218 A Brief Introduction to Modular Forms B. Ramakrishnan 1. Introduction In this article, a brief introduction to the subject of modular forms is being given. For most of the statements I have not given the proofs. There are several good books written on this subject and I have listed some of them in the references. The interested reader can look at the books for the details of proofs. I have also given some articles related to the theory of newforms (which has been presented at the end). I hope that the reader will have some feeling for the subject from these notes. I would like to thank S. D. Adhikari and S. A. Katre for a careful reading of the manuscript. 2. Modular Forms over SL2(Z) Let k be a rational integer and let denote the Poincar´eupper half- plane, consisting of complex numbers zHwith positive imaginary part. Let M (R) denote the set of all 2 2 matrices whose entries lie in the ring R. 2 × The modular group, denoted by SL2(Z) is the group defined as follows. SL (Z)= a b M (Z) ad bc = 1 . 2 c d ∈ 2 − n o It is a discrete subgroup of GL+(R)= a b M (R) ad bc > 0 . 2 c d ∈ 2 − n o GL+(R) acts on as follows. For γ = a b GL+(R) and z , 2 H c d ∈ 2 ∈H az + b γ z = . · cz + d This is called the M¨obius transformation. It is an easy exercise to check that Im(z) Im(γz)= . cz + d 2 | | + So, if z and γ GL2 (R), then γz . i.e., is preserved by ∈ H ∈ ∈ H H + the M¨obius transformation. We also extend the action of γ GL2 (R) to Q d/c by the same rule. ∈ \Further, {− } for γ = a b GL+(R) we define c d ∈ 2 a d γ = and γ (− )= , ·∞ c · c ∞ 219 220 So, we have the action of GL+(R) on Q. 2 H∪{∞}∪ We first give the formal definition of a modular forms below and the explanations shall follow the definition. Definition 2.1. A function f : C is said to be a modular function H −→ (resp. form) of weight k for the full modular group SL2(Z), if it satisfies the following: (i) f is a meromorphic (resp. analytic) function. (ii) f az+b = (cz + d)kf(z), for all a b SL (Z). cz+d c d ∈ 2 (iii) f(z) has the following Fourier expansion. ∞ n 2πiz f(z)= a(n)q (q = e ),m0 > 0 n=−m X 0 ∞ resp. f(z)= a(n)qn . ! nX=0 A modular form is said to be a cusp form if further a(0) = 0 in (iii) above. Notation. The set of all modular forms (resp. cusp forms) of weight k for the group SL2(Z) is denoted by Mk (resp. Sk). The condition (ii) above can be written as: −k az + b a b (cz + d) f =: f γ(z)= f(z) for all γ = SL2(Z). cz + d k c d ∈ In terms of this new stroke notation, we want the following. f γ1 γ2(z)= f γ1γ2(z). k k k a b Z In other words, defining j (γ, z ) = cz + d for γ = c d SL2( ), we want the equality ∈ j(γ γ , z)= j(γ , γ z) j(γ , z), 1 2 1 2 · 2 which can easily be verified. The factor j(γ, z) is called the automorphy fac- tor. Let us now analyze the condition (iii). Consider the change of variable z e2πiz =: q. 7−→ This takes the Poincar´eupper half-plane to the punctured open disk centred at the origin. We agree to take the point at to the origin under this map. The real line maps onto the boundary of∞ the disk. Since we have the condition that f is meromorphic on , under this transformation, f has a Laurent expansion around the origin,H say ∞ f(z)= b(n)qn. −∞ X INTRODUCTIONTOMODULARFORMS 221 We say that f is meromorphic at (resp. holomorphic at ) if b(n) = 0 for n 0 (resp. for n < 0). Here∞n 0 means that only∞ finitely many coefficients≪ b(n) for n< 0 can be non-zero.