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Modular Functions and Modular Forms (Elliptic Modular Curves) Modular Functions and Modular Forms (Elliptic Modular Curves) J.S. Milne Version 1.20 November 23, 2009 This is an introduction to the arithmetic theory of modular functions and modular forms, with a greater emphasis on the geometry than most accounts. BibTeX information: @misc{milneMF, author={Milne, James S.}, title={Modular Functions and Modular Forms (v1.20)}, year={2009}, note={Available at www.jmilne.org/math/}, pages={192} } v1.10 May 22, 1997; first version on the web; 128 pages. v1.20 November 23, 2009; new style; minor fixes and improvements; added list of symbols; 129 pages. Please send comments and corrections to me at the address on my website http://www.jmilne.org/math/. The photograph is of Lake Manapouri, Fiordland, New Zealand. Copyright c 1997, 2009 J.S. Milne. Single paper copies for noncommercial personal use may be made without explicit permission from the copyright holder. Contents Introduction . 1 Riemann surfaces 1, The general problem 1, Riemann surfaces that are quotients of D 1, Modular functions. 2, Modular forms. 3, Affine plane algebraic curves 3, Projective plane curves. 4, Arith- metic of Modular Curves. 5, Elliptic curves. 5, Elliptic functions. 5, Elliptic curves and modular curves. 6, Relevant books 7 I The Analytic Theory 9 1 Preliminaries . 9 Continuous group actions. 9, Riemann surfaces: classical approach 11, Riemann surfaces as ringed spaces 12, Differential forms. 13, Analysis on compact Riemann surfaces. 14, Riemann- Roch Theorem. 16, The genus of X 18, Riemann surfaces as algebraic curves. 19 2 Elliptic Modular Curves as Riemann Surfaces . 21 The upper-half plane as a quotient of SL2.R/ 21, Quotients of H 22, Discrete subgroups of SL2.R/ 24, Classification of linear fractional transformations 26, Fundamental domains 28, Fun- damental domains for congruence subgroups 30, Defining complex structures on quotients 31, The complex structure on .1/ H 31, The complex structure on H 32, The genus of X. / 33 n n 3 Elliptic Functions . 36 Lattices and bases 36, Quotients of C by lattices 36, Doubly periodic functions 36, Endomor- phisms of C= 37, The Weierstrass }-function 38, The addition formula 40, Eisenstein series 40, The field of doubly periodic functions 40, Elliptic curves 41, The elliptic curve E./ 41 4 Modular Functions and Modular Forms . 43 Modular functions 43, Modular forms 44, Modular forms as k-fold differentials 45, The dimension of the space of modular forms 46, Zeros of modular forms 48, Modular forms for .1/ 49, The Fourier coefficients of the Eisenstein series for .1/ 50, The expansion of and j 52, The size of the coefficients of a cusp form 53, Modular forms as sections of line bundles 53, Poincare´ series 55, The geometry of H 57, Petersson inner product 58, Completeness of the Poincare´ series 59, Eisenstein series for .N / 59 5 Hecke Operators . 62 Introduction 62, Abstract Hecke operators 65, Lemmas on 2 2 matrices 67, Hecke operators for .1/ 68, The Z-structure on the space of modular forms for .1/ 72, Geometric interpretation of Hecke operators 75, The Hecke algebra 76 II The Algebro-Geometric Theory 81 6 The Modular Equation for 0.N / ......................... 81 7 The Canonical Model of X0.N / over Q ...................... 85 Review of some algebraic geometry 85, Curves and Riemann surfaces 87, The curve X0.N / over Q 89 8 Modular Curves as Moduli Varieties . 90 iii The general notion of a moduli variety 90, The moduli variety for elliptic curves 91, The curve Y0.N /Q as a moduli variety 92, The curve Y.N / as a moduli variety 93 9 Modular Forms, Dirichlet Series, and Functional Equations . 94 The Mellin transform 94, Weil’s theorem 96 10 Correspondences on Curves; the Theorem of Eichler-Shimura . 98 The ring of correspondences of a curve 98, The Hecke correspondence 99, The Frobenius map 99, Brief review of the points of order p on elliptic curves 100, The Eichler-Shimura theorem 100 11 Curves and their Zeta Functions . 