a cuspidal level EFIbicmr.pku.edu.cn/~lxiao/2020fall/Lecture1.pdfLecture 1 l adicrepresentations...
Transcript of a cuspidal level EFIbicmr.pku.edu.cn/~lxiao/2020fall/Lecture1.pdfLecture 1 l adicrepresentations...
Lecture 1 l adicrepresentationsoflocalGaloisgroupsSep21
o Introduction
God Fermat'sLastTheorem
If l 35 is a prime then altbe D has no nonzerointegersolutionStep i Supposethere's a solin a b c toFermat'sequation
ConsidertheFreycurve y x x al x be elliptic curve E overQzl
satisfying i E hassemistablereductionwithdiscriminant 0 16 abcconductor N productofprimefactorspofabc odd
2 Fee Gato Aut Ell Gk Fe isunramifieolatea.LYabc
Steps TaniyamaShimuraWei1 Conjecture provedby AndrewWiles
E is associated with a cuspidal modulareigenform ffg an7 oflevelNandweight2i e Vpt N ap p 1 EFI
stinstoy x al x be in lx.DE p2 is
Step3 Serre Ribet'slevel lowering
Since Fed is unramifiedateachoddplN7 a cuspidalmodulareigenform f oflevel To 2 weight 2
s t antf's an f mod l t CnN 1
BUT there'sNOcuspidalmodularformoflevel ToG aweight2
bkXoG p e Ho pl rip 0
IdeaforprovingStep2
ByEichler Shimura weight2 cuspidal eigenform f j GaloisrepresentationPfWilestheorem
Mainidea f is pIlI mod l N mod l
y
f a si PfDeformatinringofmodularforms UniversaldeformationofGaloisreps
T deformationringR
ModularityLiftingTheorem RETTheexistenceof f followsfrom afundamentalworkofLanglandsTunnell
on solvablebasechange
I l adic representations l is aprimenumber
Notation Let 4k be a Galoisexthoffieldsie algebraic seperablenormal VaEL itsminimalpolynomial over K splitsover L
andallrootshavemultiplicityonewhen 4k isfinite Gyk Gal 4k denotesitsGaloisgroupIn general Gal yn bin Gal mga profinitegroup
4Mlk compacttotallydisconnected
MlkfiniteGalois
Cn formalGaloistheory Intermediatefields M closedsubgroup
K E M E L ofGal LIKM 1 Gal YmH e l H
Write KMP separableclosureof K E algebraicclosureof KSimplywrite GK Gal Ksm
Definition AGaloisrepresentation is a continuousgrouphomomorphismGk Gln RwhereR is a topologicalring e g R E Qe Ie QI Fe
Example Kyd GK If forgeGK G Sen G
If p Gk Gln R GL V for R aRe algebra
f p f gewrite pens VCn for P Xcynd n E7L
Lemme Let Tbe a compacttopologicalgroup e.g a Galoisgroup and
let P T Gln Qe be a continuoushomomorphismThenthereexists a finiteextension 4Qe suchthat p t CGln L
uptoconjugation wemayassumethat pCt CGln QRemart Easiertoworkwithrephswithcuffs in LorOL bk OTE isnotcomplete
Proof pH iscompact Hausdorff
ByBaire CategoryTheoremHI intersectionofcountablymanyopendensesubsetsofPG is stilldense inpG
V ft nGULDNow P Ty finite
TFclosedinph
If p t nGln L doesnotcontain anopensubsetitscomplement isopen dense
By CH wegetthecomplementofRHS isdenseSo pf nGln L contains anopensubset is anopensubgp forsome LYet pH PCMnGLna isfinite
So bymaking L abitlarger containingcorsetsof PK pHrankget p i T Gln L
To conjugateimageof pintoGln Q need tofind an Q lathi A E L stableunderTthen if A g OE forgCGln L then p T gGln Q g
Note p GLnon E T isopen finiteindex
Pick representatives x or ofthecosetsset A p t OE Ii di oOEn D
fieldDefinition If p T Gln h GL v is a representation andthere's a Tstablefiltration
f I p l I i fO Vo EV E E Vr V
sit Vinlui is an irreducible rep'nSet pss OfVitiIvi calledthesemisimplificationof pThis is independentofthechoiceoffiltration
Examplepig alot bday
thenpss a d
whenk L a finiteextensionofQp maytake a T stable Q latticeA E VThen we obtain a residualrepresentation Fa T GL Iq A
Remark ThesemisimplificationofPI is independentofthechoiceofthe latticeAwithdiscretetopology
Proposition Brauer Niscott let p p T Gluck betworepresentations
Supposeeither a VgET det x In flop det x In Big e kCx
or b charh o or o AgET tf Cgs tr pigsThen p pass
dark n suffices
Proof Mayassumethat bothp ep are semisimple
Let Vi Vr beall distinct irreduciblefactors appearing in P pwithmultiplicitymi ami respectively
Sufficestoshowthatmic miClaim Fix i ei e kCt sit ei is identity on Vi buttrivial onVjforj tiPf Let R Im f kCt QIEnd
semisimplek algebra i.e nilpotentideal o
R is semisimple finitedim'llk
ByArtin Wedderburntheorem R FIMyDj G Vr VrAs Vi's are distinctsimpleR modules NyDj has auniquesimplemod
ry j g my
s r theaitionistermwise DCaselat Pickeiasgiven then det x I e Vj g x 1
dim i
jahftofeiinf.fr xd.MY i jI
bqueso det x I f Ei Dm dimvi.ydimfi mihd.my
forallge detlx.tt pace mi mi
case b t pilei mi dimVi so needchark dimpi toconclude D
2 Galvisgroupoflocalfields willfocusonthiscase
Definition Inthislectureseries localfield finiteextensionK ofOtporofFpktK 20k 0k14gr Kk Hq 9 p
Fix a systemofrootsofunity 5 G ppnsit 5mm L when nlm
keep
l wildinertia pro pgroupk a meansthattheconjugationactionofktame kwfnfok.pk f GyeonGalfktamelkur isby1 1124
tip l cyclotomiccharacterKur Khun ptn1 Gii Ger Ik
GE Gha IifGal Pottle fig242 72Ul ul
4 Frobks 11
normalizetogeometricFrobenius inverseof xi
x9tgGal Ktamelkur is 1124 IMGlip l
S Hp
g I sits srt gtfo 5 Top
c s i g tindependentofthechoiceoffo
calledatamegeneratorbutdependsonthechoiceoff
whytwistbyu Compare 1cg vs I Igf forFaliftoffinGakktate
Igf Hiro Ig Efron offs F Groat 5n AntoT
anothernthrootofuniformiger
Write itse Gal Ktamelkur I G 744 Thisonlydependsonchoices
ofGem m