ModularFunctionsandModularForms - James Milne

1 Modular Functions and Modular FormsJ. S. MilneAbstract. 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 revisionsan sen dcomments an dcorrections to me at surfaces1 The general problem1 Riemann surfaces that are quotients affine algebraic curves3 Plane projective of Modular curves and modular Preliminaries8 Continuous group surfaces: classical approach10 Riemann surfaces as ringed spaces11 Differential on compact Riemann genus surfaces as algebraic Elliptic Modular Curves as Riemann Surfaces19c 1997 Milne . You may make one copy of these notes for your own personal S.

2 MILNEThe upper-half plane as a quotient of SL2(R)19 Quotients ofH21 Discrete subgroups of SL2(R)22 Classification of linear fractional transformations24 Fundamental domains26 Fundamental domains for congruence subgroups29 Defining complex structures on quotients30 The complex structure on (1)\H 30 The complex structure on \H 31 The genus ofX( )323. Elliptic functions35 Lattices and bases35 Quotients ofCby lattices35 Doubly periodic functions35 Endomorphisms ofC/ 36 The Weierstrass -function37 The addition formula39 Eisenstein series39 The field of doubly periodic functions40 Elliptic curves40 The elliptic curveE( )404. Modular Functions and Modular Forms42 Modular functions42 Modular forms43 Modular forms ask-fold differentials44 The dimension of the space of modular forms45 Zeros of modular forms47 Modular forms for (1)48 The Fourier coefficients of the Eisenstein series for (1)49 The expansion of andj51 The size of the coefficients of a cusp form52 Modular forms as sections of line bundles52 Poincar eseries54 The geometry ofH56 Petersson inner product57 Completeness of the Poincar eseries58 Eisenstein series for (N)585.

3 Hecke Operators62 Introduction62 MODULAR FUNCTIONS AND MODULAR FORMSiiiAbstract Hecke operators65 Lemmas on 2 2 matrices67 Hecke operators for (1)68 TheZ-structure on the space of modular forms for (1)72 Geometric interpretation of Hecke operators75 The Hecke algebra766. The Modular Equation for 0(N)817. The Canonical Model ofX0(N)overQ86 Review of some algebraic geometry86 Curves and Riemann surfaces88 The curveX0(N)overQ908. Modular Curves as Moduli Varieties92 The general notion of a moduli variety92 The moduli variety for elliptic curves93 The curveY0(N)Qas a moduli variety94 The curveY(N) as a moduli variety959. Modular Forms, Dirichlet Series, and Functional Equations96 The Mellin transform96 Weil s theorem9810.

4 Correspondences on Curves; the Theorem of Eichler-Shimura100 The ring of correspondences of a curve100 The Hecke correspondence101 The Frobenius map101 Brief review of the points of orderpon elliptic curves102 The Eichler-Shimura theorem10211. Curves and their Zeta Functions104 Two elementary results104 The zeta function of a curve over a finite field105 The zeta function of a curve overQ106 Review of elliptic curves107 The zeta function ofX0(N): case of genus 1108 Review of the theory of curves109 The zeta function ofX0(N): general case111 The Conjecture of Taniyama and Weil111 Notes113 Fermat s last theorem113 Application to the conjecture of Birch and Swinnerton-Dyer11412.

5 Complex Multiplication for Elliptic Curves115 Abelian extensions inK116 Elliptic curves overC117 Algebraicity ofj117 The integrality ofj118 Statement of the main theorem (first form)120 The theory ofa-isogenies120 Reduction of elliptic curves121 The Frobenius map122 Proofofthemaintheorem122 The main theorem for orders123 Points of orderm124 Adelic version of the main theorem124 Index125 MODULAR FUNCTIONS AND MODULAR FORMS1 IntroductionIt is easy to define modular functions and forms, but less easy to say why theyare important, especially to number theorists. Thus I will begin with a rather longoverview of the a connected Hausdorff topological space. Acoor-dinate neighbourhoodofP Xis a pair (U, z)withUan open neighbourhood ofPandza homeomorphism ofUonto an open subset of the complex plane.

