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Home , Modular form

Introduction to Modular Forms

Lectures by Dipendra Prasad

Written by Sagar Shrivastava

School and Workshop on Modular Forms and Black Holes (January 5-14, 2017) National Institute of Science Education and Research

1 Langlands program identifies relations between theory, Repre- sentation theory, Harmonic analysis and . In modern terminology, we use the collective term of Automorphic Forms. Typically, modular forms (automorphic forms) are a certain class of functions on spaces of the form SL2(Z)\SL2(R). (General automorphic forms are functions on spaces of the form G(Z)\G(R) where G(Z) is the of ‘integral’ points and G(R), the group of real points of G.)

Poincar´eUpper Half Plane

Recall the Poincar´eupper half plane H = {z ∈ C| Im(z) > 0}, which is a complex manifold of dimension 1 on which we can talk of holomorphic func- tions. The group SL2(R) acts on H by linear fractional transformations:   def az + b a b gz = ; g = ∈ SL ( ). cz + d c d 2 R

It can be checked that the above defines an action of SL2(R) on H, in other words: g1(g2z) = (g1g2)z for all g1, g2 ∈ SL2(R), z ∈ H. This action is transitive on H, as given z = x + iy ∈ H, consider the : y1/2 xy−1/2 g = x, y ∈ , y > 0 ⇒ g ∈ SL ( ), g i = x + iy. z 0 y−1/2 R z 2 R z

Also, if g ∈ SL2(R) with gi = i, then we can see that g ∈ SO(2), the subgroup of SL2(R) consisting of the matrices:  cos θ sin θ ; − sin θ cos θ ∼ thus we have an of spaces: SL2(R)/SO(2) = H For a point z ∈ H, let Im(z) be its imaginary part. For several calcula- tions below, the relationship between Im(z) and Im(gz) will be very useful. This will also prove that SL2(R) preserves H. Note that: az + b (az + b)(cz¯ + d) Im(gz) = Im = Im = |cz+d|−2Im(adz+bcz¯). cz + d |cz + d|2 But Im(adz + bcz¯) = (ad − bc)Im(z) = Im(z), since det(g) = 1. Hence a b Im(gz) = |cz + d|−2Im(z) for g = ∈ SL ( ). c d 2 R

2 So Im(z) > 0 ⇒ Im(gz) > 0. Thus the group SL2(R) acts on H (by linear fractional transformations). The subgroup of SL2(R) consisting of matrices with integer coefficients is by definition SL2(Z). It is called the “Full ”. Besides Γ = SL2(Z), certain of its subgroups have special significance. Let N be a positive integer, we define    def a b Γ(N) = ∈ SL ( )|a ≡ d ≡ 1(modN), b ≡ c ≡ 0(modN) . c d 2 Z

In other words, Γ(N) consists of 2 × 2 integral matrices of determinant 1 which are congruent modulo N to the identity matrix. Γ(N) is a subgroup of SL2(Z), actually a normal subgroup, because it is the kernel of the homo- morphism from SL2(Z) to SL2(Z/NZ) obtained by reducing entries modulo N. (This also implies that Γ(N) has finite index in SL2(Z)). A subgroup of SL2(Z) is called a “ of level N” if it contains Γ(N). Notice that a congruence subgroup of level N also has level N 0 (if N | N 0), so we choose the N to be the smallest such number. (Any congruence subgroup has finite index in SL2(Z), and hence its will have finite volume.) Similarly we define the following subgroups of SL2(Z) which play very prominent roles in the theory of modular forms:    def a b Γ (N) = ∈ SL ( )|c ≡ 0 (mod N) , 0 c d 2 Z    def a b Γ (N) = ∈ SL ( )|a ≡ d ≡ 1 (mod N), c ≡ 0 (mod N) . 1 c d 2 Z

Modular forms for SL2(Z)

A modular form f : H −→ C of weight k for SL2(Z) is a function on H with the following properties:

1. f : H −→ C is a ( in case of a modular function).

a b 2. f az+b  = (cz + d)kf(z), ∀z ∈ , ∈ SL ( ). cz+d H c d 2 Z 1 1 Note that T = ∈ SL ( ), so f(z + 1) = f(z). Hence these 0 1 2 Z

3 functions will have a Fourier expansion:

∞ X 2πinz f(z) = ane . n=−∞

P∞ 2πinz P∞ n 2πiz 3. an = 0, ∀n < 0 i.e. f(z) = n=0 ane = n=0 anq , q = e .

