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On the infinite fern of Galois representations of unitary type. (Sur la fougère infinie des représentations galoisiennes de type unitaire.) (English. French summary) Zbl 1279.11056
Summary: Let \(E\) be a CM number field, \(p\) an odd prime totally split in \(E\), and let \(X\) be the \(p\)-adic analytic space parameterizing the isomorphism classes of \(3\)-dimensional semisimple \(p\)-adic representations of \(\text{Gal}(\overline E/E)\) satisfying a selfduality condition “of type U(3)”. We study an analogue of the infinite fern of Gouvêa-Mazur in this context and show that each irreducible component of the Zariski-closure of the modular points in \(X\) has dimension at least \(3\,[E~:~\mathbb Q]\). As important steps, and in any rank, we prove that any first order deformation of a generic enough crystalline representation of \(\text{Gal}(\overline {\mathbb Q}_p / \mathbb Q)\) is a linear combination of trianguline deformations, and that unitary eigenvarieties are étale over weight space at the non-critical classical points. As another application, we give a surjectivity criterion for the localization at \(p\) of the adjoint’ Selmer group of a \(p\)-adic Galois representation attached to a cuspidal cohomological automorphic representation of \(\text{GL}_n(\mathbb A_E)\) of type \(U(n)\) (for any \(n\)).

11F80 Galois representations
11R33 Integral representations related to algebraic numbers; Galois module structure of rings of integers
12F10 Separable extensions, Galois theory
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