C∆E6. ⊕ ⊕ ⊕ 2 M30 = CE30 C∆E18 C∆ E6. ⊕ ⊕ Another basis for the same space (which is better because it is expressed in terms of E4, E6 and ∆):

5 3 2 M30 = CE6 ∆E6 ∆ E6. ⊕ ⊕

Note that these forms are linearly independent (why?). Since dimM30 = 3, they form a basis.

R Suppose that f (z) is a non-zero weakly-modular form of weight 0. Then f (γz) = f (z). By the valence formula, f has the same number of zeros as poles. This number is called the valence of f , and hence the nam for the theorem.

Example 1.5 Deﬁne the function 3 E4 1 + 1 j(z) = = ··· = q− + 744 + 196884q + . ∆ q + ··· ··· It is a weakly modular form of weight 0, with a simple pole at ∞. Since E4(ρ) = 0 (why?), then j(z) has a triple zero at ρ. 1 Now take c C. Then j(z) c = q− + 744 c + is another modular function. Since ∆ ∈ − − ··· has no zeros in H, then j(z) is holomorphic on H and thus j(z) c having one pole menas that it must have exactly one zero. Therefore there is a bijection: −

j : Γ H C. \ →

12 Chapter 1. Modular Forms for SL2(Z) 1.6 A product formula for ∆(z) Consider the weight-2 Eisenstein series 1 0 G2(z) = ∑∑ 2 , c d (cz + d) which does not converge absolutely. It is not a weight-2 modular form (there are no such other than 0), but we still have the formula:

∞ n G2(z) = 2ζ(2)E2(z), E2(z) = 1 24 ∑ σ1(n)q . − n=1

The function E2(z) is holomorphic on H, and E2(z + 1) = E2(z). However:

2 12 z− E2( 1/z) = E2(z) + . − 2πiz People sometimes calls such functions quasi-modular.

1.6.1 The Dedekind-eta function Deﬁne η(z) as:

∞ η(z) = q1/24 ∏(1 qn). n=1 −

The function η(z) is holomorphic and non-vanishing on H (to check it, it is enough to check that n ∑q converges absolutely and uniformly on compact subsets of H. Theorem 1.6.1 p η( 1/z) = z/iη(z). − Proof. Note that: ! d 2πi ∞ 2πinqn 2πi ∞ nqn logη(z) = + − = 1 24 dz 24 ∑ 1 qn 24 − ∑ 1 qn n=1 − n=1 − ∞ ∞ ! πi nm πi = 1 24 ∑ n ∑ q = E2(z). 12 − n=1 m=1 12

Using quasi-modularity of η(z), we deduce: p dlog(η( 1/z)) = dlog z/iη(z) . − Therefore p η( 1/z) = C z/iη(z). − Setting z = i one gets C = 1, as we wanted. From this we obtain the sought product formula for ∆: 1.6 A product formula for ∆(z) 13

Theorem 1.6.2

∞ ∆ = q ∏(1 qn)24. n=1 −

Proof. Note that:

∞ η24(z) = q ∏(1 qn)24, n=1 − so η24(z + 1) = η24(z). Moreover,

η24( 1/z) = z12η24(z). − 24 24 24 Since η (z) = q + and η (z) S12 = C∆, we deduce that η = ∆. ··· ∈

Congruence subgroups Fourier expansion at inﬁnity Expansions at cusps

2. Modular Forms on Congruence Groups

2.1 Congruence subgroups

Let N 1 be an integer. In this section we will consider subgroups of SL2(Z) that are especially nice to≥ work with. There are other subgroups that are interesting but beyond the scope of this course. Deﬁnition 2.1.1 The principal congruence subgroup of level N is

1 0 Γ(N) = γ SL2(Z) γ = (mod N) . { ∈ | ∼ 0 1 }

Note that Γ(1) = SL2(Z), so we are strictly generalizing Chapter 1. There are too few principal congruence subgroups (only one for each N 1), so it is desirable to consider more general subgroups. ≥