If further, a0 = 0, then f is called a form. Remark: Condition 3 is equivalent to say that f remains bounded at i∞, i.e. limt→∞ f(it) < ∞, and the cuspidality condition is equivalent to say that f(i∞) = 0. Observe that z 7→ e2πiz = q gives rise to the identification:

∼ ∗ H/Z = D = {q | |q| < 1, q 6= 0} . So if f is a modular form, f considered as a function on D∗ can be ex- tended to a holomorphic function on D. It is a if it extends to D holomorphically and vanishes at q = 0. Since SL2(Z) = hS,T i (as we will see later) where  0 1 1 1 S = ,T = , −1 0 0 1 for the modularity of a function f on H, we “just” need to check that, −1 f(z + 1) = f(z) and f( ) = zkf(z). z Modular forms are an interplay between continuous and discrete as they are continuous functions with discrete symmetries. The set of all modular forms of weight k for SL2(Z) is a finite dimen- sional vector space, and is denoted by Mk(SL2(Z)). Similarly, the set of all cusp forms of weight k for SL2(Z) is also a vector space, and is denoted by Sk(SL2(Z)).

Fundamental Domain (F)

A fundamental domain is a set of equivalence classes on H under the action of SL2(Z), thus with the property: [ H = γF γ∈SL2(Z)

4 with intersections allowed only on the boundary points of γF, i.e.

◦ \ ◦ (γ1F) (γ2F) = φ if γ1 6= γ2, γ1, γ2 ∈ SL2(Z)

◦ where (γ1F) is the interior of γ1F

Theorem. A fundamental domain for the action of SL2(Z) in H is  1 F = z | |z| ≥ 1, |Re(z)| ≤ 2

Proof. We begin by proving that given z ∈ H, there exists γ ∈ SL2(Z) such that γz ∈ F. This will follow from the following claim. Claim : Given z ∈ H, there exists γ ∈ SL2(Z) such that Im(γz) is maxi- mum possible and for such a γ, |γz| ≥ 1. The existence of γ ∈ SL2(Z) such that Im(γz) is maximum possible follows from the fact that

Im(γz) = |cz + d|−2Im(z), and the fact that for a given z ∈ H, |cz + d| tends to infinity as (c, d) (which are pair of coprime integers) tends to infinity, i.e., when |c| + |d| tends to infinity. Note that the set of coprime integers (c, d) is a single orbit for the action of SL2(Z) with the isotropy subgroup of the point (1, 0) being the group of upper triangular integral unipotent matrices. Assume that γ◦ ∈ Γ is such that Im(γ◦z) is the maximum possible, and 1 assume without loss of generality that |Re(γ◦z)| ≤ 2 . If γ◦z 6∈ F ⇒ |γ◦z| < 1. Im(γ◦z) In this case Im(Sγ z) = 2 > Im(γ z), a contradiction, hence we have ◦ |γ◦z| ◦ that γ◦z ∈ F.

5 Corollary. Γ = SL2(Z) = hS,T i.

Proof. The proof above shows that any element of PSL2(Z) = SL2(Z)/ ± 1 is a product of S,T , since for any γ ∈ Γ, z0 an interior point of F, γ · z0 can be brought back to F using the group generated by S,T . (To prove this, we will need to use the easy fact that SL2(Z) has no stabilizer in the interior of F .)

Eisenstien Series

For f a modular form of weight k for SL2(Z), considering the action of −1 0  k 0 −1 ∈ SL2(Z) on f, we get f(z) = (−1) f(z), hence for f to be non- zero, k should be an even positive integer. Let k be an even positive integer greater than 2. For z ∈ H, we define X 1 Gk(z) = . def (mz + n)k 2 (m,n)∈Z (m,n)6=(0,0) It can be seen that this double sum is absolutely convergent (for k ≥ 4), and is uniformly convergent in any compact subset of H, hence Gk(z) is a holomorphic function on H. Clearly, Gk(z) = Gk(z + 1), and the Fourier expansion of Gk has no-non negative terms as the limit of Gk(z) is finite as z → i∞: X −k lim Gk(z) = n = 2ζ(k). z→i∞ n6=0 Finally, we have that X 1 X zk G (−1/z) = = = zkG (z). k (n − m/z)k (nz − m)k k 2 2 (m,n)∈Z (m,n)∈Z (m,n)6=(0,0) (m,n)6=(0,0)

Hence Gk(z) satisfies the criterion for being a modular form, giving us first example of modular forms Gk(z) ∈ Mk(Γ). With some computation, one shows that Gk(z) has the following q-expansion: ∞ ! 2k X G (z) = 2ζ(k) 1 − σ (n)qn , k B k−1 k n=1 th where Bk is the k Bernoulli coefficient obtained as an expansion of ∞ x X xk = B ex − 1 k k! l=0

