Bernstein eigenvarieties

TL;DR

This paper introduces Bernstein eigenvarieties and related spaces, exploring their geometry and establishing local-global compatibility results, including classicality and progress on Breuil's socle conjecture in non-trianguline cases.

Contribution

It constructs Bernstein eigenvarieties and analyzes their geometry, extending previous results and providing new local-global compatibility theorems without trianguline assumptions.

Findings

Local geometry described by algebraic schemes related to Grothendieck-Springer resolution

Classicality results established without trianguline assumptions at p

Progress on Breuil's locally analytic socle conjecture in non-trianguline cases

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 (generalizing results of Breuil-Hellmann-Schraen) we show that their local geometry can be described by certain algebraic schemes related to the generalized 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.