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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 $$n$$).

##### MSC:
 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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