Table Of Contentp-ADIC FAMILIES OF GALOIS REPRESENTATIONS AND
HIGHER RANK SELMER GROUPS
7 JOE¨L BELLA¨ICHE ANDGAE¨TAN CHENEVIER
0
0
2
n
a Contents
J
2 Introduction 5
1
1. Pseudocharacters, representations and extensions 10
] 1.1. Introduction 10
T
1.2. Some preliminaries on pseudocharacters 14
N
1.3. Generalized matrix algebras 20
.
h 1.4. Residually multiplicity-free pseudocharacters 29
t
a 1.5. Reducibility loci and Ext-groups 32
m
1.6. Representations over A 39
[ 1.7. An example: the case r = 2 42
2 1.8. Pseudocharacters with a symmetry 44
v
2. Trianguline deformations of refined crystalline representations 52
0
4 2.1. Introduction 52
3 2.2. Preliminaries of p-adic Hodge theory and (ϕ,Γ)-modules 55
2
2.3. Triangular (ϕ,Γ)-modules and trianguline representations over
0
6 artinian Q -algebras 61
p
0
2.4. Refinements of crystalline representations 69
/
h 2.5. Deformations of non critically refined crystalline representations 73
t
a 2.6. Some remarks on global applications 77
m
3. Generalization of a result of Kisin on crystalline periods 79
:
v 3.1. Introduction 79
i 3.2. A formal result on descent by blow-up 81
X
3.3. Direct generalization of a result of Kisin 87
r
a
3.4. A generalization of Kisin’s result for non-flat modules 91
4. Rigid analytic families of refined p-adic representations 94
4.1. Introduction 94
4.2. Families of refined and weakly refined p-adic representations 96
4.3. Existence of crystalline periods for weakly refined families 99
4.4. Refined families at regular crystalline points 105
4.5. Results on other reducibility loci 110
5. Selmer groups and a conjecture of Bloch-Kato 114
5.1. A conjecture of Bloch-Kato 114
5.2. The quadratic imaginary case 117
6. Automorphic forms on definite unitary groups: results and conjectures 122
6.1. Introduction 122
6.2. Definite unitary groups over Q 125
6.3. The local Langlands correspondence for GL 128
m
1
2 J.BELLA¨ICHEANDG.CHENEVIER
6.4. Refinements of unramified representation of GL 129
m
6.5. K-types and monodromy for GL 134
m
6.6. A class of non-monodromic representations for a quasi-split group 135
6.7. Representations of the compact real unitary group 140
6.8. The Galois representations attached to an automorphic representation
of U(m) 141
6.9. Construction and automorphy of non-tempered representation of
U(m) 144
7. Eigenvarieties of definite unitary groups 151
7.1. Introduction 151
7.2. Definition and basic properties of the Eigenvarieties 154
7.3. Eigenvarieties attached to an idempotent of the Hecke-algebra 159
7.4. The family of G(Ap)-representations on an eigenvariety of idempotent
f
type 166
7.5. The family of Galois representations on eigenvarieties 169
7.6. The eigenvarieties at the regular, non critical, crystalline points and
global refined deformation functors 176
7.7. An application to irreducibility 183
7.8. Appendix: p-adic families of Galois representations of Gal(Q /Q )
l l
with l = p 185
6
8. The sign conjecture 196
8.1. Statement of the theorem 196
8.2. The minimal eigenvariety X containing πn 198
8.3. Proof of Theorem 8.1.2 205
9. The geometry of the eigenvariety at some Arthur’s points and higher
rank Selmer groups 209
9.1. Statement of the theorem 209
9.2. Outline of the proof 211
9.3. Computation of the reducibility loci of T 212
9.4. The structure of R and the proof of the theorem 215
9.5. Remarks, questions, and complements 217
Appendix : Arthur’s conjectures 221
A.1. Failure of strong multiplicity one and global A-packets 222
A.2. The Langlands groups 222
A.3. Parametrization of global A-packets 223
A.4. Local A-packets and local A-parameters 224
A.5. Local L-parameters and local L-packets 225
A.6. Functoriality 226
A.7. Base Change of parameters and packets 226
A.8. Base change of a discrete automorphic representation 227
A.9. Parameters for unitary groups and Arthur’s conjectural description of
the discrete spetrum 227
A.10. An instructive example, following Rogawski. 229
A.11. Descent from GL to U(m) 231
m
A.12. Parameter and packet of the representation πn 237
A.13. Arthur’s multiplicity formula for πn 241
p-ADIC FAMILIES OF GALOIS REPRESENTATIONS 3
References 246
4 J.BELLA¨ICHEANDG.CHENEVIER
Ce livre est d´edi´e `a la m´emoire de Serge Bella¨ıche, fr`ere et ami.
