Families of Galois representations and Selmer groups

"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 Q_p, namely trianguline deformations, which are naturally associated to the families above. As an application, we show how the geometry of these eigenvarieties at c̀l̀assical'' 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)."