Lecture 19 Geometricmodularforms

Dec 21Theorem X N GINN parametrizesall elliptic curves over togetherwith

an N torsionpoint

Explicitly for 2Efg Ito Zz f CEeProof Note if Ezi E q zz Ez E 2 2z

themapmustbegivenbymultiplicationbysome WEEthen 1 1 CtdZ w c dz E

Z't a be ctdz

Yet Tv fumod 2 7Lzi e F mod I Izadz 1 mud NI NIE1 D o mod N D

Theorem Yoln b ToCN parametrizesallellipticcurves over1CtogetherwithacyclicsubgpoforderN Explicitly for 2 ctry Ee 6 2 02 z 742Eez

Fact Wehavea universalfamilyofellipticcurvesE

YEEsilly.cn is locallyfreeofrank1 over E

I jezewsutin w t Ryy EEREH.cnY.CN bhHn 2 hasthedifferentialsheafon Etisthetrivialsheaf

w is a linebundle on Yi N

Geometricinterpretationf modularformsVersion 1 Ez OneachEz there's a canonicaldifferentialfan de

ferenhalfamdEt 7 2e

Y N As we usetheisomorphism Ez EzT Ii w adz c et

For amodularformfofweightb levelTIN we defineEz 1 1 ftz de kItsEei sifczgxodeixok_tlEEIdbtdIjIfczsdexokS.oL Sectionof REGI.hu is equivariantforT.CN action

Z 1 of DEOK

fxodiok fetffy.CN k

Version2 whenhis even note fC dzkkonly is invariantunderh N actionIndeed ffafzt.IT dfaIIbYhk Ccztdshfczs.f jdtbIDoI Ih f d k

So a modularform fmaybeviewedasaseetinfczsdEHZEHYYCN.rf.cn kBmt Kodaira Spencerisomorphism w 2 Ryfµy

Extensionto cusps X ftp.yfnlufcusps b nmuPkQYifn11

puniversalellipticaureC

Then E YE ZE sit over cusps wehave generalizedellipticcurvesat Je t Je for 2EC Ez lookslike or

UIYin XinEsg Gm or Em eafstairrepdi.com

Esmis a groupvariety overXelN

gray elywon YilN extends to wi erfsnyx.cn

Kodaira Spencerisomorphism w2 Nµy logcusp

differentialsheafwithlogpoleatthecusp

i e if 9 isthelocalparametuatacuspafflog has localbasisdftasopposedtdg

As Xin isawweidx.qylbg4 rx.cn ink

Geometricinterpretation Mahim H X N wh

Ul vanishingatcuspsshkilND HYX.CN w hfd

whenk z KS Isom MATCH HYXi N R'xWkycUl

szkilND HOCX.CN asHeckeoperators SayptN if cyclicsubgpoforderp

XftolpN Z Ez.subgptv74z.EE

x It I tX N X N Z Pt Estate Ek Cutch

9Version1 Consider E Ez isogenyI

E e

XIN

OnXoCpN we havetwolinebundles Tfw wza naturalmap wz q w

1naive youwww.IEEHYxcni wxok 9 isHYxCn wxok

Ip H XIN W I H XolpN bWz DH XopN h W

T H x n wit

Define tp p pai.sk ToND aSkToCNDExercise Verifythatthis is thesame astheearlierdefinitionRmtForNN level weusethesamesetup butthemap it is a littledifficult

towrite in termsof zh anettofECNofexaetorderNVersion2 whenh 2 E PnCptfEYffff

IT TE order

p.aeX CERT ftp.pntcpkpRealizing f Tim HYX.CN as

ThenTp HYX.CN.sh.cn E sHYxfNnhtoCpD.N E tifx.ln Nx.im

lls

tilxim.ci2 E.tifxaiiniiiocpi.wEgEIoEtiCx.iii.wioy

Forweightk z we alsohavethefollowingconstruction

ftp.tifx.lnl.sh.cnv

tifxfinhtocpD.ITIsHofx.Cns.Nx.cmYHifxIniiajE.tHxa

intropisia thx said

ftp.Y J.CN JATIN nTolp J.CNUpshot

Softy Jj abelianvarietywithHeckeations

HodgeTheoryForanysmoothproj came X la H x 2 CI HIX HKxSo H x 2 Cl thx HKX.IT

In particular H X N 2 a SktCNN SKITCorollary EigenvaluesofTp are algebraicintegers whenh

This is because Tpv H x N 2 th x N 2

hascharpolynomial in 21 1Cansimilarlydefine forplN Up HYX.CN in i HYX.CN ng

X NN nToGoND E 9 E

Y ti I tX 1N Xi N E E

na J N J N

f de 74Na Sd X N X N

EP B E d P

na Ld J N Ji NMoregenerally considerthenaturalmap

If II I End Jim End Hain acommutativealgebra finiterankover 2

Let h.IN imagefthismap whichisfinite12

TheoremThere is a perfectpairing h Nic x SKIN Eh fl a a Chefs

Proof nondegeneracy at Sa If fesdr.cm sit thekiln a chefsYetforh Tu a TnLlp an f o f o

f f f fHereTn is def'dby Tnm Tn Tm whengcdcn.in 1

Tph TpTph i XCpl.ph phzifptNTpkUphifpNnondegeneraeyathilNa If Teh.CN a is suchthat a Tf o tf

Then a Tnf a TnTfl anTfTf off 7 o D

Corollary H X Maio Cl Salman SachN1T u v

h.lnlxoqflatofrkzh.CN xoCh.ln xoEfreeofrk t feeofranktfinited.milArtinian

So HfXdn5Q isfeeofrankzoverh.CN QRmt Theintegralversionmightbebad

normalized eigencuspfoms f Eangf in SCN X

Ibijection Ihomomorphisms tfftp.fapptNdfih.IN sQlf 1C HCUp applN

f HkdXCd def74ve

eigensparequotientsof J.IN LetIf kerlhfihilN ilQlfi 1

Salt D tfDefineAg JdNYIfIµ H xcnn.tl IfH

Then Ag is an abelianvariety over ofdim IQlfi.IQTheGala'srep'sassociated tof is VelAf XiyuRemark f is always atotallyrealoracMfieldm

f ay lolally fat'otallginaginagguadratriextaofatotally

realfield