On the classicality theorem and its applications to the automorphy lifting theorem and the Breuil-M$\mathrm{\acute{e}}$zard conjecture in some $\mathrm{GL}_2(\mathbb{Q}_{p^2})$ cases

Abstract

In this paper, we study locally analytic vectors in the "partially" completed cohomology of Shimura varieties associated with some rank $2$ unitary groups over a totally real field $F^+$ such that $F^+_v = \mathbb{Q}_{p^2}$ for some $p$-adic places $v$ and prove a certain classicality theorem. This is a partial generalization and modification of Lue Pan's work in the modular curve case by using the works of Caraiani-Scholze, Koshikawa and Zou on mod $l$ cohomology of Shimura varieties. As applications, we prove the automorphy lifting theorem and the Breuil-M$\mathrm{\acute{e}}$zard conjecture in some $\mathrm{GL}_2(\mathbb{Q}_{p^2})$ cases. We will assume a technical regularity condition on Serre weights of residual representations, but we don't assume any technical condition on the properties of liftings of residual representations at $p$-adic places except Hodge-Tate regularity. It should be noted that previously, such results were known only when we assumed that $F^+_v$ is equal to $\mathbb{Q}_p$ for any $p$-adic place $v$ of $F^+$ so that we can use the $p$-adic Langlands correspondence of $\mathrm{GL}_2(\mathbb{Q}_p)$. Moreover, we propose a conjectural strategy to prove such results in some $\mathrm{GL}_2(\mathbb{Q}_{p^f})$ cases.