Introductiontomodular curves andSiegelmodularvarieties

1 Adelicdescriptionofmodularair

Let N34 consider h N 9 CSh Z Nlc N d i

Thenthemodular curve is Yi N e p Nth whereG ZEE Im t o

LetAf finiteadelesofIQtheoremThereis an isomorphism

pWL E g Gk Hf µ

where ft El IR T.TN f Ebd CGb74 c d i ENE

TplIpNeed a blackboin StrongApproximation Sk Q isdense in 5kHAD

In general if G is a simplyconnectedsimplegroupover anumberfield Fand v is aplanof F s t G Fv isnotcompactthen G F isdense in GHEILadelesawayfromv

E.g G In Span DENIsignage EH hermitian

spareforFFCoe Sk Ag E S4 Q TIN

IfGhb I canfirstmodifyanelementinGkHtt intoSbRemart 9Thisargumentneedtheclassgroupof 2 tobetrivial

ProofofheoremII Consider Ghia tht Gk Att pinByCor everycosetcanberepresentedby z D E ft x Gb AtfTheambiguitylies in MTN GHQ nTIN f Ebd cGb 2 I NIe i dSothattheabovedoublecoset is FINALSbl2 hasindex2 in Gk 2 as det Gk I 2E ft

rightahhhh

2 Moduli interpretations

Recall Anellipticcurve E over E takestheformof E Q Est 12 ItButTEJ is uniquelydetermineduptotheactionofslate

Theorem2 AssumeN 4 Yi N isthemodulispaceofellipticcurveswith an Ntorsionpointthat is thereexists Eun universalelliptic awe ftp.hfezrtfonrthmeang.bffteweinlles.ee In

feetonsfta togetherwith inKnx EINYi N y an embeddingofgroupscheme

s t forevery E schemes togetherwithan ellipticcurve E Is a an embedding i 74NasueINthereexists auniquemorphism a S i Y s t E aEminand i a imir

Proof Over TEff we define Et 147L Z i iuniv 74NE E IN1 TV

Then keyvariesholomorphically as a moresThen E quotientofthefamilybytheh N cutin D

Willgivea slightlydifferentapproachtogettheadeliaparametrization

F E Tatemoduleof E BEEEnIf E is over a Q schemeS F E is an etale E sheafofrank2

Claim If Ela is anellipticcurve giving anembedding i 74Nz s ECN isequivalentto

a TIN orbit of isomorphisms 7202 TEUrins

Proof Note I E EIN Stabilizerof i ispreciselyRTNBernanke Language issue If E is over a localnoetherianQ schemes wewillneedtotake

a H S s stable TIN orbitof isomorphisms E

Or in afancierlanguage Isom E T E is an etaleGL E torsor

f language

a level N structure is a sectionof Isom Ea TE r.TNAnotherproofofTheorem 1Let E i be an ellipticcurve over1Cwith anembedding i 74Na EEN

Asabove I correspondsto a TIN orbitof isomorphismsy 7202 7 E

Theelliptic une Ele hasthreefeaturesBettihomology Hite eEtoilehomology Hitfe z pfe

comparison HiEid 2 I Hitfe

deRhamfiltration o HoleR'e HtpEik HCEOe o

Lie o

Companion

dundization o weye HYNEK thEfe E HYMER

Hoff're't sE HIREYE

TheoremIThere is a bijection Yi n E GH tht GkAthynProof Startingwith E y We choose an isomorphism p H Ela Q Q02

Then we can construct elements in GkAg and tj as followsAGE 72

020 10 Is thetfe.TLxOzQ It Eid 2 Ag f AI

givesan elementofEGWAgWeye E H Ela H Ei I 2

gives anelement ee P e canprovethat itdoesnotbelongtoB

So weget Ey mo t.gg CGtxGk AgThisassociationdepends on choiceofy intheFIN orbit ggmodTIN

choiceofisomp if p hop thengj e h gf h e

Puttingthesetogethergives amapYinka old IS GkAflhitsConversely given T.gg Eff Gk Ag

we can define E Ee ice

sthere's a somewhatnatural isomP H Ee Q

2

1 o

oil

EI checkthatthefiltrationcorrespondsto e

gf gives a naturalIsom AI e HETEe Ag byinvertingthearguentaboveUI UI702 s HitEe E