p-ADIC FAMILIES OF GALOIS REPRESENTATIONS 5
Introduction
Thisbook1takes place inanow thirty years longtrendofresearches, initiated by
Ribet ([Ri]) aiming at constructing ”arithmetically interesting” non trivial exten-
sions between global Galois representations (either on finite p∞-torsion modules,
or on p-adic vector spaces) or, as we shall say, non-zero elements of Selmer groups,
by studying congruences or variations of automorphic forms. As far as we know,
despite of its great successes (to name one : the proof of Iwasawa’s main conjecture
for totally real fields by Wiles [W]), this current of research has never established,
in any case, the existence of two linearly independent elements in a Selmer group –
although well-established conjectures predict that sometimes such elements should
exist.2 The final aim of the book is, focusing on the characteristic zero case, to
understand the conditions under which, by this kind of method, existence of two
or more independent elements in a Selmer space could be proved.
To be somewhat more precise, let G be reductive group over a number field. We
assume, to fix ideas, that the existence of the p-adic rigid analytic eigenvariety
E
of G, as well as the existence and basic properties of the Galois representations
attached to algebraic automorphic forms of G are known3. Thus carries a family
E
of p-adic Galois representations. Our main result has the form of a numerical
relation between the dimension of the tangent space at suitable points x and
∈ E
the dimension of the part of Selmer groups of components of adρ that are ”seen
x
by ”, where ρ is the Galois representation carried by at the point x.
x
E E
Such a result can be used both way : if the Selmer groups are known, and small,
it can be used to study the geometry of at x, for example (see [Ki], [BCh2]) to
E
prove its smoothness. On the other direction, it can be used to get a lower bound
on the dimension of some interesting Selmer groups, lower bound that depends on
the dimension of the tangent space of at x. An especially interesting case is the
E
case of unitary groups with n+2 variables, and of some particular points x
∈ E
attached to non-tempered automorphic forms4. These forms were already used in
[B1] for a unitary group with three variables, and later for GSp in [SkU], and
4
for U(3) again in [BCh1]. At those points, the Galois representation ρ is a sum
x
of an irreducible n-dimensional representation5 ρ, the trivial character, and the
cyclotomic character χ. The representations ρ we could get this way are, at least
1During the elaboration and writing of this book, Jo¨el Bella¨ıche was supported by the NSF
grant DMS-05-01023. Ga¨etan Chenevier would like to thank the I.H.E.S. for their hospitality
duringthe last three monthsof this work.
2By a very different approach, let us mention here that the parity theorem of Nekovar [Ne]
shows,inthesign +1caseandforp-ordinarymodularforms, thattherankoftheSelmergroupis
bigger than 2 if nonzero.
3Besides GL over a totally real field and its forms, the main examples in the short term of
2
such G would be suitable unitary groups and suitable forms of GSp . Concerning unitary groups
4
in m ≥ 4 variables, it is one of the goals of the book project of the GRFA seminar [GRFAbook]
to construct the expected Galois representations, which makes the assumption relevant. All our
applications tounitaryeigenvarietiesforsuchgroups(hencetoSelmergroups)will beconditional
totheirwork. However,thankstoRogawski’sworkand[ZFPMS],everythingconcerningU(3)will
be unconditional.
4In general, their existence is predicted by Arthur’sconjectures and known in some cases.
5Say,of the absolute Galois group of a quadraticimaginary field.
6 J.BELLA¨ICHEANDG.CHENEVIER
conjecturally, all irreducible n-dimensional representations satisfying some auto-
duality condition, and such that the order of vanishing of L(ρ,s) at the center of
itsfunctionalequationisodd. Ourresultthengivesalowerboundonthedimension
of the Selmer group of ρ. Let us call Sel(ρ) this Selmer group6. This lower bound
implies, in any case, that Sel(ρ) is non zero (which is predicted by the Bloch-Kato
conjecture), and if is non smooth at x, that the dimension of Sel(ρ) is at least 2.
