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 Number theory, Repre- sentation theory, Harmonic analysis and Algebraic geometry. 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)nSL2(R). (General automorphic forms are functions on spaces of the form G(Z)nG(R) where G(Z) is the group of `integral' points and G(R), the group of real points of G.) Poincar´eUpper Half Plane Recall the Poincar´eupper half plane H = fz 2 Cj Im(z) > 0g, 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 = 2 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 2 SL2(R); z 2 H: This action is transitive on H, as given z = x + iy 2 H, consider the matrix: y1=2 xy−1=2 g = x; y 2 ; y > 0 ) g 2 SL ( ); g i = x + iy: z 0 y−1=2 R z 2 R z Also, if g 2 SL2(R) with gi = i, then we can see that g 2 SO(2), the subgroup of SL2(R) consisting of the matrices: cos θ sin θ ; − sin θ cos θ ∼ thus we have an isomorphism of spaces: SL2(R)=SO(2) = H For a point z 2 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 = jcz+dj−2Im(adz+bcz¯): cz + d jcz + dj2 But Im(adz + bcz¯) = (ad − bc)Im(z) = Im(z), since det(g) = 1. Hence a b Im(gz) = jcz + dj−2Im(z) for g = 2 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 modular group". Besides Γ = SL2(Z), certain of its subgroups have special significance. Let N be a positive integer, we define def a b Γ(N) = 2 SL ( )ja ≡ 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 \congruence subgroup of level N" if it contains Γ(N). Notice that a congruence subgroup of level N also has level N 0 (if N j 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 fundamental domain 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) = 2 SL ( )jc ≡ 0 (mod N) ; 0 c d 2 Z def a b Γ (N) = 2 SL ( )ja ≡ 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 holomorphic function (meromorphic function in case of a modular function). a b 2. f az+b = (cz + d)kf(z); 8z 2 ; 2 SL ( ): cz+d H c d 2 Z 1 1 Note that T = 2 SL ( ), so f(z + 1) = f(z). Hence these 0 1 2 Z 3 functions will have a Fourier expansion: 1 X 2πinz f(z) = ane : n=−∞ P1 2πinz P1 n 2πiz 3. an = 0; 8n < 0 i.e. f(z) = n=0 ane = n=0 anq ; q = e . If further, a0 = 0, then f is called a cusp form. Remark: Condition 3 is equivalent to say that f remains bounded at i1, i.e. limt!1 f(it) < 1, and the cuspidality condition is equivalent to say that f(i1) = 0. Observe that z 7! e2πiz = q gives rise to the identification: ∼ ∗ H=Z = D = fq j jqj < 1; q 6= 0g : 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 cusp form 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 γ2SL2(Z) 4 with intersections allowed only on the boundary points of γF, i.e. ◦ \ ◦ (γ1F) (γ2F) = φ if γ1 6= γ2; γ1; γ2 2 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 j jzj ≥ 1; jRe(z)j ≤ 2 Proof. We begin by proving that given z 2 H; there exists γ 2 SL2(Z) such that γz 2 F. This will follow from the following claim. Claim : Given z 2 H; there exists γ 2 SL2(Z) such that Im(γz) is maxi- mum possible and for such a γ, jγzj ≥ 1. The existence of γ 2 SL2(Z) such that Im(γz) is maximum possible follows from the fact that Im(γz) = jcz + dj−2Im(z); and the fact that for a given z 2 H, jcz + dj tends to infinity as (c; d) (which are pair of coprime integers) tends to infinity, i.e., when jcj + jdj 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◦ 2 Γ is such that Im(γ◦z) is the maximum possible, and 1 assume without loss of generality that jRe(γ◦z)j ≤ 2 . If γ◦z 62 F ) jγ◦zj < 1. Im(γ◦z) In this case Im(Sγ z) = 2 > Im(γ z); a contradiction, hence we have ◦ jγ◦zj ◦ that γ◦z 2 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 2 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 2 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 2 H, we define X 1 Gk(z) = : def (mz + n)k 2 (m;n)2Z (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 ! i1: X −k lim Gk(z) = n = 2ζ(k): z!i1 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)2Z (m;n)2Z (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) 2 Mk(Γ): With some computation, one shows that Gk(z) has the following q-expansion: 1 ! 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 1 x X xk = B ex − 1 k k! l=0 6 P k−1 and σk−1(n) = djn d .