Companion forms for unitary and symplectic groups

0.48 MB·English

by  Toby Gee

COMPANION FORMS FOR UNITARY AND SYMPLECTIC
GROUPS
0
1 TOBYGEEANDDAVIDGERAGHTY
0
2
Abstract. Weproveacompanionformstheoremforordinaryn-dimensional

n automorphic Galois representations, by use of automorphy lifting theorems
a developed by the second author, and a technique for deducing companion
J formstheorems duetothefirstauthor. Wededuce resultsabout thepossible

3 Serre weights of mod l Galois representations corresponding to automorphic
1 representations onunitarygroups. We thenusefunctorialitytoprovesimilar
resultsforautomorphicrepresentations ofGSp4 overtotallyrealfields.

]
T
N
.
h
Contents
t
a
m 1. Introduction. 1
2. Notation 4
[
3. Galois deformations 4

1 4. Ordinary automorphic representations 12
v
5. Existence of lifts 17
4
6. Serre weights 21
4
0 7. GSp 25
4
2 References 39
.
1
0
0
1
: 1. Introduction.
v
i 1.1. The problem of companion forms was first introduced by Serre for modular
X
forms in his seminalpaper [Ser87]. Fix a prime l, algebraicclosuresQ and Q of Q
l
r
a and Ql respectively, and an embedding of Q into Ql. Suppose that f is a modular
newform of weight k ≥ 2 which is ordinary at l, so that the corresponding l-adic
Galois representation ρ becomes reducible when restricted to a decomposition
f,l
group G at l. Then the companion forms problem is essentially the question of
Ql
determining for which other weights k′ there is an ordinary newform g of weight
k′ ≥ 2 such that the Galois representations ρ and ρ are congruent modulo l.
f,l g,l
The problem is straightforwardunless the restriction to G of ρ (the reduction
Ql f,l
mod l of ρ ) is split and non-scalar, in which case there are two possible Hida
f,l
families whose corresponding Galois representations lift ρ ; the restrictions of
f,l
the corresponding Galois representations to a decomposition group at l are either
“upper-triangular”or “lower-triangular”.
2000 Mathematics Subject Classification. 11F33.
ThefirstauthorwaspartiallysupportedbyNSFgrantDMS-0841491.
1
2 TOBYGEEANDDAVIDGERAGHTY
This problem was essentially resolved by Gross and Coleman-Voloch ([Gro90],
[CV92]). In the paper [Gee07], the first author reproved these results, and gener-
alised them to Hilbert modular forms, by a completely new technique. In essence,
rather than working directly with modular forms, the method is to firstly obtain
a Galois representationwhich should correspond to a modular form in the sought-
after Hida family, and then to use a modularity lifting theorem to prove that this
Galois representation is modular. In [Gee07] the Galois representation is obtained
by using a generalisationof a lifting technique of Ramakrishna,which is provedby
purely deformation theory techniques. The modularity is then obtained from the
R=T theorem of Kisin for Hilbert modular forms of parallel weight 2 ([Kis07b]).
Thesetechniquesseemamenabletogeneralisation(tootherreductivegroupsover
more general number fields), subject to some important caveats. In particular, it
is necessary to have modularity lifting theorems available over fields in which l is
highly ramified. The current technology for modularity lifting theorems requires
one to work with reductive groups which admit discrete series, and to work over
totallyrealorCMfields;soitisimpossibleatpresenttoworkdirectlywithGL for
n
n>2. Instead,oneworkswithcloselyrelatedgroups,suchasunitaryorsymplectic
groups, which do admit discrete series.
In the present paper we make use of R = T theorems for unitary groups to
deducecompanionformstheoremsforunitarygroups(inarbitrarydimension),and
thus for conjugate self-dual automorphic representations of GL over CM fields.
n
We then deduce similar theorems for GSp by developing the relevantdeformation
4
theory and employing known instances of functoriality. The analogue for unitary
groups of the R= T theorems of [Kis07b] seem to be out of reach at present, and
weusethemaintheoremsof[Ger09]instead. Asexplainedbelow,thisinfactallows
us to provestrongertheorems than the naturalanalogueof [Kis07b] wouldpermit.
WereplacetheuseofRamakrishna’stechniquesin[Gee07]withamethodofKhare
andWintenberger,whichallowsweakerhypothesesonlocaldeformationproblems.