6 P k−1 and σk−1(n) = d|n d . We define the Ek(z) as the normalized version of Gk(z):

∞ 1 2k X E (z) = G (z) = 1 − σ (n)qn. k 2ζ(k) k B k−1 k n=1

The Eisenstein series Ek(z) have rational q-expansion coefficients. The first few Ek(z) are:

∞ X n E4(z) = 1 + 240 σ3(n)q , n=1 ∞ X n E6(z) = 1 − 504 σ5(n)q , n=1 ∞ X n E8(z) = 1 + 480 σ7(n)q , n=1 ∞ X n E10(z) = 1 − 264 σ9(n)q . n=1

It can be seen that for the Eisenstien series Ek(z), the Fourier coefficients k−1 have the bound |an| = O(n ).

Theorem. The space of modular forms (M∗) is a graded algebra, M∗ = a b hE4,E6i, i.e. Mk is generated by E4 E6, where 4a + 6b = k.

3 1728E4 Definition. j(z) = 3 2 is a modular function of weight 0 E4 −E6 Theorem. The space of modular functions of weight 0 is isomorphic to C(j).

Another way of defining modular forms a b Given g = ∈ SL ( ), z ∈ , define a factor of automorphy j(g, z) c d 2 R H to be j(g, z) = (cz + d). For a modular form f : H → C of weight k, a b f(γz) = j(γ, z)kf(z), ∀γ ∈ SL ( ). c d 2 Z

A factor of automorphy satisfies the following defining condition, called the cocycle condition:

j(g1g2, z) = j(g1, g2z)j(g2, z) ∀g1, g2 ∈ SL2(R), z ∈ H

7 Another way of defining modular forms is to consider functions of the following type: φf : SL2(R) → C, with the property:

ikθ φf (γgm) = φf (g)χk(m), γ ∈ SL2(Z), χk(m) = e , where  cos θ sin θ m = ∈ SO(2). − sin θ cos θ

For a modular form f ∈ Mk(SL2(Z)), define

−k φf (g) = f(gi)j(g, i) , g ∈ SL2(R).

Claim. For γ ∈ SL2(Z), φf (γg) = φf (g). Proof. By the cocycle condition satisfied by j(g, z) we have:

−k φf (γg) = f(γgi)j(γg, i) = f(γ(gi))j(γ, (gi))−kj(g, i)−k, but since f is a modular form of weight k, f(γz) = j(γ, z)kf(z) for all γ ∈ SL2(Z). Therefore:

−k φf (γg) = f(gi)j(g, i) = φf (g), completing the proof of the claim.

Claim. For m ∈ SO(2), φf (gm) = φf (g)χk(m). (Recall that SO(2) is the stabilizer of i.) Proof. Once again, using the cocycle condition satisfied by j(g, z), we have:

−k φf (gm) = f(gmi)j(gm, i) = f(g(mi))j(g, (mi))−kj(m, i)−k = f(g, i)j(g, i)−kj(m, i)−k (since mi = i) −k = φf (g)(−i sin θ + cos θ) −iθ −k = φf (g)(e ) ikθ = φf (g)e

= φf (g)χk(m).

8 Shimura’s Slash Notation We introduce what’s called Shimura’s slash notation which is very useful for the theory of modular forms. + Let GL2 (Q) denote the subgroup of GL2(Q) consisting of matrices with positive determinant. Define   k/2 −k a b + f(z)|[γ]k = (det(γ)) (cz + d) f(γz), ∀γ = ∈ GL2 (Q). def c d

(The determinant factor is added so that diagonal matrices act trivially.) It can be seen that: f(z)|[γ1γ2]k = (f(z)|[γ1]k)|[γ2]k, and that f is a modular form of weight k for SL2(Z) if and only if

f(z)|[γ]k = f, ∀γ ∈ SL2(Z). (besides holomorphy conditions at infinity).

Modular forms for congruence subgroups

Suppose that Γ ⊃ Γ(N) is a congruence subgroup of SL2(Z) of level N. We say that f ∈ Mk(Γ) if f is a holomorphic function on H and it satisfies:

1. f(z)|[γ]k = f, ∀γ ∈ Γ

n P N 2. ∀γ ∈ SL2(Z), f|[γ]k(q) = n≥0 anq . 1 N The second condition comes into picture as ∈ Γ(N) ⊂ Γ. Therefore, 0 1 f(z + N) = f(z), hence by condition (1), f|[γ]k(z + N) = f|[γ]k(z) for all γ ∈ SL2(Z).

