Astérisque 2009; 314 pp; softcover Number: 324 ISBN10: 285629264X ISBN13: 9782856292648 List Price: US$105 Member Price: US$86.40 Order Code: AST/324
 This book presents an indepth 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 neighborhood of a point, reducibility loci, pseudocharacters), and other aspects more specific to Galois groups of local or number fields. In particular, the authors define and study certain deformation functors of crystalline representations of the absolute Galois group of \(\mathbb{Q}p\), namely trianguline deformations, which are naturally associated to the families above. As an application, the authors 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 the authors to prove, among other things, new cases of the BlochKato conjectures (in any dimension). A publication of the Société Mathématique de France, Marseilles (SMF), distributed by the AMS in the U.S., Canada, and Mexico. Orders from other countries should be sent to the SMF. Members of the SMF receive a 30% discount from list. Readership Graduate students and research mathematicians interested in number theory. Table of Contents  Introduction
 Pseudocharacters, representations and extensions
 Trianguline deformations of refined crystalline representations
 Generalization of a result of Kisin on crystalline periods
 Rigid analytic families of refined \(p\)adic representations
 Selmer groups and a conjecture of BlochKato
 Automorphic forms on definite unitary groups: results and conjectures
 Eigenvarieties of definite unitary groups
 The sign conjecture
 The geometry of the eigenvariety at some Arthur points and higher rank Selmer groups
 Appendix: Arthur's conjectures
 Bibliography
 Index of notations