ToourknowledgetheonlyresultsoncompanionformsforgroupsotherthanGL
2
are those announced for GSp over Q in [HT08] (see also [Til09]). Our results are
4
rather stronger than those of [HT08] in severalrespects. We are able to work with
arbitrarytotallyrealfields(withnorestrictiononramificationatl),ratherthanjust
overQ,andwedonotneedanyassumptionthattheresidualGaloisrepresentation
occursat minimallevel (indeed, one may deduce results onlevelloweringfor GSp
4
from our theorem). In addition, the results of [HT08] apply only in one special
case, effectively one of 8 cases (corresponding to the 8 elements of the Weyl group
ofGSp )whereonecouldhopetoproveacompanionformstheorem;thisisinpart
4
due to the fact that their techniques only apply to Galois representations in the
Fontaine-Laffaille range. In contrast, we make no such restrictions. We hope that
these results will prove useful for generalisations of the Buzzard-Taylor method to
GSp , as part of a programof Tilouine.
4
InrecentyearstherehasbeenagooddealofinterestingeneralisationsofSerre’s
conjecture (cf. [ADP02]) and in particular in the question of determining the set
of weights of a given Galois representation (cf. [Her09]). One of us (T.G.) has
formulated a conjecture to the effect that the set of weights should be determined
completely by the existence of (local) crystalline lifts (cf. [Gee06]). In general this
seems to be a very difficult conjecture to prove, but our methods give a substan-
tial partial result; essentially we prove the conjecture (subject to mild technical
COMPANION FORMS FOR UNITARY GROUPS 3
hypotheses) for ordinary weights for unitary groups which are compact at infinity.
See section 6 for the precise statements.
We now outline the structure of the paper. In section 3 we develop the basic
deformationtheorythatweneed. Wethenrecallinsection4thenecessarymaterial
on ordinary automorphic representations on unitary groups and modularity lifting
theorems for the corresponding Galois representations; in particular we recall the
main theorem of [Ger09].
Section 5 contains our main theorems for unitary groups; the corresponding
Galoisrepresentationsareconjugateself-dualrepresentationsoftheabsoluteGalois
group of an imaginary CM field. Using the results of section 3 we give a lower
bound for the dimension of a universaldeformation ring, and the results of section
4 then permit us to prove that this universal deformation ring is finite over Z,
l
whichimplies thatithasQ -points,whichcorrespondtotheGaloisrepresentations
l
we seek. The automorphy of these Galois representations follows at once from
the modularity lifting theorems recalled in section 4. The particular universal
deformationringweconsiderisoneforrepresentationsoftheabsoluteGaloisgroup
of a totally real field, valued in a group G defined in [CHT08]. Representations
n
valued in this groupcorrespondto representationswhich are self-dualwith respect
to some pairing; this permits us to prove results for both the conjugate self-dual
representationsconsideredin section 5, and the symplectic representationsstudied
in later sections.
We remark that the Q -points of universal deformation rings that we study in
l
section 5 always correspond to ordinary crystalline representations of a certain
weight. This is incontrastto the approachof[Gee07], whichusedpotentially crys-
talline representationscorrespondingto Hilbert modular forms of parallelweight2
and non-triviallevel at l. The requiredautomorphic representationswere then ob-
tained by specialising Hida families through these points at the sought-for weight.
The difficulty with following this approach in general is that if the weight is not
sufficiently regular a specialisation of a Hida family at this weight may fail to be
an unramified principal series at places dividing l (for example, a specialisation of
a Hida family of modular forms in weight 2 can correspond to a Steinberg repre-
sentation at l). It is for this reason that we use modularity lifting theorems for
crystalline lifts instead.
In section 6 we deduce results about the possible Serre weights of mod l Galois
representationscorrespondingtoautomorphicrepresentationsofcompactatinfinity
unitary groups. In particular, we deduce that the possible ordinary weights are
determined by the existence of localcrystalline lifts. We remark that these are the
first results in anything approaching this level of generality for any groups other
than GL .
