Modular Functions and Modular Forms

134 Pages·2017·0.964 MB·English

by  Milne J.S.

Modular Functions and Modular Forms
(Elliptic Modular Curves)
J.S. Milne
Version1.31
March22,2017
This is an introduction to the arithmetic theory of modular functions and modular forms,
withagreateremphasisonthegeometrythanmostaccounts.
BibTeXinformation:
@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 May22,1997;firstversionontheweb;128pages.
v1.20 November23,2009;newstyle;minorfixesandimprovements;addedlistofsymbols;
129pages.
v1.30 April26,2010. Corrected;manyminorrevisions. 138pages.
v1.31 March22,2017. Corrected;minorrevisions. 133pages.
Pleasesendcommentsandcorrectionstomeattheaddressonmywebsite
http://www.jmilne.org/math/.
Thepictureshowsafundamentaldomainfor(cid:0)1.10/,asdrawnbythefundamentaldomain
drawerofH.Verrill.
Copyright c 1997,2009,2012,2017J.S.Milne.
(cid:13)
Singlepapercopiesfornoncommercialpersonalusemaybemadewithoutexplicitpermission
fromthecopyrightholder.
Contents
Contents 3
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
I TheAnalyticTheory 13
1 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
2 EllipticModularCurvesasRiemannSurfaces . . . . . . . . . . . . . . . . 25
3 EllipticFunctions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41
4 ModularFunctionsandModularForms . . . . . . . . . . . . . . . . . . . 48
5 HeckeOperators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67
II TheAlgebro-GeometricTheory 87
6 TheModularEquationfor(cid:0) .N/ . . . . . . . . . . . . . . . . . . . . . . 87
0
7 TheCanonicalModelofX .N/overQ . . . . . . . . . . . . . . . . . . . 91
0
8 ModularCurvesasModuliVarieties . . . . . . . . . . . . . . . . . . . . . 97
9 ModularForms,DirichletSeries,andFunctionalEquations . . . . . . . . . 101
10 CorrespondencesonCurves;theTheoremofEichler-Shimura . . . . . . . 105
11 CurvesandtheirZetaFunctions . . . . . . . . . . . . . . . . . . . . . . . 109
12 ComplexMultiplicationforEllipticCurvesQ . . . . . . . . . . . . . . . . 121
Index 131
ListofSymbols 133
3
PREREQUISITES
Thealgebraandcomplexanalysisusuallycoveredinadvancedundergraduateorfirst-year
graduatecourses.
REFERENCES
Areferencemonnnnnistoquestionnnnnnonmathoverflow.net.
In addition to the references listed on p. 12 and in the footnotes, I shall refer to the
followingofmycoursenotes(availableatwww.jmilne.org/math/).
FT FieldsandGaloisTheory,v4.52,2017.
AG AlgebraicGeometry,v6.02,2017.
ANT AlgebraicNumberTheory,v3.07,2017.
CFT ClassFieldTheory,v4.02,2013.
ACKNOWLEDGEMENTS
Ithankthefollowingforprovidingcorrectionsandcommentsforearlierversionsofthese
notes: CarlosBarros,SaikatBiswas,KeithConrad,TonyFeng,UlrichGoertz,EnisKaya,
KeenanKidwell,JohnMiller,ThomasPreuandcolleague,NousinSabet,FrancescGispert
Sa´nchez,BhupendraNathTiwari,HendrikVerhoek.
Introduction
Itiseasytodefinemodularfunctionsandforms,butlesseasytosaywhytheyareimportant,
especiallytonumbertheorists. ThusIshallbeginwitharatherlongoverviewofthesubject.
Riemannsurfaces
LetX beaconnectedHausdorfftopologicalspace. AcoordinateneighbourhoodforX is
a pair .U;z/ with U an open subset of X and z a homeomorphism from U onto an open
subsetofthecomplexplane. AcompatiblefamilyofcoordinateneighbourhoodscoveringX
definesacomplexstructureonX. ARiemannsurfaceisaconnectedHausdorfftopological
spacetogetherwithacomplexstructure.
