Companion points on the eigenvariety with non-regular weights

TL;DR

This paper proves the existence of all companion points on the eigenvariety for definite unitary groups with generic crystalline Galois representations, extending previous results to non-regular weights under the Taylor-Wiles hypothesis.

Contribution

It generalizes the existence of companion points on eigenvarieties to non-regular weights, building on prior work in regular cases.

Findings

Proves existence of all companion points in non-regular weights

Extends previous regular case results to broader weight conditions

Operates under Taylor-Wiles hypothesis

Abstract

We prove the existence of all companion points on the eigenvariety of definite unitary groups associated with generic crystalline Galois representations with possibly non-regular weights under the Taylor-Wiles hypothesis, based on the previous results of Breuil-Hellmann-Schraen in arXiv:1702.02192 in regular cases and the author in arXiv:2103.03823 in non-regular cases.