2
Finally in section7 we study the analogousquestions for automorphic represen-
tations of GSp over totally real fields. We use the known functoriality between
4
globally generic cuspidal representations of GSp and GL to apply the methods
4 4
of the earlier sections. In particular, we prove results analogousto those of section
3 for Galois representations valued in GSp , and obtain a lower bound for the di-
4
mension of a universal deformation ring as in section 5. We then prove that this
universaldeformationringisfiniteoverthecorrespondingoneforunitaryrepresen-
tations, which allows us to deduce that our symplectic universal deformation ring
4 TOBYGEEANDDAVIDGERAGHTY
is also finite over Z. Our main results for symplectic representations follow from
l
this.
We remark that in all our main theorems we actually obtain somewhat more
preciseresults;wearealsoabletocontroltheramificationofourGaloisrepresenta-
tions at places not dividing l, and we are able to choose our Galois representations
so as to correspond to points on any particular set of irreducible components of
the local deformation rings. Thus as a direct corollary of our results one obtains
strong results on level lowering and level raising for ordinary automorphic Galois
representations. Similarly, our method yields modularity lifting theorems for or-
dinary representations of GSp which are rather stronger than those of [GT05];
4
for example, we do not need to assume any form of level-lowering for GSp , we
4
work over general totally real fields, and we are not restricted to weights in the
Fontaine-Laffaille range.
We wouldlike to thank Wee Teck Gan, FlorianHerzig,Mark KisinandRichard
Taylor for helpful conversations.
2. Notation
If M is a field, we let G denote its absolute Galois group. Let ǫ denote the l-
M
adicormodl cyclotomiccharacterofG . IfM isafiniteextensionofQ forsome
M p
p, we write I for the inertia subgroup of G . We write all matrix transposes on
M M
theleft;sotAisthetransposeofA. IfRisalocalringwewritem forthemaximal
R
ideal of R. We let Zn denote the subset of elements λ∈Zn with λ ≥...≥λ .
+ 1 n
3. Galois deformations
3.1. Local deformation rings. Letl beaprimenumberandK afiniteextension
ofQ with residue field k andring of integersO. LetM be a finite extensionofQ
l p
(with ppossibly equalto l). Letρ:G →GL (k) be a continuousrepresentation.
M n
Let C be the category of complete local Noetherian O-algebras with residue field
O
k. Then the functor from C to Sets which takes A ∈ C to the set of liftings of
O O
ρ to a continuous homomorphism ρ: G →GL (A) is represented by a complete
M n
(cid:3)
local Noetherian O-algebra R . We call this ring the universal O-lifting ring of ρ.
ρ
(cid:3) (cid:3)
We write ρ : G → GL (R ) for the universal lifting. We will need to consider
M n ρ
(cid:3)
certain quotients of R .
ρ
3.1.1. The case where p 6= l. Firstly, we consider the case p 6= l. In this case, the
quotients we wish to consider will correspond to particular inertial types. Recall
that τ is an inertial type for G over K if τ is a K-representation of I with
M M
open kernel which extends to a representation of G , and that we say that an
M
l-adicrepresentationofG has type τ if the restrictionofthe correspondingWeil-
M
Deligne representation to I is equivalent to τ. For any such τ there is a unique
M
reduced,l-torsionfreequotientR(cid:3),τ ofR(cid:3) withthepropertythatifE/K isafinite
ρ ρ
extension, then a map of O-algebras R(cid:3) → E factors through R(cid:3),τ if and only if
ρ ρ
the corresponding E-representation has type τ. Furthermore, we have:
Lemma 3.1.1. For any τ, if R(cid:3),τ 6= 0 then R(cid:3),τ[1/l] is equidimensional of di-
ρ ρ
mension n2 and is generically formally smooth.
Proof. This is Theorem 2.0.6 of [Gee06]. (cid:3)
COMPANION FORMS FOR UNITARY GROUPS 5
Of course, R(cid:3),τ 6=0 if and only if ρ has a lift of type τ.
ρ
3.1.2. The case where p = l. Now assume that p = l. In this case, we wish to
consider crystalline ordinary deformations of fixed weight. We assume from now
onthatK is largeenoughthat anyembedding M ֒→K has image containedin K.
Notation. RecallthatZn isthe setofnon-increasingn-tuples ofintegers. We say
+
thatλ∈(Zn)Hom(M,K) isregular ifforeachj =1,...,n−1thereexistsτ :M ֒→K
+
with λ >λ .
τ,j τ,j+1
Letǫbethel-adiccyclotomiccharacterandletArt :M× →Wab betheArtin
M M
map (normalized to take uniformizers to lifts of geometric Frobenius).
