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