Forexample,everyconnectedopensubsetX ofCisaRiemannsurface,andtheunit
spherecanbegivenacomplexstructurewithtwocoordinateneighbourhoods,namelythe
complementsofthenorthandsouthpolesmappedontothecomplexplaneinthestandard
way. WiththiscomplexstructureitiscalledtheRiemannsphere. Weshallseethatatorus
R2=Z2 canbegiveninfinitelymanydifferentcomplexstructures.
LetX beaRiemannsurfaceandV anopensubsetofX. Afunctionf V Cissaid
W !
tobeholomorphicif,forallcoordinateneighbourhoods.U;z/ofX,
f z(cid:0)1 z.V U/ C
ı W \ !
isaholomorphicfunctiononz.V U/. Similarly,onecandefinethenotionofameromor-
\
phicfunctiononaRiemannsurface.
Thegeneralproblem
Wecannowstatethegrandioseproblem: studyallholomorphicfunctionsonallRiemann
surfaces. Inordertodothis,wewouldfirsthavetofindallRiemannsurfaces. Thisproblem
iseasierthanitlooks.
LetX beaRiemannsurface. Fromtopology,weknowthatthereisasimplyconnected
topologicalspaceX (theuniversalcoveringspaceofX/andamapp X X whichisa
z W z!
localhomeomorphism. ThereisauniquecomplexstructureonX forwhichp X X isa
z W z!
localisomorphismofRiemannsurfaces. If(cid:0) isthegroupofcoveringtransformationsof
p X X,thenX (cid:0) X:
W z! D n z
THEOREM 0.1 EverysimplyconnectedRiemannsurfaceisisomorphictoexactlyoneof
thefollowingthree:
(a) theRiemannsphere;
(b) C
I
(c) theopenunitdiskD def z C z <1 .
Df 2 jj j g
PROOF. Ofthese,onlytheRiemannsphereiscompact. Inparticular,itisnothomeomorphic
toCorD. Thereisnoisomorphismf C D becauseanysuchf wouldbeabounded
W !
holomorphic function on C, and hence constant. Thus, the three are distinct. A special
caseofthetheoremsaysthateverysimplyconnectedopensubsetofCdifferentfromCis
isomorphictoD. ThisisprovedinCartan1963,VI,(cid:144)3. Thegeneralstatementisthefamous
UniformizationTheorem,whichwasprovedindependentlybyKoebeandPoincare´ in1907.
Seemo10516foradiscussionofthevariousproofs. (cid:50)
5
The main focus of this course will be on Riemann surfaces with D as their universal
coveringspace,butweshallalsoneedtolookatthosewithCastheiruniversalcovering
space.
RiemannsurfacesthatarequotientsofD
Infact,ratherthanworkingwithD,itwillbemoreconvenienttoworkwiththecomplex
upperhalfplane:
H z C .z/>0 :
Df 2 j= g
Themapz z(cid:0)i isanisomorphismofHontoD (inthelanguageofcomplexanalysis,H
7! zCi
andD areconformallyequivalent). WewanttostudyRiemannsurfacesoftheform(cid:0) H,
n
where(cid:0) isadiscretegroupactingonH. Howdowefindsuch(cid:0)? Thereisanobviousbig
groupactingonH,namely,SL .R/. For˛ (cid:0)a b(cid:1) SL .R/andz H,let
2 D c d 2 2 2
az b
˛.z/ C :
D cz d
C
Then
(cid:18) (cid:19) (cid:18) (cid:19)
az b .az b/.cz d/ .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/,whichequals .z/becausedet.˛/ 1. Hence
= C x D (cid:0) (cid:1)= = D
.˛.z// .z/= cz d 2
= D= j C j
for˛ SL .R/. Inparticular,
2
2
z H ˛.z/ H:
2 H) 2
The matrix I acts trivially on H, and later we shall see that SL .R/= I is the full
2
(cid:0) f˙ g
groupAut.H/ofbi-holomorphicautomorphismsofH(see2.1). Themostobviousdiscrete
subgroupofSL .R/is(cid:0) SL .Z/. Thisiscalledthefullmodular group. Foraninteger
2 2
D
N >0,wedefine
(cid:26)(cid:18) (cid:19)ˇ (cid:27)
a b ˇ
(cid:0).N/ ˇa 1;b 0;c 0;d 1mod N :
D c d ˇ (cid:17) (cid:17) (cid:17) (cid:17)
It is the principal congruence subgroup of level N. There are lots of other discrete sub-
groupsofSL .R/, butthemainonesofinteresttonumbertheoristsarethesubgroupsof
2
SL .Z/containingaprincipalcongruencesubgroup.