Definition 3.1.2. Let λ be an element of (Zn)Hom(M,K). We associate to λ an
+
n-tuple of characters I →O× as follows. For j =1,...,n define
M
χλ :I → O×
j M
σ 7→ ǫ(σ)−(j−1) τ(Art−1(σ))−λτ,n−j+1.
M
τ:MY֒→K
Note that χλ can also be thought of as the restriction to I of any crystalline
j M
×
characterG →Q whoseHodge-Tateweightwithrespecttoτ :M ֒→Q isgiven
M l l
by (j−1)+λ for all τ (we use the conventionthat the Hodge-Tate weights
τ,n−j+1
of ǫ are all −1).
Let λ be an element of (Zn)Hom(M,K). We associate to λ an l-adic Hodge type
+
v in the sense ofsection2.6 of[Kis08] as follows. Let D denote the vectorspace
λ K
Kn. Let D = D ⊗ M. For each embedding τ : M ֒→ K, we let D =
K,M K Ql K,τ
D ⊗ K so that D = ⊕ D . For each τ choose a decreasing
K,M K⊗M,1⊗τ K,M τ K,τ
filtrationFiliD ofD sothatdim griD =0 unlessi=(j−1)+λ
K,τ K,τ K K,τ τ,n−j+1
for some j = 1,...,n in which case dim griD = 1. We define a decreasing
K K,τ
filtration of D by K⊗ M-submodules by setting
K,M Ql
FiliD =⊕ FiliD .
K,M τ K,τ
Let v ={D ,FiliD }.
λ K K,M
Let B denote a finite, local K-algebra and ρ : G → GL (B) a crystalline
B M n
representation. Then DB := Dcris(ρB) = (ρB ⊗Ql Bcris)GM is a free B ⊗Ql M0-
module of rank n where M is the maximal subfield of M which is unramified over
0
Q . Moreover, D is equipped with a B-linear and ϕ -semilinear morphism ϕ
l B 0 B
where ϕ denotes geometric Frobenius on M . For each embedding τ : M → K,
0 0 0
let D =D ⊗ B. Then D =⊕ D . Also, for each τ, ϕ defines
B,τ B B⊗M0,1⊗τ B,τ τ B,τ B
∼
an isomorphism of B-modules ϕ : D −→ D . Let f = [M : Q ]. Then
B B,τ B,τ◦ϕ−1 0 l
0
ϕf is a B-linear endomorphism of D which preserves each D . For each τ, the
B B B,τ
isomorphism ϕ : D → D takes φf to φf. Let P (X) ∈ B[X] denote
B B,τ B,τ◦ϕ−1 B B B
0
the characteristic polynomial of φf on D for any choice of τ.
B B,τ
Let F denote the flag variety over SpecO whose set of A-points, for any O-
algebra A, corresponds to filtrations 0 = Fil ⊂ Fil ⊂ ... ⊂ Fil = An of An by
0 1 n
locally free submodules which, locally, are direct summands and are such that Fil
j
has rank j.
Definition 3.1.3. Let E be an algebraic extension of K let B be a finite local
E-algebra. Let ρ:G →GL (B) be a continuous homomorphism. We say that ρ
M n
6 TOBYGEEANDDAVIDGERAGHTY
is ordinary of weight λ∈(Zn)Hom(M,K) if ρ is conjugate to a representationof the
+
form
ψ ∗ ... ∗ ∗
1
0 ψ2 ... ∗ ∗ 
.. .. .. .. ..
 . . . . . 
 
0 0 ... ψ ∗ 
 n−1 
0 0 ... 0 ψ 
n
 
where for each j = 1,...,n the character ψ agrees on an open subgroup of I
j M
with the character χλ introduced above.
j
Equivalently,ρisordinaryofweightλifthereisafullflag0=Fil ⊂Fil ⊂...⊂
0 1
Fil = Bn of Bn which preserved by G and such that the representation of G
n M M
ongr =Fil /Fil ispotentiallysemistableandforeachembeddingτ :M ֒→K,
j j j−1
the Hodge-Tate weight of gr with respect to τ is (j−1)+λ .
j τ,n−j+1
Lemma 3.1.4. Suppose that E is an algebraic extension of K and ρ : G →
M
GL (E) is ordinary of weight λ. Let ψ ,...,ψ :G →E× be as above. Then
n 1 n M
(1) ρ is potentially semistable.
