The Local Companion Points Conjecture

TL;DR

This work establishes a local analogue of the companion-points phenomenon for trianguline varieties by building a deformation-theoretic framework that intertwines $p$-adic Hodge theory with Weyl-group combinatorics. It develops twisted almost de Rham representations, a formally smooth morphism, and Grothendieck–Springer–based local models to analyze trianguline deformations and their weights. The main result proves the local companion-points conjecture under regularity assumptions, extending previous crystalline-regular results to a general setting and highlighting how $p$-adic Hodge data governs possible triangulations and weights. The findings illuminate the local Langlands-geometry of trianguline spaces and provide a pathway to understanding Serre-like weight attachments through local deformation theory.

Abstract

We describe the set of points of the trianguline variety over a given local Galois representation. Global analogues describing companion points in eigenvariety by [Bre14] and [HN17], can be thought of as a rational analogue to the weight part of Serre's conjecture. Along the same line, local companion points conjecture can be thought of as a rational analogue of attaching Serre weights to residual Galois representations. [BHS19] proves the conjecture assuming the given Galois representation is cristalline regular. We prove the conjecture in general cases only assuming some regularity conditions.

Theorems & Definitions (111)

• Definition 1.1
• Conjecture 1.2
• Theorem 1.3
• Lemma 2.1
• proof
• Corollary 2.2
• proof
• Remark 2.3
• Corollary 2.4
• proof