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Modular Functions and Modular Forms Modular Functions and Modular Forms J. S. Milne Abstract. These are the notes for Math 678, University of Michigan, Fall 1990, exactly as they were handed out during the course except for some minor revisions andcorrections. Please sendcomments andcorrections to me at [email protected]. Contents Introduction 1 Riemann surfaces 1 The general problem 1 Riemann surfaces that are quotients of D.1 Modular functions. 2 Modular forms. 3 Plane affine algebraic curves 3 Plane projective curves. 4 Arithmetic of Modular Curves. 5 Elliptic curves. 5 Elliptic functions. 5 Elliptic curves and modular curves. 6 Books 7 1. Preliminaries 8 Continuous group actions. 8 Riemann surfaces: classical approach 10 Riemann surfaces as ringed spaces 11 Differential forms. 12 Analysis on compact Riemann surfaces. 13 Riemann-Roch Theorem. 15 The genus of X.17 Riemann surfaces as algebraic curves. 18 2. Elliptic Modular Curves as Riemann Surfaces 19 c 1997 J.S. Milne. You may make one copy of these notes for your own personal use. i ii J. S. MILNE The upper-half plane as a quotient of SL2(R)19 Quotients of H 21 Discrete subgroups of SL2(R)22 Classification of linear fractional transformations 24 Fundamental domains 26 Fundamental domains for congruence subgroups 29 Defining complex structures on quotients 30 The complex structure on Γ(1)\H∗ 30 The complex structure on Γ\H∗ 31 The genus of X(Γ) 32 3. Elliptic functions 35 Lattices and bases 35 Quotients of C by lattices 35 Doubly periodic functions 35 Endomorphisms of C/Λ36 The Weierstrass ℘-function 37 The addition formula 39 Eisenstein series 39 The field of doubly periodic functions 40 Elliptic curves 40 The elliptic curve E(Λ) 40 4. Modular Functions and Modular Forms 42 Modular functions 42 Modular forms 43 Modular forms as k-fold differentials 44 The dimension of the space of modular forms 45 Zeros of modular forms 47 Modular forms for Γ(1) 48 The Fourier coefficients of the Eisenstein series for Γ(1) 49 The expansion of ∆ and j 51 The size of the coefficients of a cusp form 52 Modular forms as sections of line bundles 52 Poincar´eseries 54 The geometry of H 56 Petersson inner product 57 Completeness of the Poincar´eseries 58 Eisenstein series for Γ(N)58 5. Hecke Operators 62 Introduction 62 MODULAR FUNCTIONS AND MODULAR FORMS iii Abstract Hecke operators 65 Lemmas on 2 × 2 matrices 67 Hecke operators for Γ(1) 68 The Z-structure on the space of modular forms for Γ(1) 72 Geometric interpretation of Hecke operators 75 The Hecke algebra 76 6. The Modular Equation for Γ0(N)81 7. The Canonical Model of X0(N)overQ 86 Review of some algebraic geometry 86 Curves and Riemann surfaces 88 The curve X0(N)overQ 90 8. Modular Curves as Moduli Varieties 92 The general notion of a moduli variety 92 The moduli variety for elliptic curves 93 The curve Y0(N)Q as a moduli variety 94 The curve Y (N) as a moduli variety 95 9. Modular Forms, Dirichlet Series, and Functional Equations 96 The Mellin transform 96 Weil’s theorem 98 10. Correspondences on Curves; the Theorem of Eichler-Shimura 100 The ring of correspondences of a curve 100 The Hecke correspondence 101 The Frobenius map 101 Brief review of the points of order p on elliptic curves 102 The Eichler-Shimura theorem 102 11. Curves and their Zeta Functions 104 Two elementary results 104 The zeta function of a curve over a finite field 105 The zeta function of a curve over Q 106 Review of elliptic curves 107 The zeta function of X0(N): case of genus 1 108 Review of the theory of curves 109 The zeta function of X0(N): general case 111 The Conjecture of Taniyama and Weil 111 Notes 113 Fermat’s last theorem 113 Application to the conjecture of Birch and Swinnerton-Dyer 114 12. Complex Multiplication for Elliptic Curves 115 Abelian extensions of Q 115 0J.S.MILNE Orders in K 116 Elliptic curves over C 117 Algebraicity of j 117 The integrality of j 118 Statement of the main theorem (first form) 120 The theory of a-isogenies 120 Reduction of elliptic curves 121 The Frobenius map 122 Proofofthemaintheorem 122 The main theorem for orders 123 Points of order m 124 Adelic version of the main theorem 124 Index 125 MODULAR FUNCTIONS AND MODULAR FORMS 1 Introduction It is easy to define modular functions and forms, but less easy to say why they are important, especially to number theorists. Thus I will begin with a rather long overview of the subject. Riemann surfaces. Let X be a connected Hausdorff