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Modular Functions and Modular Forms (Elliptic Modular Curves) J.S. Milne Version 1.31 March 22, 2017 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 (v1.31)}, year={2017}, note={Available at www.jmilne.org/math/}, pages={134} } 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. v1.30 April 26, 2010. Corrected; many minor revisions. 138 pages. v1.31 March 22, 2017. Corrected; minor revisions. 133 pages. Please send comments and corrections to me at the address on my website http://www.jmilne.org/math/. The picture shows a fundamental domain for 1.10/, as drawn by the fundamental domain drawer of H. Verrill. Copyright c 1997, 2009, 2012, 2017 J.S. Milne. Single paper copies for noncommercial personal use may be made without explicit permission from the copyright holder. Contents Contents3 Introduction..................................... 5 I The Analytic Theory 13 1 Preliminaries ................................. 13 2 Elliptic Modular Curves as Riemann Surfaces................ 25 3 Elliptic Functions............................... 41 4 Modular Functions and Modular Forms ................... 48 5 Hecke Operators ............................... 67 II The Algebro-Geometric Theory 87 6 The Modular Equation for 0.N / ...................... 87 7 The Canonical Model of X0.N / over Q ................... 91 8 Modular Curves as Moduli Varieties..................... 97 9 Modular Forms, Dirichlet Series, and Functional Equations . 101 10 Correspondences on Curves; the Theorem of Eichler-Shimura . 105 11 Curves and their Zeta Functions .......................109 12 Complex Multiplication for Elliptic Curves Q . 121 Index 131 List of Symbols 133 3 PREREQUISITES The algebra and complex analysis usually covered in advanced undergraduate or first-year graduate courses. REFERENCES A reference monnnnn is to question nnnnn on mathoverflow.net. In addition to the references listed on p. 12 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, v4.52, 2017. AG Algebraic Geometry, v6.02, 2017. ANT Algebraic Number Theory, v3.07, 2017. CFT Class Field Theory, v4.02, 2013. ACKNOWLEDGEMENTS I thank the following for providing corrections and comments for earlier versions of these notes: Carlos Barros, Saikat Biswas, Keith Conrad, Tony Feng, Ulrich Goertz, Enis Kaya, Keenan Kidwell, John Miller, Thomas Preu and colleague, Nousin Sabet, Francesc Gispert Sanchez,´ Bhupendra Nath Tiwari, Hendrik Verhoek. 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 for X is a pair .U;z/ with U an open subset of X and z a homeomorphism from 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, every 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 2 2 R =Z can be given infinitely many different complex structures. Let X be a Riemann surface and V 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, 1 f z z.V U/ C ı W \ ! is a holomorphic function on z.V U/. Similarly, one can define the notion of a meromor- \ phic function on a Riemann surface. The general problem We can now state the 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/ and a map p X X which is a z W z ! local homeomorphism. There is a unique complex structure on X for which p X X is a z W z ! local isomorphism of Riemann surfaces. If is the group of covering transformations of p X X, then X X: W z ! D n z THEOREM 0.1 Every 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. Of these, only the Riemann sphere is compact. In particular, it is not homeomorphic to C or D. There is no isomorphism f C D because any such f would be a bounded W ! holomorphic function on C, and hence constant. Thus, the three are distinct. A special case of the theorem says that every simply connected open subset of C different from C is isomorphic to D. This is proved in Cartan 1963, VI, 3. The general statement is the famous Uniformization Theorem, which was proved independently by Koebe and Poincare´ in 1907. See mo10516 for a discussion of the various proofs. 2 5 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 z i The map z is an isomorphism of H onto D (in the language of complex analysis, H 7! z i and D are conformallyC equivalent). We want to study Riemann surfaces of the form H, n where is a discrete group acting on H. How do we find such ? There is an obvious big a b group acting on H, namely, SL2.R/. For ˛ SL2.R/ and z H, let D c d 2 2 az b ˛.z/ C : D cz d C Then  az b à Â.az b/.cz d/à .adz bcz/ .˛.z// C C xC = C x : = 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 equals .z/ because det.˛/ 1. Hence = C x D = = D .˛.z// .z/= cz d 2 = D = j C j for ˛ SL2.R/. In particular, 2 z H ˛.z/ H: 2 H) 2 The matrix I acts trivially on H, and later we shall see that SL2.R/= I is the full f˙ g group Aut.H/ of bi-holomorphic automorphisms of H (see 2.1). The most obvious discrete subgroup of SL2.R/ is SL2.Z/. This is called the full modular group. For an integer D N > 0, we define  à ˇ a b ˇ .N / ˇ a 1; b 0; c 0; d 1mod N : D c d ˇ Á Á Á Á It is the principal congruence subgroup of level N . There are lots of other discrete sub- groups of SL2.R/, but the main ones of interest to number theorists are the subgroups of SL2.Z/ containing a principal congruence subgroup. Let Y.N / .N / H and endow it with the quotient topology. Let p H Y.N / denote D n W ! 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 /. ı Thus f f p defines a one-to-one correspondence between holomorphic functions on 7! ı U Y.N / and holomorphic functions on p 1.U / invariant under .N /, i.e., such that g. z/ g.z/ for all .N /: D 2 The Riemann surface Y.N / is not compact, but there is a natural way of compactifying it by adding a finite number of points. For example, Y.1/ is compactified by adding a single point. The compact Riemann surface obtained is denoted by X.N /. 6 Modular functions. A modular function f .z/ of level N is a meromorphic function on H invariant under .N / and “meromorphic at the cusps”. Because it is invariant under .N /, it can be regarded as a meromorphic function on Y.N /, and the second condition means that it is 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 /: X For the full modular group, it is easy to make explicit the condition “meromorphic at the cusps” (in this case, cusp). To be invariant under the full modular group means that  az b à Âa bà f C f .z/ for all SL2.Z/: cz d D c d 2 C 1 1 Since 0 1 SL2.Z/, we have that f .z 1/ f .z/, i.e., f is invariant under the action 2 C D 2iz .z;n/ z n of Z on C. The function z e is an isomorphism C=Z C 0 , 7! C 7! ! X f g and so every f satisfying f .z 1/ f .z/ can be written in the form f .z/ f .q/, 2iz C D D q e .
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