Lecture20 Eichler Shimura relations

Dec 23

Now maketheconstruction over or7L

Y.IN representsthefundis MIN Sahai SetsN 4 Eellipticcurvets

Salle.it z pn aECN3

YnlN is representedby an affinecurve spualf calledtheopenmodularcurve

X N compatificationof yin calledthe compatifiedmoddename

Ci XIN Y N

Fet Ey.cn E Esm

her he bing.Y.gsiiimeEesTgmpMsYheI E.ee

Yin X N e X N Esmzewsation

Define wi ertcsyx.cn

SWITCH i H X n kfcStill Kodaira Spencer isomorphismholds

Nx.cnbyc w

In particular SGTINDEHYXDN.dx.cn X Niv nTo DU b

Heckeoperators comesfrom X.CN XIN

algebraically Te Up Ld ICN JINLet f be a normalizedcuspidaleigenformsofwt 2 level NNDefine Af Jalm If Jin forIg Ker hln E

ThentheGaloisreph associated toAg is

A Ve Ag YojiThinkof a representationwithcoeffinf Qe Qlfk

f ale

Theorem Eichler Shimura TrlpffFrobp ap if ptNStep Understandthe HeckecorrespondenceTp

X NN nTofpParametizing E C P Eelliptic curve

csubgpoforderp.PEEofexad orde.lv

it t I aXIN XIN EP Ek P14CThen Tp is givenby JIN Jacxfr.cnnroCpD N QpWeneedto understand the specialfiberofthismorphism

Stepe Specialfiberof XChin nToCpDFact Themodularproblemworks over Ip XIXGTInhtdpDEXGTHDntolpD XT.nlT NTI

µ Frob i ob idHINT

X N X N X N o o

M't

tanation Let jrd gss

supersingularpoints x.TN

Erd theptorsionsubgpof Edsits in a canonicalexactsequenceIf gordmutt gord gord.it o

YIN s.t.ateahfp pointxEY.TN rd theexactsequencebecomesc pepo E

DCpo Hap so

o Eordmittep gordy gordie't o

roughlyafamilyoffp roughly afamilyof74ozRemade Cord ErdPhaskernelexactlyErdmittep

or

cord't Eord gordimultqy

ji Y.CN i Y Tin nToCpDmµ extendsto X.TN XCNNnToCp

Ji l 1Erdp l o gord p gord.multq.gg

j Y.INT'd KT N nToCp extends to X.INT XGINnToCpsZEord p l s eordfgord.muhqg.IMp Erdwordmultepy

Fact X TCN n ToCpi Imj u ImjaIa Ia

Then y.Tnjrdy.TN d gord

a i IYAHNnToCpf

wda Er eoryeordmuhcpeoroyfi.my

Y N HS HSordCp gordip

Step3 InducedmaponJacobians

IX N Fp

1 frostyjiu torus J r stopFp JIA JIG

IT ifJx.c.mg E

So Tp Lp Frob Frob

Thus weverifythat Frobftp.Frob plps o

ThisgivestheEichler Shimurarelation D

Fact On Skt N Q JTpactionissemisimple ptvHYX.cn wk butUpaetiusistypicallynot pln

Salt Nio Ot Sk TIN DT actsbymultb I

ShaikhQ µ.sk 1ilNDhpgpw2lpadsbymult.byXpUI vector

spare over ftp.ptN

f fReigenform canassociateaGabisrephLet 5 fpprimeplnlulo.lt Assumethat ltNConsider 0 ftp.pfN EndfgftilNb0

ocomparingtoearlierhimdenotesthesubalginchedingupaihinsforplN

Leth.IN denoteitsimage calledtheHeckealgebrawhich is finiteandflat aeroh N is a completenoetheriansemi localringandhilntmma.iedeaehiCNm

Then h.in ImfEfTp ptNJ aEnd SnIkEEenlargingE SKINN hplµnifnecessyay Tf End

multiplicity

fGalas GUEappearinginskfr.cm InthiseveryTpactsbyscalarTYE

Pictureof Spechin EIa i

p Pf

nine1 SpecF E

reductionmodeNode mail idealsBofh.CN 5fmax'lidealofh.fEH

Ip

In A't S tLL IE

ftp.hcpfsystemofeigenvaluesftp.v sfsystemofmodleigenvahesfIppfnPtv f I

GalvisrephspGal GUO fsemisimple residualGaloisrep'm

crystallineofwtco.tn e lg F Gal s GWF

f P i s

notinjective

Proposition Let f a.frdenoteallsemisimpleresidualrepresentationsappearing inSalt mForeachpi theassociatedmail idealofh N isMpi Tp tipifrobp o

Then hi N II hi DmpA n

SakCmO Serkin Dmp

MoreoverSpNiv O

mpE p.ttssqSh N E

ftp.hlperobpD

pwRemarkIn somecases one canshowthat5h47N Dmp is afinitefeemodule overh Niggbutnotalways

Now wefix an absolutelyirreducibleresidualGaloisrepn f Gal s GUERp s universaldeformation rigforf as rephsofGod is

To proof in nextlectureOFTp ptn Rp s hasdenseimageTp 1 tr puniuffrobp

ofTp tiptop PtNI RpsNow OfTp tiptop plant h Dmp

f fatuzation Ih NmpoERf'sEuniversalpropertyofRf's

pappearinginShkinbc wehaveliftspGatosGtepss pMoreover wemaythenview Skft N O as anRpsmodule

Veryimportantremark earlier hi N zMp CoTp trp bp ptv

I l p p p i ltallowtomissanyfinitelymanyprimeshere

i e candefineMpusinglessprimes p

Moreprecisely Rfh

deformationringthat is crystallineate wt HTwts ohDH

Rpsfgeays.co.kridealforcrystallinedeformations

Fact SerTin Gmpis an RhDmodule i.e IET hDannihilatesSkftasDup

b c itannihilatesSkfTiGvl.dmp E.JTechnicalissue willconsider Sh h N drip HomSplti n Gmp 0 instead

47985Cook

Recall pips deformationringoff togetherwithframeatTI

Rp 4h

defamationring fpthat iscrystallinef HTwts o hDate aframeatTAs f is absolutelyirreducible we haveDT r T

Rf s Rp s Opa Thenaturalmap isRps pips

pip Mslook

pipit's Offit Rpt fix x3 11

Similarly we define Serft n 6inpSmfh n OIy.xopy.ggcok Pip

v

pyfgays.cohD Sh T N DmpIx Xz TI

IRE Rp ftp.etays.co

hD

Wewill laterconsiderthisstructure spfpppj.ays.cohD Sh NNdigit

I p n

f Ricky YrIkz r

SpfRio