(2) If each ψ is crystalline (which occurs if and only if ψ agrees with χλ on
j j j
all of I ), then ρ is semistable.
M
(3) If each ψ is crystalline and λ is regular, then ρ is crystalline.
j
Proof. Part 2 follows from Proposition 1.28(2) of [Nek93] and part 1 follows from
part2. Part3 follows fromProposition1.26of [Nek93]and the formulaein Propo-
sition 1.24 of [Nek93]. (cid:3)
Lemma 3.1.5. Let ψ : G → E× be as above (with respect to some λ ∈
i M
(Zn)Hom(M,K)), with each ψ crystalline. Suppose that ρ : G → GL (k) is of
+ i M n
the form
µ ∗ ... ∗ ∗
1
0 µ ... ∗ ∗ 
2
.. .. .. .. ..
 . . . . . 
 
0 0 ... µ ∗ 
 n−1 
0 0 ... 0 µ 
 n
where ψ =µ for each 1≤i≤n. Suppose that for each i<j we have µ µ−1 6=ǫ.
i i i j
Then ρ has a lift to a crystalline representation ρ:G →GL (E) of the form
M n
ψ ∗ ... ∗ ∗
1
0 ψ2 ... ∗ ∗ 
 ... ... ... ... ... .
 
0 0 ... ψ ∗ 
 n−1 
0 0 ... 0 ψ 
n
 
Proof. The fact that any upper-triangularrepresentationof this formis crystalline
followseasilyasintheproofofLemma3.1.4,becausetheassumptionthatµ µ−1 6=ǫ
i j
implies that ψ ψ−1 6= ǫ. The fact that such an upper-triangular lift exists follows
i j
from the fact that H2(G ,u)=0, where u is the subspace of the Lie algebra adρ
M
consisting of strictly upper-triangular matrices. The vanishing of this cohomology
group follows from Tate local duality and the existence of a filtration on u whose
COMPANION FORMS FOR UNITARY GROUPS 7
graded pieces are one-dimensional with G acting via the characters µ µ−1 6= ǫ,
M i j
i<j (cf. Lemma 3.2.3 of [Ger09]). (cid:3)
We now recall some results of Kisin. Let λ be an element of (Zn)Hom(M,K) and
+
let v be the associated l-adic Hodge type.
λ
Definition 3.1.6. If B is a finite K-algebra and V is a free B-module of rank
B
n with a continuous action of G that makes V into a de Rham representation,
M B
thenwesaythatV is of l-adic Hodge type v ifforeachithereisanisomorphism
B λ
of B⊗ M-modules
Ql
gri(V ⊗ B )GM −∼→B⊗ (griD ).
B Ql dR K K,M
For example, if E is a finite extension of K and ρ : G → GL (E) is ordinary
M n
of weight λ, then ρ is of l-adic Hodge type v .
λ
Corollary 2.7.7 of [Kis08] implies that there is a unique l-torsion-free quotient
Rvλ,cr of R(cid:3) with the property that for any finite K-algebra B, a homomor-
ρ ρ
phism of O-algebras ζ : R(cid:3) → B factors through Rvλ,cr if and only if ζ ◦ ρ(cid:3)
ρ ρ
is crystalline of l-adic Hodge type v . Moreover, Theorem 3.3.8 of [Kis08] implies
λ
that SpecRvλ,cr[1/l] is formally smooth over K and equidimensional of dimension
ρ
n2+ 1n(n−1)[M :Q ].
2 l
Let F be the flag variety over SpecO as introduced above and let Gλ be the
closed subscheme of F × SpecRvλ,cr corresponding to filtrations Fil which
SpecO ρ
(i) are preserved by the induced action of G and (ii) are such that I acts on
M M
gr = Fil /Fil via the character χλ for each j = 1,...,n. The fact that Gλ is
j j j−1 j
a closed subscheme can proved in the same way as Lemma 3.1.2 of [Ger09]. Let
R△λ,cr be the image of
ρ
Rvλ,cr →O (Gλ[1/l]).
ρ Gλ
Inotherwords,SpecR△λ,cristheschemetheoreticimageofthemorphismGλ[1/l]→
ρ
SpecRvλ,cr. The next result follows from Lemma 3.3.3 of [Ger09].