topological space. A coor- dinate neighbourhood of P ∈ X is a pair (U, z)withU an open neighbourhood of P and z a homeomorphism of U onto an open subset of the complex plane. A complex structure on X is a compatible family of coordinate neighbourhoods that cover X.A Riemann surface is a topological space together with a complex structure. For example, any open subset X of C is a Riemann surface, and the unit sphere can be given a complex structure with two coordinate neighbourhoods, namely the complements of the north and south poles mapped onto the complex plane in the standard way. With this complex structure it is called the Riemann sphere.Weshall see that a torus can be given infinitely many different complex structures. Let X be a Riemann surface, and let V be an open subset of X. A function f : V → C is said to be holomorphic if, for all coordinate neighbourhoods (U, z)of X, f ◦ z−1 is a holomorphic function on z(U). Similarly, one can define the notion of a meromorphic function on a Riemann surface. The general problem. We can state the following grandiose problem: study all holomorphic functions on all Riemann surfaces. In order to do this, we would first have to find all Riemann surfaces. This problem is easier than it looks. Let X be a Riemann surface. From topology, we know that there is a simply connected topological space X˜ (the universal covering space of X)andamapp : X˜ → X which is a local homeomorphism. There is a unique complex structure on X˜ for which p : X˜ → X is a local isomorphism of Riemann surfaces. If Γ is the group of covering transformations of p : X˜ → X,thenX =Γ\X.˜ Theorem 0.1. A simply connected Riemann surface is isomorphic to (exactly) one of the following three: (a) C; (b) the open unit disk D =df {z ∈ C ||z| < 1}; (c) the Riemann sphere. Proof. This is the famous Riemann mapping theorem. The main focus of this course will be on Riemann surfaces with D as their universal covering space, but we shall also need to look at those with C as their universal covering space; the third type will not occur. Riemann surfaces that are quotients of D. In fact, rather than working with D, it will be more convenient to work with the complex upper half plane: H = {z ∈ C | (z) > 0}. → z−i H The map z z+i is an isomorphism of onto D (in the jargon the complex analysts use, H and D are conformally equivalent). We want to study Riemann surfaces of the 2J.S.MILNE form Γ\H, where Γ is a discrete group acting on H. How do we find such Γ’s? There ab is an obvious big group acting on H,namely,SL(R). For α = ∈ SL (R) 2 cd 2 and z ∈ H, define az + b α(z)= . cz + d Then az + b (az + b)(cz¯ + d) (adz + bcz¯) (αz)= = = . cz + d |cz + d|2 |cz + d|2 But (adz + bcz¯)=(ad − bc) · (z)= (z), 2 because det(α)=1.Hence (αz)= (z)/|cz + d| for α ∈ SL2(R). In particular, z ∈ H =⇒ α(z) ∈ H. Later we shall see that there is an isomorphism SL2(R)/{±I}→Aut(H) (bi-holomorphic automorphisms of H). There are some obvious discrete groups in SL2(R), for example, Γ = SL2(Z). This is called the (full) elliptic modular group. For any N ≥ 0, we define ab Γ(N)= | a ≡ 1,b≡ 0,c≡ 0,d≡ 1modN cd and call it the principal congruence subgroup of level N; in particular, Γ(1) = SL2(Z). There are many discrete subgroups in SL2(R), but those of most interest to number theorists are the ones containing a principal congruence subgroup as a subgroup of finite index. Let Y (N)=Γ(N)\H and endow it with the quotient topology. Let p : H → Y (N) be the quotient map. There is a unique complex structure on Y (N) such that a function f on an open subset U of Y (N) is holomorphic if and only if f ◦ p is holomorphic on p−1(U). Thus f → f ◦p defines a one-to-one correspondence between holomorphic functions on U ⊂ Y (N) and holomorphic functions on p−1(U) invariant under Γ(N), i.e., such that g(γz)=g(z) for all γ ∈ Γ(N). The Riemann surface Y (N) is not compact, but there is a natural way of com- pactifying it by adding a finite number of points. The compact Riemann surface is denoted by X(N). For example, Y (1) is compactified by adding a single point. Modular functions. A modular function f(z) of level N is a meromorphic func- tion on H invariant under Γ(N) and “meromorphic at the cusps”. Because it is invariant under Γ(N), it can be regarded as a function on Y (N), and the second con- dition means that it remains meromorphic when considered as a function on X(N), i.e., it has at worst a pole at each point of X(N) \ Y (N).
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