Hecke Operators

+ Let γ ∈ GL2 (Q), Γ ⊂ SL2(Z), a congruence subgroup. Then it can be seen Fl that the double coset ΓγΓ is a finite union of cosets: ΓγΓ = i=1 Γγi. We define certain operators on the space of modular forms, called the Hecke operators, as: l def X Tγ(f) = f|[ΓγΓ]k = f|[γi]k. i=1

9 The most conventional Hecke operators are n 0 T = SL ( ) SL ( ), n 2 Z 0 1 2 Z acting on the space of modular forms of weight k; these are a commuting set of operators. As a specific example, note that:

p−1 p 0 G 1 i G p 0 T = SL ( ) SL ( ) = SL ( ) SL ( ) . p 2 Z 0 1 2 Z 2 Z 0 p 2 Z 0 1 i=0 It is known that TmTn = Tmn, if (m, n) = 1. The space of cusp forms Sk(SL2(Z)), carries a hermitian form, called the Petersson inner product:

Z dxdy Z dxdy hf , f i = ykf (z)f¯ (z) = ykf (z)f¯ (z) . 1 2 1 2 y2 1 2 y2 SL2(Z)\H F with respect to which the Hecke operators are hermitian:

hTmf1, f2i = hf1,Tmf2i. Thus we have a commuting set of hermitian operators on a finite dimensional space of modular forms, which can then be simultaneously diagonalized for all the Hecke operators Tm, i.e., there exists a basis of modular forms fn such that: Tm(fn) = λm(n)fn, ∀m.

It can be seen that a1(fn) 6= 0, which we can then normalize to be 1, in which case, λm(n) = am(fn). As the Hecke operators are hermitian, their eigenvalues are real, thus the Fourier coefficients of Hecke eigenforms are real (assuming a1 = 1). In fact, the space of cusp forms Sk(SL2(Z)) is a vector space defined over Q (since a b the space of all modular forms is generated by E4 E6, there is a natural Q structure on the space of modular forms through the basis consisting of a b E4 E6, or through modular forms with q-expansion in Q.) This Q-structure on the space of modular forms, or on cusp forms, is invariant under the Hecke operators. Therefore the eigenvalues of Hecke operators are totally real algebraic .

10 P n Theorem. (Hecke) If f = anq is a cusp form of weight k, then |an| = k/2 k/2 O(n ), i.e. there exists c > 0 such that |an| ≤ cn for all n > 0. k/2 Proof. Observe that ψf (z) = f(z)y is a modular function, i.e. SL2(Z) invariant. Since f is a cusp form, so it vanishes at i∞ exponentially, therefore:

|ψf (z)| ≤ M for some M > 0, for all z ∈ H. (To elaborate, ψf being a modular function, one needs to prove the above only inside the fundamental domain F. The part of the fundamental domain below any horizontal line is compact, so ψf in this part of the fundamental domain is bounded; it needs to be checked that ψf is bounded on the part of the fundamental domain above any horizontal line 2πiz k/2 too. Under the mapping z → e , the function ψf (z) = f(z)y becomes −2πy k/2 the function Ff (e )y where Ff is the associated function on D and y → ∞ which by the cuspidality of f tends to zero, completing the proof of boundedness of ψf on the part of the fundamental domain above any horizontal line.) For any point y > 0, and any integer n > 0, we have: Z 1 Z 1 −2πny 1 −2πinx 1 1 −k/2 |an|e = f(x + iy)e dx ≤ |f(x+iy)|dx ≤ y M. 2π 0 2π 0 2π Therefore, M |a | ≤ y−k/2e2πny, n 2π 1 for any n > 0 and any y > 0. Taking y = n , gives M |a | ≤ nk/2e2π = cnk/2, n 2π which is the desired conclusion. Corollary. Cusp forms give rise to the “Error terms” in the Fourier ex- pansion of noncuspidal modular forms, i.e., given any modular form g ∈ Mk(SL2(Z)), g = λEk +f, where f is a cusp form. Hence an(g) = λan(Ek)+ k−1 k/2 an(f) with |an(Ek)| = O(n ) whereas |an(f)| = O(n ). Remark: The relatively simple proof of Hecke’s theorem should be con- trasted with the much deeper theorem of Deligne on Ramanujan bound — proved as a consequence of his proof of the — which gives a (k−1+)/2 bound of |an(f)| = O(n ) for any  > 0, for f any holomorphic cusp form of integral weight k. The Hecke bound is valid for half integral weight modular forms, and although Ramanujan bound is expected for half integral weight modular forms too, it is far from being a theorem.

