Modular functions and modular forms - James Milne

1 Modular functions and Modular forms (Elliptic Modular Curves) MilneVersion 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 information:@misc{milneMF,author={ Milne , James S.},title={ Modular functions and Modular forms ( )},year={2017},note={Available at },pages={134}} 22, 1997; first version on the web; 128 23, 2009; new style; minor fixes and improvements; added list of symbols;129 26, 2010. Corrected; many minor revisions. 138 22, 2017. Corrected; minor revisions. 133 send comments and corrections to me at the address on my picture shows a fundamental domain for , as drawn by the fundamental domaindrawer of H.

2 1997, 2009, 2012, 2017 paper copies for noncommercial personal use may be made without explicit permissionfrom the copyright ..5 IThe Analytic Theory131 Preliminaries ..132 Elliptic Modular Curves as Riemann Surfaces ..253 Elliptic functions ..414 Modular functions and Modular forms ..485 Hecke Operators ..67II The Algebro-Geometric Theory876 The Modular Equation for ..877 The Canonical Model ..918 Modular Curves as Moduli Varieties ..979 Modular forms , Dirichlet Series, and Functional Equations .. 10110 Correspondences on Curves; the Theorem of Eichler-Shimura .. 10511 Curves and their Zeta functions .. 10912 Complex Multiplication for Elliptic CurvesQ.. 121 Index131 List of Symbols1333 PREREQUISITESThe algebra and complex analysis usually covered in advanced undergraduate or first-yeargraduate reference monnnnn is to question nnnnn on addition to the references listed on p.

3 12 and in the footnotes, I shall refer to thefollowing of my course notes (available at ).FTFields and Galois Theory, , Geometry, , Number Theory, , Field Theory, , thank the following for providing corrections and comments for earlier versions of thesenotes: Carlos Barros, Saikat Biswas, Keith Conrad, Tony Feng, Ulrich Goertz, Enis Kaya,Keenan Kidwell, John Miller, Thomas Preu and colleague, Nousin Sabet, Francesc GispertS anchez, Bhupendra Nath Tiwari, Hendrik 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 surfacesLetXbe a connected Hausdorff topological space. Acoordinate neighbourhoodforXisa ;z/withUan open subset ofXandza homeomorphism fromUonto an opensubset of the complex plane.

4 A compatible family of coordinate neighbourhoods coveringXdefines acomplex structureonX. ARiemann surfaceis a connected Hausdorff topologicalspace together with a complex example, every connected open subsetXofCis a Riemann surface, and the unitsphere can be given a complex structure with two coordinate neighbourhoods, namely thecomplements of the north and south poles mapped onto the complex plane in the standardway. With this complex structure it is called theRiemann sphere. We shall see that a torusR2=Z2can be given infinitely many different complex a Riemann surface andVan open subset ofX. A functionfWV!Cis saidto beholomorphicif, for all coordinate ;z/ofX,f z \U/!Cis a holomorphic function \U/. Similarly, one can define the notion of ameromor-phicfunction on a Riemann general problemWe can now state the grandiose problem: study all holomorphic functions on all Riemannsurfaces.

5 In order to do this, we would first have to find all Riemann surfaces. This problemis easier than it a Riemann surface. From topology, we know that there is a simply connectedtopological spacezX(the universal covering space ofX/and a mappWzX!Xwhich is alocal homeomorphism. There is a unique complex structure onzXfor whichpWzX!Xis alocal isomorphism of Riemann surfaces. If is the group of covering transformations ofpWzX!X, thenXD simply connected Riemann surface is isomorphic to exactly one ofthe following three:(a)the Riemann sphere;(b)CI(c)the open unit diskDdefDfz2 Cjjzj< these, only the Riemann sphere is compact. In particular, it is not homeomorphictoCorD. There is no isomorphismfWC!Dbecause any suchfwould be a boundedholomorphic function onC, and hence constant.)

