### Transcription of Modular functions and modular forms - James Milne

1 **Modular** **functions** and **Modular** **forms** (Elliptic **Modular** Curves). **Milne** Version 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** ( )}, year={2017}, note={Available at }, pages={134}. }. May 22, 1997; first version on the web; 128 pages. November 23, 2009; new style; minor fixes and improvements; added list of symbols;. 129 pages.

2 April 26, 2010. Corrected; many minor revisions. 138 pages. March 22, 2017. Corrected; minor revisions. 133 pages. Please send comments and corrections to me at the address on my website 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 **Milne** . Single paper copies for noncommercial personal use may be made without explicit permission from the copyright holder. Contents Contents 3. Introduction .. 5. I The Analytic Theory 13. 1 Preliminaries .. 13. 2 Elliptic **Modular** Curves as Riemann Surfaces.

3 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. P REREQUISITES.

4 The algebra and complex analysis usually covered in advanced undergraduate or first-year graduate courses. R EFERENCES. A reference monnnnn is to question nnnnn on 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 ). FT Fields and Galois Theory, , 2017. AG Algebraic Geometry, , 2017. ANT Algebraic Number Theory, , 2017. CFT Class Field Theory, , 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 S anchez, Bhupendra Nath Tiwari, Hendrik Verhoek.

5 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.

6 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 R2 =Z2 can be given infinitely many different complex structures. Let X be a Riemann surface and V an open subset of X. A function f W V ! C is said to be holomorphic if, for all coordinate neighbourhoods.

7 U; z/ of X , 1. f z W \ U / ! C. is a holomorphic function on \ 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 Xz (the universal covering space of X / and a map pW Xz ! X which is a local homeomorphism.)

8 There is a unique complex structure on Xz for which pW Xz ! X is a local isomorphism of Riemann surfaces. If is the group of covering transformations of pW Xz ! X, then X D nXz : T HEOREM Every simply connected Riemann surface is isomorphic to exactly one of the following three: (a) the Riemann sphere;. (b) CI. def (c) the open unit disk D D fz 2 C j jzj < 1g. P ROOF. Of these, only the Riemann sphere is compact. In particular, it is not homeomorphic to C or D. There is no isomorphism f W C ! D because any such f would be a bounded holomorphic function on C, and hence constant.

9 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 Poincar e 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.

10 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 D fz 2 C j =.z/ > 0g: z i The map z 7! zCi is an isomorphism of H onto D (in the language of complex analysis, H. and D are conformally equivalent). We want to study Riemann surfaces of the **form** nH, where is a discrete group acting on H. How do we find such ? There is an obvious big group acting on H, namely, SL2 .R/. For D ac db 2 SL2 .R/ and z 2 H, let az C b .z/ D : cz C d Then az C b .az C b/.