102 Two elementary results 102, The zeta function of a curve over a finite field 103, The zeta function of a curve over Q 104, Review of elliptic curves 105, The zeta function of X0.N /: case of genus 1 106, Review of the theory of curves 107, The zeta function of X0.N /: general case 108, The Conjecture of Taniyama and Weil 109, Notes 111, Fermat’s last theorem 111, Application to the conjecture of Birch and Swinnerton-Dyer 111 12 Complex Multiplication for Elliptic Curves Q . 113 Abelian extensions of Q 113, Orders in K 114, Elliptic curves over C 115, Algebraicity of j 115, The integrality of j 116, Statement of the main theorem (first form) 118, The theory of a-isogenies 118, Reduction of elliptic curves 119, The Frobenius map 120, Proof of the main the- orem 121, The main theorem for orders 121, Points of order m 122, Adelic version of the main theorem 122 List of Symbols 123 PREREQUISITES The algebra and complex analysis usually covered in advanced undergraduate or first-year graduate courses. REFERENCES In addition to the references listed on p7 and in the footnotes, I shall refer to the following of my course notes (available at www.jmilne.org/math/). FT Fields and Galois Theory, v4.21, 2008. AG Algebraic Geometry, v5.20, 2009. ANT Algebraic Number Theory, v3.02, 2009. CFT Class Field Theory, v4.00, 2008. ACKNOWLEDGEMENTS I thank the following for providing corrections and comments for earlier versions of these notes: Carlos Barros, Ulrich Goertz, Thomas Preu and colleague, Nousin Sabet. 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 shall begin with a rather long overview of the subject. Riemann surfaces Let X be a connected Hausdorff topological space. A coordinate neighbourhood of P X is a 2 pair .U;z/ with U an open neighbourhood of P and z a homeomorphism of U onto an open subset of the complex plane. A compatible family of coordinate neighbourhoods covering X defines a complex structure on X.A Riemann surface is a connected Hausdorff topological space together with a complex structure. For example, any connected 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. We shall 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 W ! to be holomorphic if, for all coordinate neighbourhoods .U;z/ of X, f z 1 is a holomorphic ı 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 topo- logical space X (the universal covering space of X/ and a map p X X which is a local homeo- Q W Q ! morphism. There is a unique complex structure on X for which p X X is a local isomorphism Q W Q ! of Riemann surfaces. If is the group of covering transformations of p X X, then X X: W Q ! D n Q THEOREM 0.1 A simply connected Riemann surface is isomorphic to (exactly) one of the following three: (a) the Riemann sphere; (b) C I def (c) the open unit disk D z C z < 1 . D f 2 j j j g PROOF. This is the famous Riemann mapping theorem. 2 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. 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 : D f 2 j = g 1 z i The map z is an isomorphism of H onto D (in the language the complex analysts use, H 7! z i and D are conformallyC equivalent). We want to study Riemann surfaces of the form H, where n is a discrete group acting on H. How do we find such ’s? There is an obvious big group acting a b on H, namely, SL2.R/. For ˛ SL2.R/, define D c d 2 az b ˛.z/ C : D cz d C Then  az b à Â.az b/.cz d/à .adz bcz/ .˛z/ C C N C = C N : = D = cz d D = cz d 2 D cz d 2 C j C j j C j But .adz bcz/ .ad bc/ .z/, which equal .z/ because det.˛/ 1. Hence = C N D = = D .˛z/ .z/= cz d 2 = D = j C j for ˛ SL2.R/. In particular, 2 z H ˛.z/ H: 2 H) 2 Later we shall see that there is an isomorphism SL2.R/= I Aut.H/ f˙ g ! (bi-holomorphic automorphisms of H/. There are some obvious discrete groups in SL2.R/, for example, SL2.Z/. This is called the full modular group.
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