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Families of Galois representations and Selmer groups. (English) Zbl 1192.11035
Astérisque 324. Paris: Société Mathématique de France (ISBN 978-2-85629-264-8/pbk). xii, 314 p. (2009).
Authors’ abstract: “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, 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, they 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).”
Let \(E\) be a quadratic imaginary field, \(G_E=\text{Gal}(\overline{\mathbb Q}/E)\), \(p\) a prime that is split in \(E\), \[ \rho:G_E\to \mathrm{GL}_n(L) \] an \(n\)-dimensional geometric semisimple representation of \(G_E\) with coefficients in a finite extension \(L/{\mathbb Q}_p\). Under some technical assumptions on \(\rho\) and conjectures which are hoped to be proved in the future, the authors prove that the dimension of the Selmer group of \(\rho\) as \(L\)-vector space is \(\geq 1\). In some cases the dimension is \(\geq 2\).

MSC:
11F80 Galois representations
11-02 Research exposition (monographs, survey articles) pertaining to number theory
11F70 Representation-theoretic methods; automorphic representations over local and global fields
11F85 \(p\)-adic theory, local fields
11G40 \(L\)-functions of varieties over global fields; Birch-Swinnerton-Dyer conjecture
11S25 Galois cohomology
14F30 \(p\)-adic cohomology, crystalline cohomology
14G22 Rigid analytic geometry
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