2
LetY.N/ (cid:0).N/ Handendowitwiththequotienttopology. Letp H Y.N/denote
D n W !
thequotientmap. ThereisauniquecomplexstructureonY.N/suchthatafunctionf on
anopensubsetU ofY.N/isholomorphicifandonlyiff p isholomorphiconp(cid:0)1.U/.
ı
Thusf f p definesaone-to-onecorrespondencebetweenholomorphicfunctionson
U Y.N7!/ anıd holomorphic functions on p(cid:0)1.U/ invariant under (cid:0).N/, i.e., such that
(cid:26)
g.(cid:13)z/ g.z/forall(cid:13) (cid:0).N/:
D 2
TheRiemannsurfaceY.N/isnotcompact,butthereisanaturalwayofcompactifying
itbyaddingafinitenumberofpoints. Forexample,Y.1/iscompactifiedbyaddingasingle
point. ThecompactRiemannsurfaceobtainedisdenotedbyX.N/.
6
Modularfunctions.
Amodularfunctionf.z/oflevelN isameromorphicfunctiononHinvariantunder(cid:0).N/
and“meromorphicatthecusps”. Becauseitisinvariantunder(cid:0).N/,itcanberegardedasa
meromorphicfunctiononY.N/,andthesecondconditionmeansthatitismeromorphicwhen
consideredasafunctiononX.N/,i.e.,ithasatworstapoleateachpointofX.N/ Y.N/:
X
Forthefullmodulargroup,itiseasytomakeexplicitthecondition“meromorphicatthe
cusps”(inthiscase,cusp). Tobeinvariantunderthefullmodulargroupmeansthat
(cid:18) (cid:19) (cid:18) (cid:19)
az b a b
f C f.z/forall SL .Z/:
cz d D c d 2 2
C
Since(cid:0)1 1(cid:1) SL .Z/,wehavethatf.z 1/ f.z/,i.e.,f isinvariantundertheaction
0 1 2 2 C D
.z;n/ z n of Z on C. The function z e2(cid:25)iz is an isomorphism C=Z C 0 ,
7! C 7! ! X(cid:3)f g
and so every f satisfying f.z 1/ f.z/ can be written in the form f.z/ f .q/,
C D D
q e2(cid:25)iz. As z ranges over the upper half plane, q.z/ ranges over C 0 . To say that
D (cid:3) Xf g
f.z/ismeromorphicatthecuspmeansthatf .q/ismeromorphicat0,whichmeansthat
f hasanexpansion
X
f.z/ a qn; q e2(cid:25)iz;
n
D D
n(cid:21)(cid:0)N0
insomeneighbourhoodofq 0.
D
Modularforms.
Toconstructamodularfunction,wehavetoconstructameromorphicfunctiononHthatis
invariantundertheactionof(cid:0).N/. Thisisdifficult. Itiseasiertoconstructfunctionsthat
transforminacertainwayundertheactionof(cid:0).N/;thequotientoftwosuchfunctionsof
sametypewillthenbeamodularfunction.