E
This first result (the non-triviality of Sel(ρ), proved in chapter 8) extends to
any dimension n a previous work of the authors [BCh1] in which we proved that
Sel(ρ) = 0 in the case n = 1, i.e. G = U(3), and the work of Skinner-Urban [SkU]
6
in the case n = 2 and ρ ordinary. Moreover, the techniques developed in this paper
shed also many lights back on those works. For example, the arguments in [BCh1]
to produce a non trivial element in Sel(ρ) involved some arbitrary choice of a germ
of irreducible curve of at x, and it was not clear in which way the resulting
E
element depended on them. With our new method, we do not have to make these
choices and we construct directly a canonical subvectorspace of Sel(ρ).
In order to prove our second, main, result (the lower bound on dim(Sel(ρ)), see
chapter 9) we study the reducibility loci of the family of Galois representations on
. An originality of the present work is that we focus on points x at which
E ∈ E
the Galois deformation at p is as non trivial as possible (we call some of them anti-
ordinary points)7. We discovered that at these points, the local Galois deformation
is highly irreducible, that is not only generically irreducible8, but even irreducible
on every proper artinian thickening on the point x inside (recall however that
E
ρ = 1 χ ρ is reducible). In other words, the reducibility locus of the family is
x
⊕ ⊕
schematically equal to the point x. It should be pointed out here that the situation
is quite orthogonal to the one of Iwasawa’s main conjecture (see [MaW], [W]),
for which there is a big known part in the reducibility locus at x (the Eisenstein
part), and this locus cannot be controlled a priori. In our case, this fact turns
out to be one of the main ingredients in order to get some geometric control on
the size of the subspace we construct in Sel(ρ), and it is maybe the main reason
why our points x are quite susceptible to produce independent elements in Sel(ρ).
The question of whether we should (or not) expect to construct the full Selmer
group of ρ at x remains a very interesting mystery, whatever the answer may be.
Although it might not be easy to decide it even in explicit examples (say with
L(ρ,s) vanishing at order > 1 at its center), our geometric criterion reduces this
question to some computations of spaces of p-adic automorphic forms on explicit
definite unitary groups, which should be feasible. Last but not least, we hope that
it could be possible to relate the geometry of at x (which is built on spaces of
E
p-adic automorphicforms)to the L-function (or rather ap-adic L-function) of ρ, so
that our results could be used in order to prove the ”lower bound of Selmer group”
part of the Bloch-Kato conjecture. However, this is beyond the scope of this book.
6Precisely, thisis thegroup usually denoted byH1(E,ρ).
f
7A bit more precisely, among the (finite number of) points of x ∈ E having the same Galois
representation ρ , we choose one which is refined in quitea special way.
x
8Note that although we do assume in the applications of this paper to Selmer groups the
existenceofGaloisrepresentationsattachedtoalgebraic automorphicformsonU(m)withm≥4,
we do not assume that the expected ones are irreducible, but instead our arguments prove this
irreducibility for some of them.
p-ADIC FAMILIES OF GALOIS REPRESENTATIONS 7
The four first chapters form a detailed study of p-adic families of Galois repre-
sentations, especially near reducible points, and how their behavior is related to
Selmer groups. There are no references to ”automorphic forms” in them, as op-
posite to the following chapters 5 to 9 which are devoted to the applications to
eigenvarieties. In what follows, we describe very briefly the contents of each of the
different chapters by focusing on the way they fit in the general theme of the book.
As they contain quite a number of results of independent interest, we invite the
reader to consult then their respective introductions for more details.
When we deal with families of representations (ρ ) of a group G (or an
x x∈X
algebra) over a ”geometric” space X, there are two natural notions to consider.
The most obvious one is the data of a ”big” representation of G of a locally free
sheaf of -modules whose evaluations at each x X are the ρ . Another one,
X x
O ∈
visibly weaker, is the data of a ”trace map” G (X) whose evaluation at
−→ O
each x X is tr(ρ ); these abstract traces are then called pseudocharacters (or
x
∈
pseudorepresentations). As a typical example, when we are interested in the space
of all representations (up to isomorphisms) of a given group say, we usually only
get a pseudocharacter on that full space. This is what happens also for the family
of Galois representations on the eigenvarieties. When all the ρ are irreducible,
x
the two definitions turn out to be essentially the same, but the links between them
are much more subtle around a reducible ρ and they are related to the extensions
x
between the irreducible constituents of ρ , what we are interested in.
x
ThusourfirstchapterisageneralstudyofpseudocharactersT overahenselianlo-
calringA(havinginviewthatthelocalringsofarigidanalyticspacearehenselian).
