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arXiv · 2510.00281

The Local Companion Points Conjecture

Lie Qian
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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.

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Lie Qian. 2025-09-30. The Local Companion Points Conjecture. https://arxiv.org/abs/2510.00281

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