The Local Companion Points Conjecture
Number Theory
2025-10-02 v1
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.
Cite
@article{arxiv.2510.00281,
title = {The Local Companion Points Conjecture},
author = {Lie Qian},
journal= {arXiv preprint arXiv:2510.00281},
year = {2025}
}