Deﬁnition 2.1.2 A subgroup Γ SL2(Z) is a congruence subgroup if ≤

Γ(N) Γ SL2(Z). ≤ ≤ The level of a congruence subgroup Γ is the minimum N such that Γ(N) Γ. ≤ One can think of many different ways to construct congruence subgroups. There are two families that arise so frequently that have special notation for them: Example 2.1 For each N 1, deﬁne ≥ 1 Γ1(N) = (mod N) , { 0 1∗ } and also

Γ0(N) = ( 0∗ ∗ )(mod N) . { ∗ }

Lemma 2.1.1 For each N 1, ≥

Γ(n) < Γ1(N) < Γ0(N) < SL2(Z), (2.1) N φ(N) ψ(N) 16 Chapter 2. Modular Forms on Congruence Groups

where 1 1 φ(N) = N ∏ 1 , ψ(N) = N ∏ 1 + . p N − p p N p | | Proof. Exercise.

Deﬁnition 2.1.3 A function f : H C is weakly modular of weight k with respect to Γ if it → is meromorphic on H and

f kγ = f , γ Γ. | ∀ ∈

2.2 Fourier expansion at inﬁnity 1 N Let Γ be a congruence subgroup of level N. Note that the matrix 0 1 belongs to Γ(N), and thus there is a minimum h > 0 with the property that 1 h Γ. 0 1 ∈ Deﬁnition 2.2.1 The fan width of Γ is the minimum h > 0 such that 1 h Γ. 0 1 ∈

R The fan width of a congruence subgroup of level N is a divisor of N.

Write

2πiz qh = qh(z) = e h ,

and note that z qh(z) is periodic with period h. Let g be the function: 7→ 1 g = f q− . ◦ h

That is, g(qh) = f (z). Note that although qh is not invertible, the above definition makes sense, and g has a Laurent expansion. Definition 2.2.2 The q-expansion of f at infinity is the Laurent expansion:

∞ n f (z) = g(qh) = ∑ a(n)qh. n= ∞ −

2.3 Expansions at cusps We first need to define what we mean by cusps: Definition 2.3.1 The cusps are the Q-rational points the boundary of H. That is:

1 cusps = Q ∞ = P (Q). { } ∪ { } Note that if a/c Q is a reduced fraction (that is, if gcd(a,c) = 1) then there is a matrix a b ∈ γ = SL2(Z) (why?). Moreover, c d ∈ a b a ∞ = , c d c

and therefore SL2(Z) acts transitively on the set of cusps. However, when Γ is not SL2(Z) then not all cusps are equivalent under the action of Γ. 2.3 Expansions at cusps 17

Exercise 2.1 Sho that ∞ and 0 are not equivalent under Γ(N) for any N > 1.

Let s be a cusp, s = ∞. Write s = α∞ for some α SL2(Z), and consider the equation: 6 ∈ k f (αz) = j(α,z) ( f kα)(z). | Since j(α,z) = 0,∞ when z is near ∞, the behavior of f (z) near s is related to the behavior of 6 ( f kα)(z) near ∞. | Assume that f is weakly modular for the congruence subgroup Γ. Since

1 ( f kα) k(α− γα) = ( f kγ) kα = f kα, | | | | | 1 the new function f kα is invariant under the group Γ = α Γα. Since Γ(N) is normal inside | 0 − SL2(Z) (why?), we deduce that Γ0 is also a congruence subgroup of level N. Hence f kα has a | Fourier expansion at inﬁnity as in Section ?? in powers of qN. Deﬁnition 2.3.2 The expansion of f at a cusp s is the expansion:

∞ n f kα = ∑ b(n)qN. | n= ∞ − The expansions at different cusps allow us to define modular forms for arbitrary congruence subgroups. Definition 2.3.3 A function f : H C is a modular form of weight k for a congruence → subgroup Γ if: 1. f is holomorphic on H, 2. f kγ = f for all γ Γ, and | ∈ 3. f kα is holomorphic at infinity for all α SL2(Z). | ∈ A function is a cusp form of weight k for a congruence subgroup Γ if instead of 3 is satisfies: 3’. f kα vanishes at infinity for all α SL2(Z). | ∈ The space of modular forms of weight k for a congruence subgroup Γ is written Mk(Γ); the space of cusp forms of weight k for a congruence subgroup Γ is written Sk(Γ).

