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Families of Galois Representations and Selmer Groups
Joël Bellaïche, Columbia University, New York, NY, and Gaëtan Chenevier, Université Paris 13, Villetaneuse, France
A publication of the Société Mathématique de France.
cover
Astérisque
2009; 314 pp; softcover
Number: 324
ISBN-10: 2-85629-264-X
ISBN-13: 978-2-85629-264-8
List Price: US$105
Member Price: US$86.40
Order Code: AST/324
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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 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 Bloch-Kato 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 Bloch-Kato
  • 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
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