Thisisanalogoustothefollowingsituation. Let
P1.k/ .k k origin/=k(cid:2)
D (cid:2) X
andassumethatkisinfinite. Letk.X;Y/bethefieldoffractionsofkŒX;Y(cid:141). Anf k.X;Y/
2
defines a function .a;b/ f.a;b/ on the subset of k k where its denominator doesn’t
7! (cid:2)
vanish. ThisfunctionwillpasstothequotientP1.k/ifandonlyif
(cid:2)
f.aX;aY/ f.X;Y/foralla k :
D 2
Recallthatahomogeneousformofdegreed isapolynomialh.X;Y/ kŒX;Y(cid:141)suchthat
h.aX;aY/ adh.X;Y/foralla k(cid:2). Thus,togetanf satisfyingthe2condition,weneed
D 2
onlytakethequotientg=hoftwohomogeneousformsofthesamedegreewithh 0.
¤
TherelationofhomogeneousformstorationalfunctionsonP1 isexactlythesameas
therelationofmodularformstomodularfunctions.
DEFINITION 0.2 A modular form of level N and weight 2k is a holomorphic function
f.z/onHsuchthat
(a) f.˛z/ .cz d/2k f.z/forall˛ (cid:0)a b(cid:1) (cid:0).N/
D C (cid:1) D c d 2 I
(b) f.z/is“holomorphicatthecusps”.
7
For the full modular group, (a) again implies that f.z 1/ f.z/, and so f can be
C D
writtenasafunctionofq e2(cid:25)iz;condition.b/thensaysthatthisfunctionisholomorphic
D
at0,sothat
X
f.z/ a qn; q e2(cid:25)iz:
n
D D
n(cid:21)0
ThequotientoftwomodularformsoflevelN andthesameweightisamodularfunction
oflevelN.
Affineplanealgebraiccurves
Letk beafield. AnaffineplanealgebraiccurveC overk isdefinedbyanonzeropolyno-
mialf.X;Y/ kŒX;Y(cid:141). ThepointsofC withcoordinatesinafieldK k arethezeros
2 (cid:27)
off.X;Y/inK K;wedenotethissetbyC.K/. WeletkŒC(cid:141) kŒX;Y(cid:141)=.f.X;Y//and
(cid:2) D
callittheringofregularfunctionsonC. Whenf.X;Y/isirreducible(forusthisisthe
mostinterestingcase),weletk.C/denotethefieldoffractionsofkŒC(cid:141)andcallitthefield
ofrationalfunctionsonC.
We say that C f.X;Y/ is nonsingular if f, @f , @f have no common zero in the
W @X @Y
algebraicclosureofk. Apointwhereallthreevanishiscalledasingularpointonthecurve.
EXAMPLE 0.3 LetC bethecurvedefinedbyY2 4X3 aX b,i.e.,bythepolynomial
D (cid:0) (cid:0)
f.X;Y/ Y2 4X3 aX b:
D (cid:0) C C
Assumechark 2. Thepartialderivativesoff are2Y and 12X2 a d . 4X3 a/.
¤ (cid:0) C D dX (cid:0) C
Thus a singular point on C is a pair .x;y/ such that y 0 and x is a repeated root of
D
4X3 aX b. WeseethatC isnonsingularifandonlyiftherootsof4X3 aX b areall
(cid:0) (cid:0) (cid:0) (cid:0)
simple,whichistrueifandonlyifitsdiscriminant(cid:1)defa3 27b2 isnonzero:
D (cid:0)
PROPOSITION 0.4 LetC beanonsingularaffineplanealgebraiccurveoverC;thenC.C/
hasanaturalstructureasaRiemannsurface.
PROOF. LetP beapointinC.C/. If.@f=@Y/.P/ 0,thentheimplicitfunctiontheorem
¤
shows that the projection .x;y/ x C.C/ C defines a homeomorphism of an open
7! W !
neighbourhood of P onto an open neighbourhood of x.P/ in C. This we take to be a
coordinate neighbourhood of P. If .@f=@Y/.P/ 0, then .@f=@X/.P/ 0, and we use
D ¤
the projection .x;y/ y to define a coordinate neighbourhood of P. The coordinate
7!
neighbourhoods arising in this way are compatible as so define a complex structure on
C.C/. (cid:50)
Projectiveplanecurves.
