the al ibicmr.pku.edu.cn/~lxiao/2020fall/lecture23.pdf · 2020. 12. 31. · lecture 23...
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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
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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
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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
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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
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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