Lecture23 Modularity liftingtheorem It Wiles 3 5trickandfinishoftheproofJan 6

FermatLastTheorem Theequation al be I fer l 35 primehasno nonzerointegersoltiscasewhen l 3 4 are treatedbyelementarymeans

Supposethere's asofa mayassumethat a b a coprime and b even a Imod4

Steps Considerthe Freycurve E y x x al x be

Generaltheory E y x e x ez x es hasdiscriminant

DE 16 e ez ea e es e

SoforFreycurve Oe 16 all b l Elcheckthat at pIabc p z hasmultiplicativereduction

e g pl a y x2 x be

plc y x x al5 so µyat p z Use a differentWeierstrassmodel

y xy x'tb

x blmultiplicativereduction

TE Timid2XZor 0Generalrecipe conductorof E is NE squarefreeproductofallprimesdividingabckeylemma fee Gallo Aut Ell GkFe isunramifiedatoddprimesplNe

pteProof AnalogoustothecomplextheorythatE E 142 z E q2Thevalueofq isdeterminedbythej invariantofE j E

g g j of jas of j t 744J t

when Ehasmultiplicativereductionat p we canwrite

E Otp OFFIga note yfqj.eu upo zlThentheGal pactinon Ele canbeseenusing1 a pep Ele Ip get74g 1

U

Cpefte e p UH

Theinertiasubgroupcutstrivially ongte ifVpCj werenotdivisiblebyetheng'tI pureIqswillhaveSo fee is unramifiedat p non trivialcutin

Remade Similarargument butmoresophiscated showsthat

if e labc ECe corresponds to aFontaineLaffittemoduleofwto 1

Steps Modularitytheorem Wileswork

theorem E ismodular i e E is associatedto acuspidaleigenformf in Sa TowStep Continuousoddirreduciblerephsp Gatos aGk ft ismodular

Biginput LanglandsTunnell supposethatp Gato Gk E is acontinuousinedrepnwhoseimage in PGL IC is asubgroupofS4 detCpCa t

Thenthereexists a normalized cuspidaleigenformglop Ebitofweight 1 level Ti Nst.bg trace pffrobof for9TN

Coincidencefor l There's afaithfulrepresentation4 Gb Fz GHZFT E G Es.t trlycgD trig modGtf tgeGutsdet CgD detg mod3

Explicitly 4 f i 1 f 1 x 1 El

I J t kaol f o 447 f oSogiven f Hop Galo GkCFD Gk ICE eGh

7modularformglopofweight 1st by trCfCFrobot modEtf wheffnMoreover as pi is ramifiedat 3 as deff Kydmod3 whichisramified

31 level NofgConsider a weight1 level IT 3 Eisensteinseries

EH It 6 X d eint where X d f f ifif D IG

Sotheproductglzs.EE hasweight 2 e eSafTolNDMoreover aq GG EG boy trCpFrobop mod Itf

Sog z EG C SzTocNDmp

e

I canfind an eigenformflophere

f ismodular

Step2b If for our elliptic curve E Fez is irreducible then E ismodularTabsolutely irreducible

otherwiseuptoconjugation fe.siGal Ff E G Gfs

Y x Idetcomplex Asconjugal

mustmapto 1yetdetcfe.deD 1

ByStep2 a fe s ismodularmodularityliftingtheorem E is also

modular

Step Supposethat fez isreducibleTheoen Wiles 3 5trick In thiscase wehavethefollowing

Either E is alreadymodular

or a pang

E 5 irreducible

b 7 A 10 anotherellipticcurve semistable s t

FE5 E FA5

FAis irreducible

Proof Firstassumethatpens isreduciblethen Els has a nontrivialsubgroupfixedbyGato

E15 GalQEdefines a Q point on Xo45

Xo 15 y2txy y x31 2 tox to

Tact Xo15 Q C t o 8 I8 7 7422 7442

fouroftherationalpoints are cusps 0 co toTheotherfourpoints a ellipticcurves areknowntobemodular

LMFDB.org labeling 50at 50a2 50 a3 50a4

If E isnoneofabove Fes isabsolutelyirreducibleProofof b Consider Y 5 EX't5

asmodulispare p f Afp AIR elliptic curvesEC53 Af5 preservingWeilpairingThis is an innerformof XGGDassociated to

f Es Gato Gk Es E Auto XfCsssuchinnerforms are classifiedby H Gato Antos XGGha

Yetgenus x 5 genus XC5 oE EX't5 QXIsHoi PYothere are infinitemanyelliptic curves satisfying b

For b consideranothermodular curve

Y't5 3 i R ta f AIR gAB 4 EYE51GEAID gelicsulgpoforder3

SoanyQ pointof y153 willsatisfies b butnotCbsYet genus YC5,3 1

bio over E YC5 3 T 5 nToDTfByFaltingsproofofModell'sconjecture YCs3 hasonlyfinitelymanyQptsinfinitelymanyelliptic curves satisfiesCbs

Rmd Semistableconditions are openconditions

Steps E s f ESa ToN1 yet Fee Ffe is unramifiedat odd primesplNRibet's level loweringthan a gESaTOG st f g mod l

needmorebackgroundthat wecan'tcovergiventhetimeofthissemester

StephThere's no nonzero cuspfour on XoG of wt 2