Families of Galois representations and Selmer groups

Introduction
Chapter 1. Pseudocharacters, representations and extensions
1.1. Introduction
1.2. Some preliminaries on pseudocharacters
1.2.1. Definitions
1.2.2. Main example
1.2.3. The Cayley-Hamilton identity and Cayley-Hamilton pseudocharacters
1.2.4. Faithful pseudocharacters, the kernel of a pseudocharacter
1.2.5. Cayley-Hamilton quotients
1.2.6. Two useful lemmas on pseudocharacters
1.2.7. Tensor operations on pseudocharacters
1.3. Generalized matrix algebras
1.3.1. Definitions, notations and examples
1.3.2. Structure of a GMA
1.3.3. Representations of a GMA
1.3.4. An embedding problem
1.3.5. Solution of the embedding problem in the reduced and nondegenerate case
1.3.6. Solution of the embedding problem in the general case
1.4. Residually multiplicity-free pseudocharacters
1.4.1. Definition
1.4.2. Lifting idempotents
1.4.3. The structure theorem
1.5. Reducibility loci and Ext-groups
1.5.1. Reducibility loci
1.5.2. The representation i
1.5.3. An explicit construction of extensions between the i's
1.5.4. The projective modules Mi and a characterization of the image of i,j
1.5.5. Complement: Topology
1.6. Representations over A
1.7. An example: the case r=2
1.8. Pseudocharacters with a symmetry
1.8.1. The set-up
1.8.2. Lifting idempotents
1.8.3. Notations and choices
1.8.4. Definition of the morphisms i,j
1.8.5. Definition of the morphisms i,j
1.8.6. The main result
1.8.7. A special case
Chapter 2. Trianguline deformations of refined crystalline representations
2.1. Introduction
2.2. Preliminaries of p-adic Hodge theory and (,)-modules
2.2.1. Notations and conventions
2.2.2. (,)-modules over the Robba ring RA
2.2.3. Some algebraic properties of RA
2.2.4. Étale and isocline -modules
2.2.5. Cohomology of (,)-modules
2.2.6. (,)-modules and representations of Gp
2.2.7. Berger's theorem
2.3. Triangular (,)-modules and trianguline representations over artinian Q p-algebras
2.3.1. (,)-modules of rank one over RA
2.3.2. Definitions
2.3.3. Weights and Sen polynomial of a triangular (,)-module
2.3.4. De Rham triangular (,)-modules.
2.3.5. Deformations of triangular (,)-modules
2.3.6. Trianguline deformations of trianguline representations
2.4. Refinements of crystalline representations
2.4.1. Definition
2.4.2. Refinements and triangulations of (,)-modules
2.4.3. Non critical refinements
2.5. Deformations of non critically refined crystalline representations
2.5.1. A local and infinitesimal version of Coleman's classicity theorem
2.5.2. A criterion for a deformation of a non critically refined crystalline representation to be trianguline
2.5.3. Properties of the deformation functor XV,F
2.6. Some remarks on global applications
Chapter 3. Generalization of a result of Kisin on crystalline periods
3.1. Introduction
3.2. A formal result on descent by blow-up
3.2.1. Notations
3.2.2. The left-exact functor D
3.2.3. Statement of the result
3.3. Direct generalization of a result of Kisin
3.3.1. Notations and definitions
3.3.2. Hypotheses
3.3.3. The finite slope subspace Xfs
3.4. A generalization of Kisin's result for non-flat modules
Chapter 4. Rigid analytic families of refined p-adic representations
4.1. Introduction
4.2. Families of refined and weakly refined p-adic representations
4.2.1. Notations
4.2.2. Rigid analytic families of p-adic representations
4.2.3. Refined and weakly refined families of p-adic representations
4.2.4. Exterior powers of a refined family are weakly refined
4.3. Existence of crystalline periods for weakly refined families
4.3.1. Hypotheses
4.3.2. The main results
4.3.3. Analytic extension of some A[G]-modules, and proof of Lemma 4.3.3
4.4. Refined families at regular crystalline points
4.4.1. Hypotheses
4.4.2. The residually irreducible case (r=1)
4.4.3. A permutation
4.4.4. The total reducibility locus
4.5. Results on other reducibility loci
Chapter 5. Selmer groups and a conjecture of Bloch-Kato
5.1. A conjecture of Bloch-Kato
5.1.1. Geometric representations
5.1.2. Selmer groups
5.1.3. The general conjecture
5.1.4. The sign conjecture
5.2. The quadratic imaginary case
5.2.1. Assumptions and notations
5.2.2. An important example
5.2.3. Upper bounds on auxiliary Selmer groups
Chapter 6. Automorphic forms on definite unitary groups: results and conjectures
6.1. Introduction
6.2. Definite unitary groups over Q
6.2.1. Unitary groups
6.2.2. The definite unitary group U(m)
