Bernstein Eigenvarieties

Abstract

We construct parabolic analogues of (global) eigenvarieties, of patched eigenvarieties and of (local) trianguline varieties, that we call, respectively, Bernstein eigenvarieties, patched Bernstein eigenvarieties, and Bernstein paraboline varieties. We study the geometry of these rigid analytic spaces, in particular (generalising results of Breuil–Hellmann–Schraen) we show that their local geometry can be described by certain algebraic schemes related to the generalised Grothendieck–Springer resolution. We deduce several local–global compatibility results, including a classicality result (with no trianguline assumption at p), and new cases towards the locally analytic socle conjecture of Breuil in the non-trianguline case.