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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