Companion points and locally analytic socle for $\mathrm{GL}_2(L)$

TL;DR

This paper proves Breuil's conjecture for $ ext{GL}_2(L)$, establishing the existence of all companion points on the eigenvariety by connecting infinitesimal R=T results with de Rham and Hodge-Tate families.

Contribution

It confirms Breuil's locally analytic socle conjecture for $ ext{GL}_2(L)$ and introduces a method to find companion points of non-classical points.

Findings

Proves existence of all companion points on the eigenvariety.

Establishes infinitesimal R=T results for patched eigenvarieties.

Identifies companion points of non-classical points.

Abstract

Let $p>2$ be a prime number, and $L$ be a finite extension of $\mathbb{Q}_p$, we prove Breuil's locally analytic socle conjecture for $\mathrm{GL}_2(L)$, showing the existence of all the companion points on the definite (patched) eigenvariety. This work relies on infinitesimal "R=T" results for the patched eigenvariety and the comparison of (partially) de Rham families and (partially) Hodge-Tate families. This method allows in particular to find companion points of non-classical points.