Proposition 2.3.1 Suppose that f : H C satisﬁes 1 and 2 above. Suppose that f is holomorphic at inﬁnity. That is, →

∞ n f (z) = ∑ a(n)qN. n=0 Furthermore, supose that there exists a constant C such that:

a(n) < Cnr, n 0. | | ∀ ≥

Then f satisfies 3. That is, f Mk(Γ). ∈ Proof. Exercise. We end this chapter by realizing that the definition of modular forms can be checked by finitely many computations. Suppose that σ = α∞ and τ = β∞ are two cusps (here α and β are in SL2(Z)). Suppose that σ = γτ with γ Γ. ∈ 18 Chapter 2. Modular Forms on Congruence Groups

Proposition 2.3.2 If

∞ n f kα = ∑ a(n)qh, | n= ∞ − then ∞ n k 2πin j f kβ = ∑ b(n)qh, b(n) = (pm1) e h a(n), j Z. | n= ∞ ∈ −

1 Proof. By assumption α∞ = γβ∞, so α− γβ∞ = ∞, and therefore since the only matrices that ﬁx inﬁnity are of the form 1 j we have: ± 0 1 1 1 j α− γβ = , j Z. ± 0 1 ∈

This means that 1 1 j β = γ− α , ± 0 1 and therefore: 1 1 j k 2πinz 1 j f kβ = f k I kγ− kα k = ( 1) a(n)e h k (2.2) | | ± | | | 0 1 ± ∑ | 0 1 k 2πin(z+ j) = ( 1) a(n)e h . (2.3) ± ∑

Corollary 2.3.3 For each n Z, we have a(n) = 0 if and only if b(n) = 0. In particular, it is ∈ enough to check 3 or 30 for one representative from each of the equivalence classes of cusps. Lattices and tori Tori and elliptic curves Meromorphic functions on C/Λ The Weierstrass ℘-function

3. Complex tori

In this chapter we reinterpret modular forms as functions on certain very interesting geometric objects.

3.1 Lattices and tori Deﬁnition 3.1.1 A lattice is a free Z-module Λ of rank 2 inside C. Concretely, Λ = Zω1 ⊕ Zω2, where ω1 and ω2 are R-linearly independent complex numbers. We will always assume that ω1/ω2 H, which can always be accomplished by possible swapping them. ∈ As you know, in general there are many choices for a basis of a given submodule.

Proposition 3.1.1 Suppose that Λ = ω1,ω2 and Λ = ω ,ω . Then Λ = Λ if and only if h i 0 h 10 20 i 0 there exists γ SL2(Z) such that ∈ ω1 = γ ω10 ,ω20 . omega2

Proof. Exercise.

Lattices are get interesting when we quotient C out by them. Deﬁnition 3.1.2 A complex torus is the set C/Λ = z + Λ z C . It has the structure of an abelian group, and analytically it is a torus (a genus{ 1 Riemann| ∈ surface).} 20 Chapter 3. Complex tori

Proposition 3.1.2 Suppose that ϕ : C/Λ C/Λ0 is a holomorphic map. Then there exist complex numbers m and b such that: → 1. mΛ Λ , and ⊆ 0 2. ϕ(z + Λ) = mz + b + Λ0. Moreover, ϕ is invertible if and only if mΛ = Λ0.

Proof. The complex plane C is the univeral covering space of C/Λ and C/Λ0. Therefore, ϕ can be lifted to a map ϕ˜ : C C. Suppose now that λ Λ, and deﬁne → ∈ f (z) = ϕ˜(z + λ) ϕ˜(z). λ −

Then fλ is continuous and has image in Λ0. Since Λ0 is discrete, necessarily fλ is constant. Consider the derivative. For each λ Λ, we have ∈

ϕ˜ 0(z + λ) = ϕ˜ 0(z).