ρ
Lemma 3.1.7. For any finite local K-algebra B, a homomorphism of O-algebras
ζ :Rvλ,cr →B factors through R△λ,cr if and only if ζ◦ρ(cid:3) is ordinary of weight λ.
ρ ρ
Moreover, SpecR△λ,cr is a union of irreducible components of SpecRvλ,cr.
ρ ρ
3.1.3. The p = l case with a slight refinement. We continue to consider, as above,
crystalline lifts of ρ which are ordinary of a given weight λ. A necessary condition
for such lifts to exist is that ρ itself is conjugate to an upper triangular represen-
tation whose ordered n-tuple of diagonal characters, restricted to I , is given by
M
(χλ,...,χλ). Let us assume that ρ has this property. In fact, let us fix charac-
1 n
ters µ ,...,µ : G → k× with µ | = χλ and assume that ρ is conjugate to
1 n M j IM j
an upper triangular representationwhose orderedn-tuple of diagonalcharacters is
µ := (µ ,...,µ ). Note that if the characters χλ are not distinct, then we may
1 n j
havemorethanone choiceforthe orderedn-tuple(µ ,...,µ ). We nowwouldlike
1 n
to study crystalline lifts of ρ which are ordinary of weight λ and are such that for
each j, the character ψ of Definition 3.1.3 lifts µ .
j j
LetR denotetheobjectofC representingthefunctorwhichsendsanobjectA
µ O
ofC tothesetoflifts(ψ ,...,ψ )oftheorderedn-tuple(µ ,...,µ )withψ | =
O 1 n 1 n j IM
χλ for each j. The ring R is non-canonically isomorphic to O[[X ,...,X ]]. Let
j µ 1 n
(ψuniv,...,ψuniv)betheuniversalliftofthetuple(µ ,...,µ )toR . LetGλdenote
1 n 1 n µ µ
8 TOBYGEEANDDAVIDGERAGHTY
theclosedsubschemeoftheflagvarietyF× Spec(R△λ,cr⊗ R )corresponding
SpecO ρ O µ
tofiltrationswhichare(i)preservedbytheinducedactionofG and(ii)suchthat
M
b
G actsongr viathepushforwardofψunivforeachj =1,...,n. LetR△λ,crbethe
M j j ρ,µ
quotient of R△λ,cr⊗ R corresponding to the scheme theoretic image of Gλ[1/l].
ρ O µ µ
Note that we have a natural morphism Gλ[1/l] → Gλ[1/l] covering the morphism
b µ
SpecR△λ,cr →SpecR△λ,cr.
ρ,µ ρ
Lemma 3.1.8. After inverting l, the morphism SpecR△λ,cr → SpecR△λ,cr be-
ρ,µ ρ
comes a closed immersion and identifies SpecR△λ,cr[1/l] with a union of irre-
ρ,µ
ducible components of SpecR△λ,cr[1/l]. Moroever, every irreducible component of
ρ
SpecR△λ,cr arises in this way.
ρ
Proof. LetXord,cr =SpecR△λ,cr and let Xord,cr =SpecR△λ,cr. Let x be a closed
ρ µ ρ,µ
point of Xord,cr[1/l] with residue field E. Let z denote the image of x in Xord,cr.
µ
We claim that the natural map on completed local rings O∧ →O∧ is
Xord,cr,z Xord,cr,x
µ
an isomorphism.
With this in mind, let B denote a finite, local E-algebra and ζ :O∧ →B
Xord,cr,z
an E-algebra homomorphism. It follows from Lemma 3.1.7 that ζ corresponds
to a crystalline representation ρ : G → GL (B) which preserves a full flag
B M n
0⊂Fil ⊂...⊂Fil =Bn with I acting on gr via χλ. Since the characters χλ
1 n M j j j
are pairwise distinct, there is a unique such flag. Moreover, there exists a subring
A⊂B which is local and finite over O and such that the action of G on gr is
E M j
givenbyacharacterψ :G →B× whichfactorsthroughA× withψ mod m =
j M j A
µ ⊗ A/m . We see that there is a unique lifting of ζ∗ :SpecB →SpecR△λ,cr to
j k A ρ
Gλ. It follows that the homomorphism O∧ →O∧ is formally smooth
µ Xord,cr,z Xord,cr,x
µ
of relative dimension 0. It’s easy to see that both sides have the same residue field
and hence the map is an isomorphism.
