Tangent spaces on the trianguline variety at companion points

Abstract

Many results about the geometry of the trianguline variety have been obtained by Breuil–Hellmann–Schraen. Among them, using geometric methods, they have computed a formula for the dimension of the tangent space of the trianguline variety at dominant crystalline generic points, which has a conjectural generalisation to companion (i.e. non-dominant) points. In an earlier work, they proved a weaker form of this formula under the assumption of modularity using arithmetic methods. We prove a generalisation of a result of Bellaïche–Chenevier in $p$-adic Hodge theory and use it to extend the arithmetic methods of Breuil–Hellmann–Schraen to a wide class of companion points.