Abstract: Trianguline representations form the class of $p$-adic Galois representations whose associated $(\varphi,\Gamma)$-module is a successive extension of rank-one objects, and they correspond to overconvergent eigenforms. In a series of works, Breuil, Hellmann, and Schraen studied eigenvarieties by analyzing the geometry of the trianguline variety, the moduli of trianguline representations.
In this talk, I will introduce the trianguline variety for a split connected reductive group, establish some of its basic properties, and construct a local model at certain singular points, extending the work of BHS. This has many applications, including to the density of crystalline representations in the deformation space of a residual $p$-adic Galois representation.
This is joint work with Andrea Conti and Julian Quast.