SERRE'S MODULARITY CONJECTURE II

Jean-Pierre Serre

Description

SERRE'S MODULARITY CONJECTURE (II) CHANDRASHEKHAR KHARE AND JEAN-PIERRE WINTENBERGER to Jean-Pierre Serre Abstract. We provide proofs of Theorems 4.1 and 5.1 of [32]. Contents 1. Introduction 2 1.1. Some features of our work 3 1.2. Notation 4 2. Deformation rings: the general framework 5 2.1. CNLO-algebras 5 2.2. Lifts and deformations of representations of profinite groups 6 2.3. Points and tensor products of CNLO algebras 8 2.4. Quotients by group actions for functors represented by CNLO algebras 10 2.5. Diagonalizable groups 12 2.6. Truncations and chunks 12 2.7. Inertia-rigid deformations 14 2.8. Resolutions of framed deformations 16 3. Structure of certain local deformation rings 17 3.1. The case v =∞ 19 3.2. The case of v above p 21 3.3. The case of a finite place v not above p 31 4. Global deformation rings: basics and presentations 36 4.1. Basics 37 4.2. Presentations 42 5. Galois cohomology: auxiliary primes and twists 45 5.1. Generalities on twists 45 5.2. Freeness of action by twists. 46 5.3. A useful lemma 46 5.4. p > 2 47 5.5. p = 2 47 5.6. Action of inertia of auxiliary primes 52 CK was partially supported by NSF grants DMS 0355528 and DMS 0653821, the Miller Institute for Basic Research in Science, University of California Berkeley, and a Guggenheim fellowship.

• group


• modularity lifting


• wiles


• field theory


• auxiliary primes


• then all members


• solvable image when


• when doing


• results proved

Sujets

• Institut universitaire de France
• Taylor
• Field theory

Informations

• Publié par: pefav
• Nombre de lectures: 21
• Langue: English

Extrait

SERRE’SMODULARITYCONJECTURE(II)
CHANDRASHEKHARKHAREANDJEAN-PIERREWINTENBERGER
toJean-PierreSerre
Abstract.
WeprovideproofsofTheorems4.1and5.1of[32].
Contents
1.Introduction2
1.1.Somefeaturesofourwork3
1.2.Notation4
2.Deformationrings:thegeneralframework5
2.1.CNL
O
-algebras5
2.2.Liftsanddeformationsofrepresentationsofprofinitegroups6
2.3.PointsandtensorproductsofCNL
O
algebras8
2.4.QuotientsbygroupactionsforfunctorsrepresentedbyCNL
O
algebras10
2.5.Diagonalizablegroups12
2.6.Truncationsandchunks12
2.7.Inertia-rigiddeformations14
2.8.Resolutionsofframeddeformations16
3.Structureofcertainlocaldeformationrings17
3.1.Thecase
v
=
∞
19
3.2.Thecaseof
v
above
p
21
3.3.Thecaseofafiniteplace
v
notabove
p
31
4.Globaldeformationrings:basicsandpresentations36
4.1.Basics37
4.2.Presentations42
5.Galoiscohomology:auxiliaryprimesandtwists45
5.1.Generalitiesontwists45
5.2.Freenessofactionbytwists.46
5.3.Ausefullemma46
5.4.
p>
247
5.5.
p
=247
5.6.Actionofinertiaofauxiliaryprimes52
CKwaspartiallysupportedbyNSFgrantsDMS0355528andDMS0653821,the
MillerInstituteforBasicResearchinScience,UniversityofCaliforniaBerkeley,anda
Guggenheimfellowship.
JPWismemberoftheInstitutUniversitairedeFrance.
1