Therefore ϕ˜ 0(z) is holomorphic and doubly-periodic, hence bounded. By Liouville’s theorem, ϕ˜ 0 is constant. We deduce that ϕ˜(z) = mz + b, as wanted.

Corollary 3.1.3 Let ϕ : C/Λ C/Λ0 be a holomorphic map. Then ϕ is a group homomor- → phism if and only if ϕ(0) = 0, if and only if b Λ . ∈ 0

R If ϕ as above is a holomorphic group isomorphism, then necessarily mΛ = Λ0 and ϕ(z + Λ) = mz + Λ0.

Here are two examples of possible maps like the ones above.

Example 3.1 The map multiplication-by-N, usually written [N], is a homomorphism:

[N] C/Λ / C/Λ z + Λ / Nz + Λ

The kernel of [N] is the group of N-torsion points, isomorphic to Z/NZ Z/NZ. × Example 3.2 Consider Λ = Zω1 Zω2. Deﬁne τ = ω1/ω2 H, and set Λτ = Zτ Z. Then . ⊕ ∈ ⊕ C/Λ ∼= C/Λτ The previous example can be brought a little bit further as follows.

Lemma 3.1.4 The complex tori C/Λτ and C/Λτ0 are isomorphic if and only if τ = γτ0 for some γ SL2(Z). ∈ Proof. Supose that

aτ0 + b τ = γτ0 = . cτ0 + d

Let m = cτ0 + d. Then mΛτ = Z(aτ0 + b) Z(cτ0 + d). By Proposition 3.1.1, this lattice is the ⊕ same as Zτ0 Z = Λτ . The other direction is obtained by reading the equalities in reverse. ⊕ 0 3.2 Tori and elliptic curves 21

R We have just seen that there is a “natural bijection” between isomorphism classes of tori and elements τ SL2(Z) H. This innocent statement is really important. ∈ \

Example 3.3 — Eistenstein series attached to a lattice. Let Λ be a lattice. Deﬁne, for k > 2 even,

k Gk(Λ) = ∑ 0 ω− . ω Λ ∈

Note that Gk(Λτ ) = Gk(τ) is the usual Eistenstein series deﬁned in the previous chapter. The transformation law reads in this case:

k Gk(mΛ) = m− Gk(Λ).

3.2 Tori and elliptic curves The next goal is to relate C/Λ to elliptic curves. This will allow to think of modular forms as functions either on lattices or on elliptic curves.

3.2.1 Meromorphic functions on C/Λ Let C(Λ) be the ﬁeld of meromorphic functions on C/Λ. That is, meromorphic functions f : C C satisfying f (z + λ) = f (z) for all λ Λ. → ∈ Proposition 3.2.1 Let f C(Λ). Then: ∈ 1. ∑z C/Λ Resz f = 0. ∈ 2. ∑z C/Λ ordz f = 0. ∈ 3. ∑z C/Λ zordz f Λ. ∈ ∈ Proof. Consider a fundamental parallelopiped D which misses all zeroes and poles. This can be done because zeroes and poles form a discrete set. Now one can compute the quantities

1 Z 1 Z f (z) 1 Z z f (z) f (z)dz, 0 dz ,and 0 dz. 2πi ∂D 2πi ∂D f (z) 2πi ∂D f (z)

Deﬁnition 3.2.1 The order of a meromorphic function f is the numer ord( f ) of zeroes (which equals the number of poles) of f , when counted with multiplicities.

Note that the ﬁrst statement in the above proposition implies that ord( f ) 2. ≥ 3.2.2 The Weierstrass ℘-function Consider the following function:

1 1 1 0 ℘Λ(z) = 2 + ∑ 2 2 . z w Λ (z w) − w ∈ −

It is immediate to see that ℘Λ is an even function, which converges absoutely and uniformly on compact sets away from Λ. 22 Chapter 3. Complex tori

Lemma 3.2.2 The function ℘Λ is Λ-periodic.