≪ So the condition (iii) is equivalent to saying that f is meromorphic (resp. holomorphic) at . ∞ Cusps. The points Q are called the cusps. Let s Q. Then, s can ∪ {∞} a ∈ be written in the reduced form s = c , where gcd(a, c) = 1. Now complete a b a, c to get the matrix γ = c d SL2(Z). Then it is clear that s = γ . But one has a natural equivalence∈ (modulo the group SL (Z)) among·∞ the 2 cusps, namely, the cusps s1 and s2 are said to be SL2(Z) equivalent if there exists a matrix γ SL (Z) satisfying s = γ s . From the above remarks. ∈ 2 1 · 2 it is clear that all rational numbers are SL2(Z) equivalent to . This is the reason why we have only one Fourier expansion for f. ∞ Fundamental domain. The action of SL2(Z) on divides into equiv- alence classes, called orbits. Selecting one point fromH each orbitH one gets a fundamental set for SL2(Z). Modifying the concept slightly for having nice topological properties, one obtains what are called fundamental domains. A fundamental domain for SL2(Z) is given as follows. 1 1 = z z 1, − Re(z) . (1) F ∈H | |≥ 2 ≤ ≤ 2 The equivalence is defined as usual: two points z1 and z2 are said to be SL2(Z)- equivalent, if there exists a matrix γ belonging to SL2(Z) which takes one to the other. The fact that is a fundamental domain for SL2(Z) is equivalent to the following: F (i) For any z , there exists a unique z1 belonging to such that z = γz , for∈ some H γ SL (Z). F 1 ∈ 2 (ii) No two interior points of are SL2(Z) equivalent. (If we take the fundamental domain suitably,F one can omit the condition “in- terior”.) Let us briefly indicate the proof of the fact that is a fundamental F domain for SL2(Z). The modular group SL2(Z) contains the following two special matrices. 1 1 T = 0 1 ; T z = z + 1 (translation.) S = 0 −1 ; Sz = 1/z (inversion.) 1 0 − For a given point z , the idea is to apply the translations T m (m Z to ∈H ∈ get a point, say z1, inside the strip 1/2 Re(z1) 1/2. If this point z1 is inside , then we are through. Otherwise,− ≤ apply the≤ inversion map S to z F 1 to get another point z2, which will lie outside the unit circle. The process is continued till we get a point inside our region . (It is a fact that this F 222 B.RAMAKRISHNAN process ends after a finite number of steps!) This proof implies an interesting fact that the modular group SL2(Z) is generated by the two elements T and S. Weight formula. Let f be a complex valued meromorphic function defined on . For a point P , let v (f) denote the order of zero of f at P and H ∈H P v∞(f) denote the least integer n for which a(n) is non-zero in the expansion (iii) of Definition 2.1. Then we have the following theorem. Theorem 2.1. Let f be a non-zero modular function of weight k for the group SL2(Z). Then 1 1 k v∞(f)+ vρ(f)+ vi(f)+ vP (f)= , (2) 3 2 Z 12 P ∈H\SL2( ) PX6=i,ρ where ρ = 1/2+ i√3/2. − The above theorem is proved by integrating the function f ′/f along a specific contour in and using the residue theorem. We omit the details here. This theoremH has many applications as we shall see in the sequel. Remark 2.1. (1) There is no modular form of weight k< 0. (This is clear from the formula (2) as the left hand side is always non- negative.) (2) There is no non-zero modular form of odd weight. This will follow −1 0 directly using Definition 2.1. Take the matrix γ = 0 −1 . By the condition (ii) of Definition 2.1, we must have, ( 1)kf(z) = − f γ(z)= f(z). If k is odd, it follows that f(z) = 0. k C (3) If f is a modular form of weight k = 0, then f . Assume that f = 0 is a modular form of weight 0. Then by∈ weight formula, f 6 does not vanish on . Put c = v∞(f). Then, g = f c is a modular form of weightH ∪ {∞} 0 and it vanishes at (by definition),− This implies that g = 0. Therefore, f = c C. ∞ (4) Let k = 2. Then all the terms on the left∈ hand side of (2) are non-negative. So, we must have f = 0. From the above remark, it follows that the first non-trivial example of a modular form occurs only when k 4. We shall present now an example of a modular form, which plays an important≥ role in the theory of modular forms. Eisenstein series. For k 4, and z , put ≥ ∈H −k Gk(z)= (mz + n) (3) m,n∈Z (m,nX)6=(0,0) INTRODUCTIONTOMODULARFORMS 223 We will show that Gk(z) Mk. Since k 4, the series on∈ the right hand side of (3) converges absolutely and ≥ uniformly on any compact subset of .