2CHANDRASHEKHARKHAREANDJ-P.WINTENBERGER
6.Taylor’spotentialversionofSerre’sconjecture
7.
p
-adicmodularformsondefinitequaternionalgebras
7.1.Signsofsomeunramifiedcharacters
7.2.Isotropygroups
7.3.Basechangeandisotropygroups
7.4.Δ
Q
-freenessinpresenceofisotropy
7.5.Twistsofmodularformsfor
p
=2
7.6.Afewmorepreliminaries
8.Modularliftswithprescribedlocalproperties
8.1.Fixingdeterminants
8.2.Minimalat
p
modularliftsandlevel-lowering
8.3.Liftingdata
8.4.Liftingswithprescribedlocalproperties:Theorem8.4
9.
R
=
T
theorems
9.1.Taylor-Wilessystems
9.2.ApplicationstomodularityofGaloisrepresentations
10.ProofofTheorems4.1and5.1of[32]
10.1.Finitenessofdeformationrings
10.2.ProofofTheorem4.1of[32]
10.3.ProofofTheorem5.1of[32]
11.Acknowledgements
References

35750606162656668696963747777778888809192939

1.
Introduction
Wefixarepresentation
ρ
¯:
G
Q
→
GL
2
(
F
)with
F
afinitefieldofcharacter-
istic
p
,thatisof
S
-type(oddandabsolutelyirreducible),2
≤
k
(
ρ
¯)
≤
p
+1if
p>
2.Weassumethat
ρ
¯hasnon-solvableimagewhen
p
=2,and
ρ
¯
|
Q
(

p
)
is
absolutelyirreduciblewhen
p>
2.Foranumberfield
F
weset
ρ
¯
F
:=
ρ
¯
|
G
F
.
InthispartweprovideproofsofthetechnicalresultsTheorems4.1and5.1
statedin[32].WeadaptthemethodsofWiles,Taylor-WilesandKisin(see
[62],[60],[33])toprovetheneededmodularityliftingresults(seeProposition
9.2andTheorem9.7below).WealsoneedtogeneraliseslightlyTaylor’s
potentialmodularityliftingresultsin[54]and[55](seeTheorem6.1below)
tohaveitinaformsuitedtoourneeds.
ModularityliftingresultsprovedhereleadtotheproofofTheorem4.1of
[32].Modularityliftingresultswhencombinedwithpresentationresultsfor
deformationringsduetoBo¨ckle[4](seeProposition4.5below),andTaylor’s
potentialversionofSerre’sconjecture,leadbythemethodof[31]and[30]to
theexistenceof
p
-adicliftsassertedinTheorem5.1of[32](seeCorollary4.7
below).Theseliftsaremadepartofcompatiblesystemsusingargumentsof
Taylor(see5.3.3of[59])andDieulefait(see[21],[63]).

SERRE’SMODULARITYCONJECTURE3
1.1.
Somefeaturesofourwork.
Weremarkontheargumentsinthe
paperwhichdifferfromthemainreferencesweuse:
–Thedefinitionoflocaldeformationringsfollows[33],buttherearesome
noveltiesintheformalismweuseandthecalculationswemake.Wealso
followKisin’ssuggestionofworkingwithframeddeformationsatfiniteand
infiniteplace
–Weovercomenon-neatnessproblemsencounteredinprovingproperties
ofspacesofmodularformsbytheargumentsusedin[5](seeitsappendix).
–Wegiveadifferentproofofthelowerboundsondimensionsofglobal
deformationrings(seeProposition4.5),thatareneededinourmethodfor
producingcharacteristic0liftsofglobalmod
p
Galoisrepresentationswith
prescribedramificationbehaviour,thanintheliterature.Theproofismore
consistentlyrelativetothestructureoftheseglobaldeformationringsas
algebrasovercertainlocaldeformationrings.
–Forresultsaboutpresentationsofdeformationringswefixthedetermi-
nantsofthedeformationsweconsider.Thereasonwesucceedintermsof
thenumericsintheWiles’formula(see(2)ofSection4.2below)producing
apositivelowerboundonthedimensionoftheseringsisthatthereduced
tangentspacesofthedeformationringsweconsider,thatparametrisede-
formationsthatinparticularhavefixeddeterminants,areisomorphicto
theimagesofcertaincohomologygroupswithAd
0
(
ρ
¯)coefficientsinrelated
cohomologygroupswithAd(
ρ
¯)coefficients.
–Wearriveatwhathasturnedouttobethemaininnovationofthe
paper:thepatchingargumenttoprovemodularityof2-adiclifts.For
p
=2
whendoingthepatchingofdeformationandHeckeringsin
§
9.1,wefixde-
terminantslocallyatplacesintheset
S
forwhichthelocaldeformationrings
areputinthecoefficients(see
§
9.1).Asin[20],wedonotfixdeterminant
locallyatauxiliaryprimes.Wedothisbecause,astherefereepointedout
tous,for
p
=2,theusualargumentforgettingauxiliaryprimesfixingthe
determinantdoesnotwork.Wethenneedtoproveequalityofdimensions
ofringsobtainedbypatchingdeformationringsandHeckerings,fixingthe
determinant.Tobesurethatfixingthedeterminantgivetherightnumber
ofrelations,weusetwists.Weneedtoimposeafurtherconditiononthe
auxiliaryprimes
Q
n
intheTaylor-Wilespatchingargument,tomakecertain
classgroupsgrow(whichallowsustoustheformalismthatweintroduce
in
§
2.4,2.5and2.6),whichisoneofthenovelfeaturesoftheworkhere(cf.
§
5.5).WefinditmiraculousthatthemethodofWilesandTaylor-Wilesin
[62]and[60]canbemadetoworkwithmodifications
inextremis
.(Fora
sketchoftheproof,thereadermightlookat[64].)
ThroughoutthepaperthefactthatwecanprovemodularityofGalois
representationsaftersolvablebasechangeisexploitedextensivelyfollowing
theworkofSkinner-Wilesin[52].