6 Thus, the three are distinct. A specialcase of the theorem says that every simply connected open subset ofCdifferent fromCisisomorphic toD. This is proved in Cartan 1963, VI, 3. The general statement is the famousUniformization Theorem, which was proved independently by Koebe and Poincar e in mo10516 for a discussion of the various 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 universal surfaces that are quotients ofDIn fact, rather than working withD, it will be more convenient to work with the complexupper half plane:HDfz2Cj=.z/ > 0g:The mapz7!z izCiis an isomorphism ofHontoD(in the language of complex analysis,HandDare conformally equivalent).

7 We want to study Riemann surfaces of the form nH,where is a discrete group acting onH. How do we find such ? There is an obvious biggroup acting onH, namely, For D a bc d , let .z/DazCbczCd:Then=..z//D= azCbczCd D= .azCb/.cxzCd/jczCdj2 D=.adzCbcxz/jczCdj2:But=. bc/ =.z/, which equals=.z/because det. /D1. Hence=..z//D=.z/=jczCdj2for In particular,z2HH) .z/2H:The matrix Iacts trivially onH, and later we shall see Igis the bi-holomorphic automorphisms ofH(see ). The most obvious discretesubgroup This is called thefull Modular group. For an integerN > 0, we define .N/D a bc d a 1; b 0; c 0; d 1modN :It is theprincipal congruence subgroupof levelN. There are lots of other discrete sub-groups , but the main ones of interest to number theorists are the subgroups a principal congruence.

8 N/nHand endow it with the quotient topology. LetpWH! quotient map. There is a unique complex structure that a functionfonan open holomorphic if and only iff pis holomorphic onp !f pdefines a one-to-one correspondence between holomorphic functions onU holomorphic functions onp under .N/, , such thatg. all 2 .N/:The Riemann not compact, but there is a natural way of compactifyingit by adding a finite number of points. For example, compactified by adding a singlepoint. The compact Riemann surface obtained is denoted levelNis a meromorphic function onHinvariant under .N/and meromorphic at the cusps . Because it is invariant under .N/, it can be regarded as ameromorphic function , and the second condition means that it is meromorphic whenconsidered as a function , , it has at worst a pole at each point :For the full Modular group, it is easy to make explicit the condition meromorphic at thecusps (in this case, cusp).

9 To be invariant under the full Modular group means thatf azCbczCd all a bc d :Since 1 10 1 , we have , ,fis invariant under the ;n/7!zCnofZonC. The functionz7!e2 izis an isomorphismC=Z!CXf0g,and so be written in the .q/,qDe2 iz. Aszranges over the upper half plane, overCXf0g. To say meromorphic at the cusp means thatf .q/is meromorphic at0, which means thatfhas an N0anqn; qDe2 iz;in some neighbourhood construct a Modular function, we have to construct a meromorphic function onHthat isinvariant under the action of .N/. This is difficult. It is easier to construct functions thattransform in a certain way under the action of .N/; the quotient of two such functions ofsame type will then be a Modular is analogous to the following situation.

10 KXorigin/=k and assume thatkis infinite. ;Y/be the field of fractions ofk X;Y . ;Y/defines a ;b/7! ;b/on the subset ofk kwhere its denominator doesn tvanish. This function will pass to the and only ; ;Y/for alla2k :Recall that a homogeneous form of degreedis a ;Y/2k X;Y such ; ;Y/for alla2k . Thus, to get anfsatisfying the condition, we needonly take the quotientg=hof two homogeneous forms of the same degree withh relation of homogeneous forms to rational functions onP1is exactly the same asthe relation of Modular forms to Modular form of levelNand weight2kis a holomorphic that(a)f. all D a bc d 2 .N/I(b) holomorphic at the cusps .7 For the full Modular group, (a) again implies , and sofcan bewritten as a function ofqDe2 iz; says that this function is holomorphicat0, so 0anqn; qDe2 iz:The quotient of two Modular forms of levelNand the same weight is a Modular functionof plane algebraic curvesLetkbe a field.