There is no mention of a Galois group in all this chapter, and those results can be
applied to any group or algebra. Most of our work is based on the assumption
that the residual pseudocharacter T¯ (that is, the pseudocharacter one gets after
tensorizing T by the residue field of A) is without multiplicity, so it may be re-
ducible, which is fundamental, but all its components appear only once. Under
this hypothesis, we prove a precise structure theorem for T, describe the groups of
extensions between the constituents of T¯ we can construct from T, and define and
characterize the reducibility loci of T (intuitively the subscheme of SpecA where T
has a given reducibility structure). We also discuss conditions under which T is, or
cannot be, the trace of a true representation. This chapter provides the framework
of many of our subsequent results.
In the second chapter we study infinitesimal (that is, artinian) families of p-
adic local Galois representations, and their Fontaine’s theory, having in mind to
characterize abstractly those coming from eigenvarieties. A key role is played by
the theory of (ϕ,Γ)-modules over the Robba ring and Colmez’ notion of triangu-
line representation [Co2]. We generalize some results of Colmez to any dimension
and with artinian coefficients, giving in particular a fairly complete description of
the trianguline deformation space of a non critical trianguline representation (of
any rank). For the applications to eigenvarieties, we also give a criterion for an
infinitesimal family to be trianguline in terms of crystalline periods.
In the third chapter, we generalize a recent result of Kisin in [Ki] on the ana-
lytic continuation of crystalline periods in a family of local Galois representations.
This result was proved there for the strong definition of families, namely for true
8 J.BELLA¨ICHEANDG.CHENEVIER
representations of Gal(Q /Q ) on a locally free -module, and we prove it more
p p
O
generally for any torsion free coherent -module. Our main technical tool is a
O
method of descent by blow-up of crystalline periods (which turns out to be rather
general) and a reduction to Kisin’s case by a flatification argument.
Inthefourthchapter, wegive ourworkingdefinitionof ”p-adicfamilies ofrefined
Galois representation”, motivated by the families carried by eigenvarieties, and we
applytothemtheresultsofchapters 2and3. Inparticular, weareableinfavorable
cases to understand their reducibility loci in terms of the Hodge-Tate-Sen weight
maps, and to prove that they are infinitesimally trianguline.
In the fifth chapter we discuss our main motivating conjecture relating the di-
mension of Selmer groups of geometric semi-simple Galois representations to the
order of the zeros of their L-functions at integers. We are mainly interested in
“one half” of this conjecture, namely to give a lower boundon the dimension of the
Selmer groups, as well as in a very special case of it that we call the sign conjecture.
As was explained in [B1], an important feature of the method we use is that we
need as an input some results (supposedly simpler) about upper bounds of other
Selmer groups. For the sign conjecture, we only need the vanishing of Sel(χ) (for
a quadratic imaginary field) which is elementary. However, we need more upper
bounds results for our second main theorem, and we cannot prove all of them in
general. Thus we formulate as hypotheses the results we shall need, which will
appear as assumptions in the results of chapter 9. Using results of Kisin and Kato,
we are able to prove all that we need in most cases when n = 2, and in all the cases
for n = 1.
Thesixthchapter containsalltheresultsweneedabouttheunitarygroups,their
automorphic forms, and the Galois representations attached to them, and includes
a detailled discussion of Langlands and Arthur’s conjectures. In particular, we
formulate there the two hypotheses (AC(π)) and (Rep(m)) that we use in chapters
8 and 9.
In the seventh chapter, we introduce and study in details the eigenvarieties of
definite unitary groups and we prove the basic properties of the (sometimes conjec-
tural) family of Galois representations that they carry. We essentially rely on one
of us’ thesis [Ch1]), and actually go a bit further on several respects. It furnishes a
lot of interesting examples where all the concepts studied in this book occur, and
provides also an important tool for the applications to Selmer groups. The first
half of this chapter only concerns eigenvarieties and may be read independently,
whereas the second one depends on chapters 1 to 5.
Finally, in chapters 8 and 9 we prove our main results, and we refer to those
chapters for precise statements.