4CHANDRASHEKHARKHAREANDJ-P.WINTENBERGER
ThedebtthatthispaperowestotheworkofWiles,Taylor-Wiles,Skinner-
Wiles,Diamond,FujiwaraandKisin(see[62],[60],[52],[16],[33])onmodu-
larityliftingtheorems,andtheworkofTayloronthepotentialversionof
Serre’sconjecture(see[54],[55])willbereadilyvisibletoreaders.
1.2.
Notation.
Welet
p
bearationalprime.Let
E
beafiniteextension
of
Q
p
andcall
O
theringofintegersof
E
.Let
π
beauniformizerof
O
and
let
F
betheresiduefield.
For
F
anumberfield,
Q
⊂
F
⊂
Q
,wewrite
G
F
fortheGaloisgroup
of
Q
/F
.For
v
aprime/placeof
F
,wemeanby
D
v
(resp.,
I
v
when
v
is
afiniteplace)adecomposition(resp.,inertia)subgroupof
G
F
at
v
.We
denoteby
N
(
v
)thecardinalityoftheresiduefield
k
v
at
v
.Wedenoteby
F
v
acompletionof
F
at
v
anddenoteby
O
F
v
theringofintegersof
F
v
,and
sometimessuppress
F
fromthenotation.Wedenote
O
F
p
=Π
v
O
F
v
with
theproductoverplaces
v
of
F
aboveaprime
p
of
Q
.Foreachplace
p
of
Q
,
wefixembeddings
ι
p
of
Q
initscompletions
Q
p
.
Denoteby
χ
p
the
p
-adiccyclotomiccharacter,and
ω
p
theTeichmu¨llerlift
ofthemod
p
cyclotomiccharacter
χ
p
(thelatterbeingthereductionmod
p
of
χ
p
).Byabuseofnotationwealsodenoteby
ω
p
the
ℓ
-adiccharacter
ι
ℓ
ι
p
−
1
(
ω
p
)
foranyprime
ℓ
:thisshouldnotcauseconfusionasfromthecontextitwill
beclearwherethecharacterisvalued.Wealsodenoteby
ω
p,
2
afundamental
characteroflevel2(valuedin
F
p
∗
2
)of
I
p
:itfactorsthroughthequotientof
I
p
thatisisomorphicto
F
p
∗
2
.WedenotebythesamesymbolitsTeichmu¨ller
lift,andalsoallits
ℓ
-adicincarnations
ι
ℓ
ι
p
−
1
(
ω
p,
2
).Foranumberfield
F
wedenotetherestrictionofacharacterofGa