A local model for the trianguline variety and applications

Abstract

We describe the completed local rings of the trianguline variety at certain points of integral weights in terms of completed local rings of algebraic varieties related to Grothendieck's simultaneous resolution of singularities. We derive several local consequences at these points for the trianguline variety: local irreducibility, description of all local companion points in the crystalline case, combinatorial description of the completed local rings of the fiber over the weight map, etc. Combined with the patched Hecke eigenvariety (under the usual Taylor-Wiles assumptions), these results in turn have several global consequences: classicality of crystalline strictly dominant points on global Hecke eigenvarieties, existence of all expected companion constituents in the completed cohomology, existence of singularities on global Hecke eigenvarieties.

References in corpus (3)

• On the automorphy of $l$-adic Galois representations with small residual image
• Une interprétation modulaire de la variété trianguline
• Paraboline variation of $p$-adic families of $(\phi ,\Gamma )$-modules

Cited by in corpus (9)

• On $p$-adic $L$-functions for Hilbert modular forms
• Deformation theory of the trivial mod $p$ Galois representation for ${\mathrm{GL}}_n$
• Zariski density of crystalline points
• Density of automorphic points in deformation rings of polarized global Galois representations
• A $p$-adic adjoint $L$-function and the ramification locus of the Hilbert modular eigenvariety
• Towards the finite slope part for ${\mathrm{GL}}_n$
• Smoothness of definite unitary eigenvarieties at critical points
• Companion points and locally analytic socle for ${\mathrm{GL}}_2(L)$
• ${\mathrm{GL}}_2({\mathbb{Q}}_p)$-ordinary families and automorphy lifting