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MODULAR FORMS and the FOUR SQUARES THEOREM Contents 1 MODULAR FORMS AND THE FOUR SQUARES THEOREM AARON LANDESMAN Contents 1. Introduction 1 2. Definition of modular Forms 1 2.1. Preliminaries 2 2.2. A First Definition of Modular Forms 3 2.3. The action of SL2(Z) on H 4 2.4. Three Other Ways to Think about Modular Forms 5 3. The Space of modular Forms 7 3.1. Eisenstein Series 7 3.2. Notation for Spaces of Modular Forms 9 3.3. A Bound on the Dimension of Modular Forms of a Fixed Weight 10 3.4. A Complete Characterization of Modular Forms 12 4. Modular Forms of Higher Levels 13 4.1. Congruence Subgroups 13 4.2. A More Refined Definition of Modular Forms 14 5. The space M2(Γ0(4)) 15 5.1. Computing the Fourier Series of G2(τ) 15 5.2. The almost-invariance of G2 17 5.3. An Almost Invariant SL2(Z) function from G2: 21 5.4. A Bounding Theorem 21 5.5. The elements of M2(Γ0(4)) 23 6. Theta Functions 25 7. The Sum of Four Squares Theorem 27 References 28 1. Introduction The main goal of this paper is to introduce modular forms and to obtain an explicit formula for the number of ways to write a positive integer as a sum of four squares. Along the way, we shall also explicitly describe all modular forms with respect to the principal congruence subgroup SL2(Z): 2. Definition of modular Forms Modular forms are naturally viewed in several different contexts: as holomorphic functions satisfying transformation equations, as sections of line bundles on Rie- mann surfaces, and as functions on lattices. In the first section, we shall describe 1 2 AARON LANDESMAN these different perspectives for thinking about modular forms and explain how they relate. But first we must build up the necessary background. 2.1. Preliminaries. Definition 2.1.1. Let F be a ring. The nth general linear group over F; GLn(F ) is the group of invertible n × n matrices with coefficients in F; viewed as a group via multiplication and inversion of matrices. Definition 2.1.2. Again, with F a ring, the nth special linear group over F; no- tated SLn(F ) is the group of n×n matrices with coefficients in F with determinant 1. Definition 2.1.3. Let F be a ring and let Z(n; F ) be the subgroup of diagonal matrices of GLn(F ): The Projective linear group of dimension n over F , P GLn(F ) = GLn(F )=Z(n; F ): Remark 2.1.4. Let In denote the n × n identity matrix. Note that in the case that F = R or C; we have Z(n; F ) = F · In; is formed by diagonal matrices. In the further case that n = 2; which we shall mostly be dealing with in this paper, ∼ ∼ we have that P GL2(R) = GL2(R)=Z(2; R); P GL2(C) = GL2(C)=Z(2; C); since the matrices I2; −I2 are the only two elements of Z(2;F ) with determinant 1. Definition 2.1.5. Let C^ denote the Riemann sphere, also known as the one point compactification of C: The group of fractional linear transformations, notated ^ ^ az+b F LT; are rational functions f : C ! C which are of the form f(z) = cz+d : They form a group under composition. This is a group because it is the image of the natural surjection GL2(C) ! F LT: Note that the point of Cb not in C is labeled 1 ax+b and any FLT of the form 0 is labeled 1: Lemma 2.1.6. The group of fractional linear transformations is naturally isomor- phic to P GL2(C) Proof. Observe that we have a natural map φ : GL2(C) ! F LT; mapping the a b matrix to the function f(z) = az+b : Clearly Z(2; ) 2 ker φ. Hence, we c d cz+c C ∼ have an induced map φ : P GL2(C) = GL2(C)=Z(2; C) ! F LT: This map is clearly surjective by choosing the same (a; b; c; d) for both groups. It only remains to check only the identity acts trivially, which holds because the identity is the only element mapping to the fractional linear transformation with a zero at z = 0 and a pole at z = 1: Remark 2.1.7. As described above, we obtain that P GL2(C) acts naturally on the Riemann sphere by fractional linear transformations. Lemma 2.1.8. Let Aut(C^) denote the set of invertible meromorphic maps f : C^ ! C^: We have Aut(C^) =∼ F LT: Proof. Clearly all fractional linear transformations are automorphisms, with their inverse given by their inverse in the group of fractional linear transformations, which ^ we saw above was P GL2(C): Conversely, suppose T 2 