11 L-functions P∞ n For f ∈ Sk(SL2(Z)), f(z) = n=1 anq , define the following function of the complex variable s: ∞ X an L(f, s) = , ns n=1 called the L-function associated to the cusp form. By Hecke’s estimate on Fourier coefficients, these functions L(f, s) are analytic functions of s in the half-region of the complex plane Re(s) > k/2+1. Hecke proved that they have analytic continuation to the whole complex plane as an entire function, and satisfy a functional equation. If f is an eigenform for all the Hecke operators Tp, and a1 = 1, then it is known that the coefficients are multiplicative, i.e., amn = anam, whenever (m, n) = 1. In fact, we have the following formula Y 1 L(f, s) = . ap pk−1 p (1 − ps + p2s ) For a cusp form f of weight k, we define

def −s Λf (s) = (2π) Γ(s)L(f, s)

Theorem. Λf (s) has analytic continuation on the whole complex plane, and it satisfies the following functional equation:

k/2 Λf (s) = (−1) Λf (k − s) Remark : In our lectures, we have not emphasized L-functions so much, but for many people, these give the most important aspect of modular forms. There are many L-functions associated to modular forms, for example the so- called symmetric power L-functions, whose analytic continuation and func- tional equation are important open problems. It is a theorem of Hecke that the theorem above has a converse: any L-function with analytic continuation, and the above functional equation, and some estimates on growth properties arises from modular forms for SL2(Z).

Elliptic Curves

An over a field K is a projective non-singular curve of one over K with a specific base point. If the Char(K) 6= 2, 3, then we have the short Weierstrass form of the elliptic curve as y2 = x3 + Ax + B

12 which has discriminant = −16(4A3 +27B2). The above curve is non singular if the discriminant is nonzero. Definition. A lattice in the complex plane C is a discrete subgroup of the form Λ = Zω1 + Zω2, where ω1 and ω2 are linearly independent over real numbers. Two lattices Λ and Λ0 are said to be equivalent if there exists λ ∈ C − {0}, with λΛ = Λ0. A complex T is a quotient C/Λ of the complex plane with the lattice Λ It can be shown that any complex analytic isomorphism between two tori actually corresponds to an equivalence between the associated lattices. Any complex torus T = C/Λ can be given an algebraic structure using the Weierstrass ℘ and ℘0 functions:

02 3 ℘ = 4℘ − g2℘ − g3, where X 1 X 1 g = 60 , g = 140 . 2 λ4 3 λ6 λ∈Λ\0 λ∈Λ\0 This allows one to identify the set of complex tori and elliptic curves over C. The following lemma brings out the first relationship between elliptic curves and ‘modular varieties’ which are the ‘homes’ for modular forms. There is a similar relationship in much greater generality for abelian varieties with level structures and a given endomorphism rings, and points on certain ‘Shimura varieties’.

Lemma. Points in SL2(Z)\H are in bijective correspondence with the iso- morphism classes of Elliptic curves over C. The correspondence is given by τ ∈ H → Eτ = C/(Z + τZ). Given an Elliptic curve E over Q, we can reduce it mod p. If the reduced curve E is non-singular, then we call p a prime of good reduction for E, otherwise we call p a prime of bad reduction. For a prime p, if E has a good reduction mod p, define an integer

ap(E) = p + 1 − #E(Fp).

One can define ap(E) even for primes of bad reduction, but we shall not do so here. The number of points of E over Fp as p varies, can be packaged in what’s called the Hasse-Weil L-function of E, as Y Y 1 L(E, s) = Lp(E, s) = . ap(E) p p p (1 − ps + p2s )

13 The following conjecture (now a theorem of Wiles) brings out the second relationship — a much deeper one than the first one above — between elliptic curves and modular forms.

Conjecture (Shimura-Taniyama-Weil). L(E, s) is the L-function of a mod- ular form on Γ0(N) of weight 2, and as a consequence, there exists a bijective correspondence between cusp forms on Γ0(N) (for some integer N) of weight 2 with integral Fourier coefficients, and Elliptic curves (up to isogeny) over Q. As is well-known, the Shimura-Taniyama-Weil conjecture was proved by A. Wiles and others in the mid 90’s, and as a consequence Fermat’s Last Theorem was proved.

References

[1] Jean-Pierre Serre, A course in arithmetic, Springer Science & Business Media, Volume 7, 2012.

[2] Neal Koblitz, Introduction to elliptic curves and modular forms, Springer Science & Business Media, Volume 97, 2012.

[3] Daniel Bump, Automorphic forms and representations, Cambridge uni- versity press, Volume 55, 1998.

[4] Joseph H Silverman The arithmetic of elliptic curves, Springer Science & Business Media, Volume 106, 2009.

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