Proof. Note that the derivative of ℘Λ is 1 ℘λ0 (z) = 2 ∑ 3 , − w Λ (z w) ∈ − which is clearly Λ-periodic. Set f (z) =℘Λ(z + w1) ℘Λ(z), where w1 Λ. Then f 0(z) = 0, so − w ∈ f is constant, say f (z) = c. To determine c, set z = and note that since ℘Λ is even, we get − 2 c =℘Λ(w1/2) ℘Λ( w1/2) = 0. − − Therefore f (z) = 0, and thus ℘Λ is Λ-periodic.

The lemma gives that ℘λ (z) belongs to C(Λ). In fact, C(Λ) is generated by ℘Λ and ℘Λ0 , but we are not going to prove this here.

Proposition 3.2.3 The Laurent expansion of ℘Λ(z) at z = 0 is

1 2m ℘Λ(z) = 2 + ∑ (2m + 1)G2m+2(Λ)z , z m 1 ≥ and it has radius of convergence equal to the lattice point closest to the origin.

Proof. See [DS05].

These expansions allow us to ﬁnd algebraic relations between ℘Λ and ℘Λ0 . Since 1 ℘ = + 3G (Λ)z2 + O(z4) Λ z2 4 and 2 ℘ = − + 6G (Λ)z + O(z3), Λ0 z3 4 we deduce: 4 (℘ )2 = + O(z 2) = 4(℘ )3 + O(z 2). Λ0 z6 − Λ − We can work with a couple more terms of the expansions, to get:

2 3 2 (℘0 ) = 4(℘Λ) 60G4(Λ)℘Λ 140G6(Λ) + F(z), F(z) = O(z ). Λ − − Finally, note that F(z) is Λ-periodic so by Liouville’s theorem it must be constant 0.

Proposition 3.2.4 Let g2(Λ) = 60G4(Λ) and g3(Λ) = 140G6(Λ). Then: 1. The point (℘Λ(z),℘Λ0 (z)) lies on the elliptic curve

2 3 EΛ : Y = 4X g2(Λ)X g3(Λ). − − 2 2. EΛ can be written as Y = 4(X e1)(X e2)(X e3), where − − −

ei =℘Λ(wi/2), w3 = w1 + w2.

are the 2-torsion points. 3.2 Tori and elliptic curves 23

e2 e3 = e1 + e2

0 w1 e1

Figure 3.1: The 2-torsion points of EΛ

Proof. It only remains to prove the second statement. Since ℘Λ0 is odd and periodic, we get:

℘0 (wi/2) =℘0 ( wi/2) = ℘0 (wi/2), Λ Λ − − Λ so ℘Λ0 (wi/2) is 0. Since ℘Λ takes the value ei twice and ℘Λ has order 2, it does not take the value ei at any other points outside the 2-torsion.

Bibliography

[DS05] Fred Diamond and Jerry Shurman. A ﬁrst course in modular forms. Volume 228. Graduate Texts in Mathematics. Springer-Verlag, New York, 2005, pages xvi+436. ISBN: 0-387-23229-X (cited on pages 6, 22).

Index

A G automorphy factor...... 7 general linear group ...... 6

B H

Bernoulli numbers ...... 9 holomorphic at ∞ ...... 8

C L lattice...... 19 complex torus ...... 19 level of a congruence subgroup ...... 15 congruence subgroup...... 15 cusp form ...... 9 cusp form of weight k for a congruence subgroup...... 17 M cusps ...... 16 meromorphic at ∞...... 8 meromorphic modular forms ...... 9 modular form ...... 9 E modular form of weight k for a congruence subgroup ...... 17 expansion of f at a cusp ...... 17 multiplication-by-N ...... 20

F O fan width ...... 16 order of a meromorphic function...... 21 28 INDEX

P principal congruence subgroup of level N 15 projective special linear group ...... 7

Q quasi-modular ...... 12

R regular ...... 9

S space of cusp forms of weight k for a congruence subgroup ...... 17 space of modular forms of weight k for a congruence subgroup ...... 17 special linear group ...... 6

U upper half-plane ...... 6

V valence ...... 11 vanishes at ∞ ...... 9

W weakly modular of weight k with respect to Γ 16 weakly-modular...... 7 weight-k slash operator ...... 7