Modular functions and modular forms

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. April 26, 2010. Corrected; many minor revisions. 138 pages. March 22, 2017. Corrected; minor revisions.

2 133 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 .. 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.

3 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. 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 ).

4 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. 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.

5 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.

6 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 .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.

7 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. 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.)

8 There is no isomorphism f W C ! D because any such f would be a bounded 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 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.

9 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/.cxz Cd/ =.adz C bcx . z/. =..z// D = D= D : cz C d jcz C d j2 jcz C d j2. But =.adz C bcx z / D.

10 Ad bc/ =.z/, which equals =.z/ because det. / D 1. Hence =..z// D =.z/=jcz C d j2. for 2 SL2 .R/. In particular, z 2 H H) .z/ 2 H: The matrix I acts trivially on H, and later we shall see that SL2 .R/=f I g is the full group of bi-holomorphic automorphisms of H (see ). The most obvious discrete subgroup of SL2 .R/ is D SL2 .Z/. This is called the full Modular group. For an integer N > 0, we define . a b ..N / D a 1; b 0; c 0; d 1 mod N : c d . 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.