This book presents an in-depth study of the families of
Galois representations carried by the p-adic eigenvarieties
attached to unitary groups. The study encompasses some
general algebraic aspects (properties of the space of representations
of a group in the neighbourhood of a point, reducibility
loci, pseudocharacters), and other aspects more
specific to Galois groups of local or number fields. In particular,
we define and study certain deformation functors
of crystalline representations of the absolute Galois group
of Qp, namely trianguline deformations, which are naturally
associated to the families above. As an application, we show
how the geometry of these eigenvarieties at "classical" points
is related to the dimension of certain Selmer groups. This,
combined with conjectures of Langlands and Arthur on the
discrete automorphic spectrum of unitary groups, allows us
to prove, amongst other things, new cases of the Bloch-Kato
conjectures (in any dimension).
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