As a parallel goal, we made considerable efforts all along the redaction of this
booktodevelopconceptsandtechniquesadaptedtoaproperstudyofeigenvarieties.
We hope that the reader will enjoy playing with them as much as we did.
p-ADIC FAMILIES OF GALOIS REPRESENTATIONS 9
The leitfaden is as follows :
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9
It is a pleasure to thank Laurent Berger, Brian Conrad, Laurent Clozel, Pierre
Colmez, Jean-Francois Dat, Guy Henniart, Michael Harris, Kiran Kedlaya, Colette
Moeglin and Claudio Procesi for many useful conversations.
Last but not least, the authors would like to thank Cl´ementine, Sarah and
Valeria: this book is also theirs.
10 J.BELLA¨ICHEANDG.CHENEVIER
1. Pseudocharacters, representations and extensions
1.1. Introduction. This section is devoted to the local (in the sense of the ´etale
topology) study of pseudocharacters T satisfying a residual multiplicity freeness
hypothesis. Two of our main objectives are to determine when those pseudochar-
acters come from a true representation and to prove the optimal generalization of
“Ribet’s lemma” for them.
Let us precise our main notations and hypotheses. Throughout this section, we
will work with a pseudocharacter T : R A of dimension d, where A is a local
−→
henselian commutative ring of residue field k where d! is invertible and R a (not
necessarily commutative) A-algebra9. To formulate our residual hypothesis, we
assume10 that T k :R k k is the sum of r pseudocharacters of the form trρ¯
i
⊗ ⊗ −→
wheretheρ¯’sareabsolutelyirreduciblerepresentationsofR k definedoverk. Our
i
⊗
residually multiplicity freehypothesisisthattheρ¯’saretwobytwononisomorphic.
i
In this context, “Ribet’s lemma” amounts to determine how much we can deduce
about the existence of non-trivial extensions between the representations ρ¯ from
i
the existence and irreducibility properties of T. Before explaining our work and
results in more details, let us recall the history of those two interrelated themes :
pseudocharacters and the generalizations of “Ribet’s lemma”.
We begin with the original Ribet’s lemma ([Ri, Prop. 2.1]). Ribet’s hypotheses
are that d = r = 2, andthat Ais a complete discrete valuation ring. He works with
a representation ρ : R M (A), but that is no real supplementary restriction
2
−→
since every pseudocharacter over a strictly complete discrete valuation ring is the
trace of a true representation11. Ribet proves that if ρ K (K being the fraction
⊗
field of A) is irreducible, then a non-trivial extension of ρ¯ by ρ¯ (resp. of ρ¯ by ρ¯ )
1 2 2 1
arises as a subquotient of ρ. This seminal result suggests numerous generalizations
: wemay wishtoweaken thehypothesesonthedimensiond,thenumberofresidual
factorsr,theringA,andformoregeneralA,toworkwithgeneralpseudocharacters
instead of representations. We may also wonder if we can obtain, under suitable
hypotheses, extensions between deformations ρ and ρ of ρ¯ and ρ¯ over some
1 2 1 2
suitable artinian quotient of A, not only over k.
AbigstepforwardismadeinthepapersbyMazur-WilesandWiles([MaW],[W])
on Iwasawa’s main conjecture. As their works is the primary source of inspiration
for this section, let us explain it with some details (our exposition owes much
to [HaPi]; see also [BCh2, 2]). They still assume d = r =2, but the ring A now is
§
anyfiniteflatreducedlocalA -algebraA, whereA isacompletediscretevaluation
0 0
ring. Though the notion of pseudocharacter at that time was still to be defined,
their formulation amount to consider a pseudocharacter (not necessarily coming
from arepresentation) T : R A, whereR is the groupalgebra of a global Galois
−→
group. The pseudocharacter is supposed to be odd which implies our multiplicity
free hypothesis. They introduce an ideal I of A, which turns out to be the smallest
9In the applications, R will be the group algebra A[G] where G a group, especially a Galois
group. However, it is important to keep this degree of generality as most of the statements
concerning pseudocharacters are ring theoretic.
10Since it is allowed to replace k by a separable extension, this assumption is actually not a
restriction.
11We leave the proof of this assertion to the interested reader (use the fact that the Brauer
group of any finite extension of K is trivial, e.g. by [S1, XII §2, especially exercise 2]).
