Lecture16 Galois representations appearing ingeometryDec 7

Let k be a fieldX proper

smoothvariety1hForevery l charkis HietXpegQe ith etalecohomology o e i e zdinX

U sepGallk kwhen chark o assumingno issuewithcardinality andfi E

Then H XpQe Hising HEY eas Qevectorspares

But HietXaQe hastheadditionalGaloisactin

whenh is a finitefield f Fg etfH XaQe 99g geometricFrobenius7

finitedimensional vectorspace

Deligne'sWeilConjecture Thecharacteristicpolynomial

det x id Gg H XpQe x tan ix t ao E Il x

has 2 coefficients

is independentoftheprime l includingthedegree i e dimHitXaQeforanyzero a cQT andanyembedding Etc I ate9ik

callsuch a a Weil fitnumberConjecture Sqcutin on HietXf Qe is semisimple i.e noJordanblocks

BmtLefschetztrueformula if ai x denotestheg eigenvalues onHit XityQe

I s f f g g gdimX

etenvm.xat.am c.si 5

when k Qp notethat we nowallow l p as charQp o

Case1 Itp X admits apropersmoothmodel over Ipi e I H X

Ipsromper 1SpecIp c SpecQp

Then HiaHapQe Hieiftp.QeU U

Galp

GdFp

SotheGaloisrep'nHitGapQe ofGal isunramifiedtheFrobeniuseigenvalues are 9 2Weilnumbers

theeigenvalues are independent of l

Case2 ftp Xpropersmoothgeneral SpecQpGal G Hit Xa Qe

corollaryofWeightmonodromyConjectureLet r N V denotetheWeilDelignerep'mattachedtoHitHoiQer N V is independentof lAs g Ng pN N sends g a eigenspareto f Feigenspare

N isnilpotentGeneralfait JacobsonMorozov V can be decomposed as thedirectsumof

Pi n Nes NB N_Nap N INT nI sNnp to

0

Ian N a in N I un N l un N I un

Pi in NR n i N'pin NYP.in NmPixn iN

PNPin.zN N2Pinz I N Pi n z

Po 0KerN

suchthateachPi is f stableTheWMCsaysthattheg eigenvaluesoneachNIPi.im are gilitm

2J Weilnumbers

sayNipitmhasDeligneweight itm gAseachN chain is centered at i wesay r N V haspureDeligneweight i

Case3 f Qp l p A propersmoothoverXpGallopGHit HqQp Hi is a crystalline rep'ns

Dais Hi 94 geomFob

Theorem det x id g DaisfHi c IG isequalto detfx.id ftrobpitietffo.io

All 9 eigenvalues are p Weilnumbers forage

It's aconjecturethattheyactin is semisimpleThere's a canonicalisomorphismDais Hi EHIRHop offiltered Qp usTheHodgeTateweightsof Dais Hi are preciselythosej's sit

Hiftp IT to withmultiplicitybeingthedimensions

Eg ifAtpis anabelianvarietyofdin n thenforH H'etAapQpDais H Qp HTwts fo n times 1 ntimes

all 4 eigenvalueshaveabsolutevalueTp

Caselt h p l p X propersmooth Qp

There are waystoobtain aWDrepnoutof HietHojoQpIt isconjecturedthatthisWDrepn isthesame astheonefrom l adictheory

Now let k Q X a propersmoothvariety Q

HICx Hit Xa Qe 5Galo

Foreachprimep Hie xGa

pHit Xa Qe

So we can viewHi X as a repn f Gal ethenitenjoysthepropertiesforX e as discussedabove

Then Hei X teprime forms a compatiblesystemofGaloisrepns asdefinedbelow

Define Let L be a numberfieldand

foreachprime XofL let Pa Galo Gln LI be acont semisimplerepnca wesaythatIpal isweaklycompatible if afinitesetofplanes Sof Qsit if pets andHp thenA is unramified atp

and charp ohp x E LIxIEIsLx is independentof 1

b Wecall aweaklycompatiblesystem1pal compatible ifeis f p s t Xlp therestrictionpalGal p is deRhain

whenpetsthis isequivalentto requireAGal ptobecrystallineCii VpsitHp theset HTDarplodpl is independentofpCiii If petS andHp then f todayis crystalline

and det x id ol Daidpxloalo.pl charpCFrobp x

iv let c a complexconjugation then tr fCc E7L is independentof bc A compatiblesystem1pal is strictlycompatible if

my system f dlyany ifif Vp theFobsemisimpleassociatedWDreph WDPps ofWasp

is independentof dtheorem If Hap issmoothprojectivethenIHit Xa Qe is a compatiblesystem10

If Xia is an abelianvariety thenit isstrictlycompatible a semisimplicity isautomatic

Conjecture

a Any a g_l adicrepresentation ispartof a strictlycompatiblesystemmeans Pe is anramifiedoutside afinitesetofplanes

Pelad e is deRham PelGal iscrystalline

z weaklycompatible strictlycompatible comesfromsummandofsometHie X fFontaineMayurConjecture Everyalgebraic l adicrepresentation comesfromgeometry

Dimensionnunneries

ByConjecture in any algebraic L adicrep'nshouldcomefromgeometrySofixingramificationandHTwts it isexpectedthatsuchrephis

SeethisfromdeformationtheoryFix p Go Gk Fe F

teldimover 0 isexpected to be

Xp 0 if HTwts detpcc nicecase

I DL reldimeo

negative otherwise nsfyhngofobhafq.gg

FLdet det det at v lXpe IIeXp TutXp dim o I f ifdetfed

1 odd

q g ifdelpas i evendim p to

distinctHTwts foreveryvflsameHTwts