Tdonotmatch

Say Mik02E Hitler E e 7502

Consider M02 Hit EtFL

C E Ee MMaMMzH 2 MMzHftfee.TL

Hg is a subgroupof EeSet E E IG thenthequasi isogeny

E E Eeinduces the'tE E Htt E E Hitler E

M02E AqEf HftfeeAg

Eg iswhatweneeded D

3 Moduliproblems over Q andIcpBlackBoxTheorem AssumeN34

ThefunctorMaSahibwe

Setsisomclassesof E.g Eellipticcurve Is

S M S OneachconnectedcomponentofS fixingageometricpoint5

y FIE is a I S s stableTIN orbitofis representableby a

Geometrically

connected smooth ame y N overisomorphism

1 by ally 1Remarks This isequivalenttotheearliermoduliproblembutwe caneasilymodifyFTDto any

opencompartsubgroupK E GK E Exercise ReformulatethemoduliproblemforKeGHgBut one needstobecarefulwhentalkingaboutintegralversions

Letpbe a primenumber

Icp localizationatCp ICP Tef If AIM 724 zQLetKPE GkTEP be an opencompactsubgroup

K GbZp KP so nolevel pCandefine a functor Mkp Schleif Sets

isomclassesof Eg Eellipticcurve IsS M S OneachconnectedcomponentofS fixingageometricpoint5

y724 FEI is a a s s stableKPorbitofisoms

It is represented by a schemesmooth over74ps

4 Siegelmodulispace

Fix V I 28 0 Rei togetherwith a symplecticform

1 1 V V i 2 givenbymatrix fEgG Gsp v GSpg isthegroupsitforevery2 algebra S

GSpg S theGhg s c e 5 thxhyf chxy foreveryxyE V I1 Spag

GSpgc Gm 1

Fix anopencompactsubgroupK EGSpgEtheorem Mumford Forksufficientlysmall thefollowingfunctor

Mk Schwe

SetsA Xy A is anabelianvarietyover S

X A Au is a principalpolarization

S MK sychoosing a geompointsoneachconncompofS

I vg g pond ampof

7 2 7 A is a S 5 stableKorbitofisoms

sit VE x Vz 7,4 forsomeCEE

A x A T.INis representableby a smoothvariety Shk G over IQ

Now weexplaintheterms

EveryabelianvarietyA has a dualabelianvarietyAVGiven a polarization 1 A Au we get

A n xACn Afn xAhn apen bilinearsymplectic

Takingtheinverselimit TTA x'TA fifth ICDbut724 is non canonicallyisomorphicto72 hencethesimilitudefactor c

Theorem Let tyg fZ ESymn E InnZ o or Im E so Imetanstotallypositive

Then She G a Whig GCAfykProof Asbefore each A X g EShK G E gives

Bettihomology H Ala 2

Etatehomology HitAla E f AHodgefiltration 0 Waye HIYA1a tiene 10

115

Hi Ala 2 zaMumford'sAVbook If wewriteA EMA for a lattice then

H Aki 2 A A R Edhas anaturalcomplexstructure FAppapT

firing i t t i

sit Xp Ju Ju Xp un and Xp u Ju ofor a 10Principalpolarization X is perfect inducing A Homla 2

P 1 per gupto11Togetthecomplexuniformization choose an isom

a Hi Alaba X v 4 compatiblewiththesymplecticforms

ThenYa If A HIYAAH Hi Aid.TLx0ztAf

VxotAfginesanekmentofGSpgCtAf

Hodgefiltration war HI Ale A a Lien

giesA R a complexstatureA RSoHodgefiltration a complexstructure on A IR

4CSpy IR J I Ulf J ispositivedefEBD ornegdefbio x isupto11

get EBD GIgEff exercise