AprojectiveplanecurveC overk isdefinedbyanonconstanthomogeneouspolynomial
F.X;Y;Z/. Let
P2.k/ .k3 origin/=k(cid:2);
D X
andwrite.a b c/fortheequivalenceclassof.a;b;c/inP2.k/. AsF.X;Y;Z/ishomoge-
neous,F.cx;WcyW;cz/ cm F.x;y;z/foreveryc k(cid:2),wherem deg.F.X;Y;Z//. Thus
D (cid:1) 2 D
itmakessensetosayF.x;y;z/iszeroornonzerofor.x y z/ P2.k/. ThepointsofC
W W 2
with coordinatesinafieldK k arethezerosofF.X;Y;Z/inP2.K/. Wedenotethisset
(cid:27)
byC.K/. Welet
kŒC(cid:141) kŒX;Y;Z(cid:141)=.F.X;Y;Z//
D
8
andcallitthehomogeneouscoordinateringofC. WhenF.X;Y;Z/isirreducible,kŒC(cid:141)
isanintegraldomain,andwedenotebyk.C/thesubfieldofthefieldoffractionsofkŒC(cid:141)of
elementsofdegreezero(i.e.,quotientsofhomogeneouspolynomialsofthesamedegree).
Wecallk.C/thefieldof rationalfunctionsonC:
A plane projective curve C is the union of three affine curves C , C , C defined
X Y Z
by the polynomials F.1;Y;Z/, F.X;1;Z/, F.X;Y;1/ respectively, and we say that C is
nonsingularifallthreeaffinecurvesarenonsingular. Thisisequivalenttothepolynomials
@F @F @F
F; ; ;
@X @Y @Z
having no common zero in the algebraic closure of k. When C is nonsingular, there is a
naturalcomplexstructureonC.C/,andtheRiemannsurfaceC.C/iscompact.
THEOREM 0.5 EverycompactRiemannsurfaceS isoftheformC.C/forsomenonsingular
projectivealgebraiccurveC,andC isuniquelydetermineduptoisomorphism. Moreover,
C.C/isthefieldofmeromorphicfunctionsonS:
Unfortunately,C maynotbeaplaneprojectivecurve. Thestatementisfarfrombeing
truefornoncompactRiemannsurfaces, forexample, HisnotoftheformC.C/forC an
algebraiccurve. Seep.23.
ArithmeticofModularCurves.
The theorem shows that we can regard X.N/ as an algebraic curve, defined by some
homogeneouspolynomial(s)withcoefficientsinC. Thecentralfactunderlyingthearithmetic
of the modular curves (and hence of modular functions and modular forms) is that this
algebraic curve is defined, in a natural way, over QŒ(cid:16) (cid:141), where (cid:16) exp.2(cid:25)i=N/, i.e.,
N N
D
thepolynomialsdefiningX.N/(asanalgebraiccurve)canbetakentohavecoefficientsin
QŒ(cid:16) (cid:141),andthereisanaturalwayofdoingthis.
N
ThisstatementhasasaconsequencethatitmakessensetospeakofthesetX.N/.L/of
pointsofX.N/withcoordinatesinanyfieldcontainingQŒ(cid:16) (cid:141). However,thepolynomials
N
defining X.N/ as an algebraic curve are difficult to write down, and so it is difficult to
describe directly the set X.N/.L/. Fortunately, there is another description of X.N/.L/
(hence of Y.N/.L/) which is much more useful. In the remainder of the introduction, I
describethesetofpointsofY.1/withcoordinatesinanyfieldcontainingQ:
Ellipticcurves.
