$\mathrm{GL}_2(\mathbb{Q}_p)$-ordinary families and automorphy lifting

TL;DR

This paper establishes automorphy lifting theorems for certain $p$-adic Galois representations over CM fields, using $R= ext{T}$-type results over $ ext{GL}_2(Q_p)$-ordinary families, and explores related local conjectures.

Contribution

It proves automorphy lifting results for specific Galois representations and advances understanding of Breuil's locally analytic socle conjecture in non-trianguline cases.

Findings

Automorphy lifting theorems for conjugate self-dual Galois representations.

Results on Breuil's locally analytic socle conjecture.

Establishment of an $R= ext{T}$-type theorem over $ ext{GL}_2(Q_p)$-ordinary families.

Abstract

We prove automorphy lifting results for certain essentially conjugate self-dual $p$-adic Galois representations $\rho$ over CM imaginary fields $F$, which satisfy in particular that $p$ splits in $F$, and that the restriction of $\rho$ on any decomposition group above $p$ is reducible with all the Jordan-H\"older factors of dimension at most $2$. We also show some results on Breuil's locally analytic socle conjecture in certain non-trianguline case. The main results are obtained by establishing an $R=\mathbb{T}$-type result over the $\mathrm{GL}_2(\mathbb{Q}_p)$-ordinary families considered by Breuil-Ding.