ForeachclosedpointinXord,cr[1/l],thereisatmost1closedpointofXord,cr[1/l]
µ
lying over it. From this, and the claim just established, we deduce the first two
statements. The final statement is clear. (cid:3)
3.1.4. The p = l case in non-fixed weight. In this section we assume that p = l,
(cid:3)
that ρ:G →GL (k) is the trivial homomorphism. Let R denote the universal
M n ρ
O-liftingringofρandletΛ =O[[I (l)n]]whereforagroupH,H(l)denotesits
M Mab
pro-l completion. Then Λ represents the functor C → Sets sending an algebra
M O
A to the set of ordered n-tuples (χ ,...,χ ) of characters χ : I → A× lifting
1 n j Mab
the trivial character modulo m . Let ρ(cid:3) denote the universal lift of ρ to R(cid:3) and
A ρ
let (χuniv,...,χuniv) denote the universal n-tuple of characters I →Λ×.
1 n Mab M
(cid:3) (cid:3)
Let R = R ⊗ Λ . Let G denote the closed subscheme of the flag variety
ρ,ΛM ρ O M
(cid:3)
F × SpecR corresponding to filtrations which are (i) preserved by the
SpecO ρ,ΛbM
induced action of G and (ii) such that I acts on gr via the pushforward of
M M j
χuniv. Let R△ be the quotient of R(cid:3) corresponding to the scheme theoretic
j ρ,ΛM ρ,ΛM
image of the morphism
(cid:3)
G[1/l]→SpecR .
ρ,ΛM
COMPANION FORMS FOR UNITARY GROUPS 9
(cid:3)
If E is a finite extension of K, a homomorphism of O-algebras ζ : R → E
ρ,ΛM
factors through R△ if and only if ζ ◦ρ(cid:3) is conjugate to an upper triangular
ρ,ΛM
representation whose ordered n-tuple of diagonal characters, restricted to I , is
M
the pushforward of (χuniv,...,χuniv).
1 n
3.2. Global deformation rings.
3.2.1. The group G . Let n be a positive integer, and let G be the group scheme
n n
over Z which is the semidirect product of GL ×GL by the group {1,j}, which
n 1
acts on GL ×GL by
n 1
j(g,µ)j−1 =(µtg−1,µ).
There is a homomorphismν :G →GL sending (g,µ) to µ and j to −1. Write g0
n 1 n
for the trace zero subspace ofthe Lie algebraof GL , regardedas a Lie subalgebra
n
of the Lie algebra of G .
n
Definition 3.2.1. Let F+ be a totally real field, and let r : G → G (L) be a
F+ n
continuous homomorphism, where L is a topological field. Then we say that r is
odd if for all complex conjugations c ∈G , ν◦r(c )=−1.
v F+ v
3.2.2. Bigness. Recall definition 2.5.1 of [CHT08].
Definition 3.2.2. Let k be an algebraic extension of the finite field F . We say
l
that a finite subgroup H ⊂GL (k) is big if the following conditions are satisfied.
n
• H has no quotient of l-power order.
• H0(H,g0(k))=(0).
n
• H1(H,g0(k))=(0).
n
• Forallirreduciblek[H]-submodulesW ofg0(k)wecanfindh∈H andα∈
n
k suchthat the α-generalisedeigenspaceV ofh inkn is one-dimensional
h,α
and furthermore π ◦W ◦i 6= (0). Here π : kn → V is the h-
h,α h,α h,α h,α
equivariantprojectionofkn to V ,andi is the h-equivariantinjection
H,α h,α
of V into kn.
h,α
We call a finite subgroup H ⊂ G (k) big if H surjects onto G (k)/G0(k) and H ∩
n n n
G0(k) is big.
n
3.2.3. Deformationproblems. LetF/F+beatotallyimaginaryquadraticextension
of a totally real field F+. Let c denote the non-trivial element of Gal(F/F+). Let
k denote a finite field of characteristic l and K a finite extension of Q , inside a
l
fixedalgebraicclosureQ , with ring ofintegers O andresidue field k. Assume that
l
K contains the image of every embedding F ֒→ Q and that the prime l is odd.
l
Assume that every place in F+ dividing l splits in F. Let S denote a finite set
of finite places of F+ which split in F, and assume that S contains every place
dividing l. Let S denote the set of places of F+ lying over l. Let F(S) denote
l
the maximal extensionof F unramifiedawayfrom S. Let G =Gal(F(S)/F+)
F+,S
and G = Gal(F(S)/F). For each v ∈ S choose a place v of F lying over v and
F,S
let S denote the set of v for v ∈ S. For each place v|∞ of F+ we let c denote a
v
e
choice of a complex conjugation at v in G . For each place w of F we have a
G e-conjugacy class ofehomomorphisms GF+,S→ G . For v ∈ S we fix a choice
F,S Fw F,S
of homomorphism G →G .