AnellipticcurveE overafieldk (ofcharacteristic 2;3)isaplaneprojectivecurvegiven
¤
byanequation:
Y2Z 4X3 aXZ2 bZ3; (cid:1)defa3 27b2 0:
D (cid:0) (cid:0) D (cid:0) ¤
WhenwereplaceX withX=c2 andY withY=c3,somec k(cid:2),andmultiplythroughbyc6,
2
theequationbecomes
Y2Z 4X3 ac4XZ2 bc6Z3;
D (cid:0) (cid:0)
andsoweshouldnotdistinguishthecurvedefinedbythisequationfromthatdefinedbythe
firstequation. Notethat
j.E/def1728a3=(cid:1)
D
9
isinvariantunderthischange. Infactonecanshow(withasuitabledefinitionofisomorphism)
0
thattwoellipticcurvesE andE overanalgebraicallyclosedfieldareisomorphicifand
0
onlyifj.E/ j.E /.
D
Ellipticfunctions.
WhatarethequotientsofC? AlatticeinCisasubsetoftheform
(cid:3) Z! Z!
1 2
D C
with! and! complexnumbersthatarelinearlyindependentoverR. ThequotientC=(cid:3)is
1 2
(topologically)atorus. Letp C C=(cid:3)bethequotientmap. ThespaceC=(cid:3)hasaunique
W !
complexstructuresuchthatafunctionf onanopensubsetU ofC=(cid:3)isholomorphicifand
onlyiff p isholomorphiconp(cid:0)1.U/:
ı
TogiveameromorphicfunctiononC=(cid:3)wehavetogiveameromorphicfunctionf on
Cinvariantundertheactionof(cid:3),i.e.,suchthatf.z (cid:21)/ f.z/forall(cid:21) (cid:3). Define
C D 2
(cid:18) (cid:19)
1 X 1 1
}.z/
D z2 C .z (cid:21)/2 (cid:0)(cid:21)2
(cid:21)2(cid:3);(cid:21)¤0 (cid:0)
ThisisameromorphicfunctiononC,invariantunder(cid:3),andthemap
Œz(cid:141) .}.z/ }0.z/ 1/ C=(cid:3) P2.C/
7! W W W !
isanisomorphismoftheRiemannsurfaceC=(cid:3)ontotheRiemannsurfaceE.C/,whereE
istheellipticcurve
Y2Z 4X3 g XZ2 g Z3
2 3
D (cid:0) (cid:0)
with
X 1 X 1
g 60 ; g 140 .
2D (cid:21)4 3D (cid:21)6
(cid:21)2(cid:3);(cid:21)¤0 (cid:21)2(cid:3);(cid:21)¤0
Thisexplainedin(cid:144)3ofChapterI.
Ellipticcurvesandmodularcurves.
Wehaveamap(cid:3) E.(cid:3)/ C=(cid:3)fromlatticestoellipticcurves. WhenisE.(cid:3)/isomorphic
toE.(cid:3)0/? If(cid:3)0 7!c(cid:3)forsDomec C,then
D 2
Œz(cid:141) Œcz(cid:141) C=(cid:3) C=(cid:3)0
7! W !
isanisomorphism,Infactonecanshow
E.(cid:3)/ E.(cid:3)0/ (cid:3)0 c(cid:3),somec C(cid:2):
(cid:25) ” D 2
Suchlattices(cid:3)and(cid:3)0 aresaidtobehomothetic. ByscalingwithanelementofC(cid:2),wecan
normalizeourlatticessothattheyareoftheform
(cid:3).(cid:28)/defZ 1 Z (cid:28);some(cid:28) H:
D (cid:1) C (cid:1) 2
Twolattices(cid:3).(cid:28)/and(cid:3).(cid:28)0/arehomotheticifandonlyifthereisamatrix(cid:0)a b(cid:1) SL .Z/
c d 2 2
suchthat(cid:28)0 a(cid:28)Cb. Wehaveamap
D c(cid:28)Cd
(cid:28) E.(cid:28)/ H ellipticcurvesoverC = ;
7! W !f g (cid:25)
10