6.2.3. Automorphic forms and representations
6.3. The local Langlands correspondence for GLm
6.4. Refinements of unramified representations of GLm
6.4.1. The Atkin-Lehner rings
6.4.2. Computation of some Jacquet modules
6.4.3. Unramified representations
6.4.4. Refinements
6.4.5. Accessible refinements of almost tempered unramified representations
6.5. K-types and monodromy for GLm
6.5.1. An ``ordering'' relation on Irr(GLm(F))
6.5.2. Types
6.6. A class of non-monodromic representations for a quasi-split group
6.6.1. Review of normalized induction and the Jacquet functor over a base ring
6.6.2. Non Monodromic Strongly Regular Principal Series
6.6.3. The locus of non monodromic principal series is constructible
6.7. Representations of the compact real unitary group
6.8. The Galois representations attached to an automorphic representation of U(m)
6.8.1. Settings and notations
6.8.2. Statement of the assumption Rep(m)
6.9. Construction and automorphy of a non-tempered representation of U(m)
6.9.1. The starting point
6.9.2. Hecke characters
6.9.3. Construction of nl, for l split in E
6.9.4. Construction of nl, for l inert or ramified in E
6.9.5. Construction of s
6.9.6. Assumption AC()
Chapter 7. Eigenvarieties of definite unitary groups
7.1. Introduction
7.2. Definition and basic properties of the Eigenvarieties
7.2.1. The setting
7.2.2. p-refined automorphic representations
7.2.3. Eigenvarieties as interpolations spaces of p-refined automorphic representations
7.3. Eigenvarieties attached to an idempotent of the Hecke-algebra
7.3.1. Eigenvarieties of idempotent type
7.3.2. Review of the construction of the eigenvariety (Ch)
7.3.3. Step I. The family of the U–stable principal series of a Iwahori subgroup
7.3.4. Step II. p-adic automorphic forms
7.3.5. Step III. Classical versus p-adic automorphic forms
7.3.6. Step IV. Fredholm series and construction of the eigenvariety
7.4. The family of G(Afp)-representations on an eigenvariety of idempotent type
7.4.1. The family of local representations on X
7.4.2. The non monodromic principal series locus of X
7.5. The family of Galois representations on eigenvarieties
7.5.1. Setting
7.5.2. The family of Galois representations on X
7.5.3. Properties of T at the primes l=p in S
7.5.4. Properties of T at the prime v
7.6. The eigenvarieties at the regular, non critical, crystalline points and global refined deformation functors
7.6.1. Some global deformation functors and a general conjecture
7.6.2. An automorphic special case
7.6.3. R=T at the regular non critical crystalline points of minimal eigenvarieties
7.7. An application to irreducibility
7.8. Appendix: p-adic families of Galois representations of Gal( Q l/ Q l) with l=p
7.8.1. Some preliminary lemmas on nilpotent matrices
7.8.2. Preliminaries on general families of pseudocharacters
7.8.3. Grothendieck's l-adic monodromy theorem in families
7.8.4. p-adic families of WF-representations
Chapter 8. The sign conjecture
8.1. Statement of the theorem
8.2. The minimal eigenvariety X containing n
8.2.1. Definition of n and X
8.2.2. Normalization of the Galois representation on X
8.2.3. The faithful GMA at the point z
8.2.4. Properties at l of ExtT(i,j)
8.2.5. Properties at v and of ExtT(1,i)
8.2.6. Symmetry properties of T
8.3. Proof of Theorem 8.1.2
8.3.1.
8.3.2. Some remarks about the proof
Chapter 9. The geometry of the eigenvariety at some Arthur points and higher rank Selmer groups
9.1. Statement of the theorem
9.2. Outline of the proof
9.3. Computation of the reducibility loci of T
9.4. The structure of R and the proof of the theorem
9.5. Remarks, questions, and complements
9.5.1. The case of a CM field E and the sign of Galois representations
9.5.2. When is TA the trace of a representation over A?
9.5.3. Other remarks and questions
Appendix: Arthur's conjectures
A.1. Failure of strong multiplicity one and global A-packets
A.2. The Langlands groups
A.3. Parameterization of global A-packets
A.4. Local A-packets and local A-parameters
A.5. Local L-parameters and local L-packets
A.6. Functoriality
A.7. Base change of parameters and packets
A.8. Base change of a discrete automorphic representation
A.9. Parameters for unitary groups and Arthur's conjectural description of the discrete spectrum
A.10. An instructive example, following Rogawski
A.11. Descent from GLm to U(m)
A.12. Parameter and packet of the representation n
A.13. Arthur's multiplicity formula for n
Bibliography
Index of notations