6 AcomplexstructureonXis a compatible family of coordinate neighbourhoods that surfaceis a topological space together with a complex example, any open subsetXofCis a Riemann surface, and the unit spherecan be given a complex structure with two coordinate neighbourhoods, namely thecomplements of the north and south poles mapped onto the complex plane in thestandard way. With this complex structure it is called theRiemann that a torus can be given infinitely many different complex a Riemann surface, and letVbe an open subset ofX. A functionf:V Cis said to beholomorphicif, for all coordinate neighbourhoods (U, z)ofX,f z 1is a holomorphic function onz(U). Similarly, one can define the notion ofameromorphicfunction on a Riemann general can state the following grandiose problem: study allholomorphic functions on all Riemann surfaces.

7 In order to do this, we would firsthave to find all Riemann surfaces. This problem is easier than it a Riemann surface. From topology, we know that there is a simplyconnected topological space X(the universal covering space ofX)andamapp: X Xwhich is a local homeomorphism. There is a unique complex structure on Xfor whichp: X Xis a local isomorphism of Riemann surfaces. If is the groupof covering transformations ofp: X X,thenX= \ simply connected Riemann surface is isomorphic to (exactly)one of the following three:(a)C;(b)the open unit diskDdf={z C||z|<1};(c)the Riemann is the famous Riemann mapping main focus of this course will be on Riemann surfaces withDas their universalcovering space, but we shall also need to look at those withCas their universalcovering space; the third type will not surfaces that are quotients fact, rather than working withD, it will be more convenient to work with the complex upper half plane:H={z C| (z)>0}.

8 The mapz z iz+iis an isomorphism ofHontoD(in the jargon the complex analystsuse,HandDare conformally equivalent). We want to study Riemann surfaces of \H, where is a discrete group acting onH. How do we find such s? Thereis an obvious big group acting onH,namely,SL2(R). For = abcd SL2(R)andz H, define (z)=az+bcz+ ( z)= az+bcz+d = (az+b)(c z+d)|cz+d|2 = (adz+bc z)|cz+d| (adz+bc z)=(ad bc) (z)= (z),because det( )=1. Hence ( z)= (z)/|cz+d|2for SL2(R). In particular,z H= (z) we shall see that there is an isomorphismSL2(R)/{ I} Aut(H)(bi-holomorphic automorphisms ofH). There are some obvious discrete groups inSL2(R), for example, = SL2(Z). This is called the(full) elliptic modular anyN 0, we define (N)= abcd |a 1,b 0,c 0,d 1modN and call it theprincipal congruence subgroupof levelN; in particular, (1) = SL2(Z).

9 There are many discrete subgroups in SL2(R), but those of most interest to numbertheorists are the ones containing a principal congruence subgroup as a subgroup offinite (N)= (N)\Hand endow it with the quotient topology. Letp:H Y(N) be the quotient map. There is a unique complex structure onY(N) such thata functionfon an open subsetUofY(N) is holomorphic if and only iff pisholomorphic onp 1(U). Thusf f pdefines a one-to-one correspondence betweenholomorphic functions onU Y(N) and holomorphic functions onp 1(U) invariantunder (N), , such thatg( z)=g(z) for all (N).The Riemann surfaceY(N) is not compact, but there is a natural way of com-pactifying it by adding a finite number of points.

10 The compact Riemann surface isdenoted byX(N). For example,Y(1) is compactified by adding a single functionf(z) of levelNis a meromorphic func-tion onHinvariant under (N) and meromorphic at the cusps . Because it isinvariant under (N), it can be regarded as a function onY(N), and the second con-dition means that it remains meromorphic when considered as a function onX(N), , it has at worst a pole at each point ofX(N)\Y(N).In the case of 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 fullMODULAR FUNCTIONS AND MODULAR FORMS3modular group means thatf az+bcz+d =f(z) for all abcd SL2(Z).