Fve F,S
10 TOBYGEEANDDAVIDGERAGHTY
If R is a ring and r : G → G (R) is a homomorphism with r−1(GL (R)×
F+,S n n
GL (R))=G , we will make a slight abuse of notation and write r| (respec-
1 F,S GF,S
tively r| for w a place of F) to mean r| (respectively r| ) composed with
GFw GF GFw
the projection GL (R)×GL (R)→GL (R).
n 1 n
Fix a continuous homomorphism
r¯:G →G (k)
F+,S n
suchthatG =r¯−1(GL (k)×GL (k))andfixacontinuouscharacterχ:G →
F,S n 1 F+,S
O× such that ν ◦ r¯ = χ. Assume that r¯| is absolutely irreducible. As in
GF,S
Definition 1.2.1 of [CHT08], we define
• a lifting of r¯ to an object A of C to be a continuous homomorphism
O
r :G →G (A) lifting r¯and with ν◦r =χ;
F+,S n
• two liftings r, r′ of r¯ to A to be equivalent if they are conjugate by an
element of ker(GL (A)→GL (k));
n n
• adeformation ofr¯toanobjectAofC tobeanequivalenceclassofliftings.
O
(cid:3)
For each place v ∈S, let R denote the universal O-lifting ring of r¯| and
r¯|GFve GFve
(cid:3)
let Rve denote a quotient of Rr¯|GFve which satisfies the following property:
(*) let A be an object of C and let ζ,ζ′ : R(cid:3) → A be homomorphisms
O r¯|GFve
corresponding to two lifts r and r′ of r¯| which are conjugate by an
GFve
element ofker(GLn(A)→GLn(k)). Thenζ factorsthroughRve if andonly
if ζ′ does.
We consider the deformation problem
S =(F/F+,S,S,O,r¯,χ,{Rve}v∈S)
(see sections 2.2 and 2.3 of [CHT08] forethis terminology). We say that a lifting r:
(cid:3)
G →G (A)isoftypeS ifforeachplacev ∈S,thehomomorphismR →A
F+,S n r¯|GFve
corresponding to r|GFve factors through Rve. We also define deformations of type S
in the same way.
Let Def be the functor C → Sets which sends an algebra A to the set of
S O
deformations of r¯to A of type S. By Proposition 2.2.9 of [CHT08] this functor is
represented by an object Runiv of C .
S O
Lemma 3.2.3. Let M be a finite extension of Q for some prime p and ρ:G →
p M
GL (k) a continuous homomorphism. If p 6= l, let τ be an inertial type for G
n M
over K and let R be a quotient of R(cid:3),τ corresponding to a union of irreducible
ρ
components. If p=l, assume that K contains the image of every embedding M ֒→
K, letλ∈(Zn)Hom(M,K) andletRbeaquotientofRvλ,cr corresponding toaunion
+ ρ
of irreducible components. Then R satisfies property (*) above.
Proof. We consider the case p = l; the other case is similar. Let Rvλ,cr[[X]] =
ρ
Rvλ,cr[[X : 1 ≤ i,j ≤ n]] and consider the lift of ρ to Rvλ,cr[[X]] given by
ρ ij ρ
(1 +(X ))ρ(cid:3)(1 +(X ))−1. This lift gives rise to an O-algebra homomorphism
n ij n ij
R(cid:3) → Rvλ,cr[[X]]. We claim that this homomorphism factors through Rvλ,cr.
ρ ρ ρ
This follows from the fact that Rvλ,cr[[X]] is reduced and l-torsion-free and every
ρ
Q pointofthis ringgivesrisetoaliftofρwhichiscrystallineofl-adicHodgetype
l
v . Let α denote the resulting O-algebra homomorphism Rvλ,cr → Rvλ,cr[[X]]
λ ρ ρ