Aut(C): Then, T must have precisely one pole, and precisely 1 zero. Suppose the pole is p and the zero is q: z−p Note that we must have p; q distinct. Then, the meromorphic function S = T · z−q ; has no zeros and no poles. Note this is written as a product of meromorphic MODULAR FORMS AND THE FOUR SQUARES THEOREM 3 functions, with removable singularities filled in. That is, in S(z); we cancel off common factors from the numerator and denominator. Then, S is a function on the Riemann sphere with no poles or zeros. Hence, by Liouville's theorem, it must cz−cp be constant, and so S = c; which implies T = z−q ; which can be rewritten in the form of a linear transformation by multiplying the numerator and denominator by the same constant so that the resulting determinant will be 1. Notation 2.1.1. Let Im(z) denote the imaginary part of z: Definition 2.1.9. The upper half plane is the subset H = fz 2 CjIm(z) > 0g: a b Lemma 2.1.10. Let g = 2 SL ( ): Then, for z 2 BC; Im(gz) = Im(z) : c d 2 R jcz+dj2 Proof. Observe that az + b g(z) = cz + d (az + b)(d + cz) = jcz + dj2 bd + acjzj2 + Re(z)(ad + bc) + i(ad − bc)Im(z) = jcz + dj2 bd + acjzj2 + Re(z)(ad + bc) + iIm(z) = : jcz + dj2 Im(z) Hence, Im(gz) = jcz+dj2 : Corollary 2.1.11. Let Aut(H) denote the group of meromorphic functions send- ing H to H with a meromorphic inverse on H: The restriction of fractional linear transformations to H define a subgroup of Aut(H). Proof. Clearly, and FLT maps R [ f1g ! R [ f1g; and by the previous lemma, it maps i to something with positive imaginary part. Therefore, it must map all of H ! H; and since it is an automorphism of C^ it restricts to an automorphism of H: 2.2. A First Definition of Modular Forms. The main idea of modular forms is that they have a certain invariance under composition with fractional linear transformations. Definition 2.2.1. Let Γ = SL2(Z) act on the upper half plane via fractional linear az+b transformations. Suppose γ 2 Γ; γ(z) = cz+c : A weakly modular function of weight k is a complex meromorphic function f : H ! H; satisfying f(γ(z)) = (cz + d)kf(z): Proposition 2.2.2. The only weakly modular function of odd weight k is 0. Proof. Let f be such a weakly modular function. Then, applying the matrix −I; we see f(z) = (−1)kf(z) = −f(z) if k is odd. This implies f(z) = 0 identically on H: We'd like to say that a modular function is simply a weakly modular function that is also meromorphic at infinity, where infinity is thought of as lying very far in the imaginary direction. We can also translate H to the unit disk by a linear 4 AARON LANDESMAN fractional transformation, which translates R [ f1g to the boundary of the unit disk. Now we develop a tiny bit a Fourier analysis to make this notion precise. Notation 2.2.1. We shall use D to denote the complex unit disk D = fz 2 Cjjzj < 1g, and D∗ to denote the punctured complex unit disk, D∗ = D − f0g: Lemma 2.2.3. Any function f : H ! C satisfying f(z) = f(z + 1) can be written as f(z) = g(e2πiz) for g : ∆∗ ! C holomorphic. Proof. Since the upper half plane is simply connected, we have a well defined log- arithm log : H ! D∗ ⊂ C: Locally, log has an inverse. Define the coordinate 1 2πiq q(z) = 2πi log z. Then, at least locally, we can write z = e : Therefore, locally we can write g(q) = f(e2πiq) = f(z): This is well defined on the strip 0 ≤ Re(z) < 1: However, since f(z) = f(z + 1); we can holomorphically extend this function by defining f(z) = f(z − bzc): So, g(q) is holomorphic because it is locally holomor- phic. 0 −1 1 1 Notation 2.2.2. Let S = ;T = as elements of SL ( ). 1 0 0 1 2 Z Remark 2.2.4. Note that any weakly modular function f satisfies f(z) = f(z + 1) because f(z +1) = f(T z) = 1kf(z) = f(z) by definition of weakly modular. There- fore, we can write f(z) = f(e2πiq) = g(q): D∗ ! C: Because g(q) is holomorphic, on D∗ and meromorphic at 0; we can expand it as a Laurent series around 0; P1 n g(q) = i=−∞ anq : Definition 2.2.5. A function f : H ! C satisfying f(z) = f(z +1) is meromorphic P1 n at infinity if we write f(z) = g(q) = i=−∞ anq as above, we have an = 0 for all n ≤ N; for some N 2 BZ: Equivalently, we can say that g(q) has a pole or a removable singularity at 0.
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