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
Kojiro Matsumoto
TL;DR
The paper develops a classicality theorem for locally analytic vectors in the partially completed cohomology of rank-2 unitary Shimura varieties with certain p-adic places, generalizing Pan’s modular-curve approach using perfectoid techniques and geometric Sen theory. It then uses this classicality to establish automorphy lifting and Breuil–Mézard results in GL_2(Q_{p^2}) cases, without requiring ordinary or Fontaine–Laffaille liftings at p, but assuming Hodge–Tate regularity and Serre-weight constraints. Central to the argument are geometric and arithmetic locally analytic de Rham complexes, Fontaine operators, and a big R = T framework complemented by a comparison of completed cohomologies across unitary Shimura varieties to propagate automorphic information. The work also outlines conjectural strategies for GL_2(Q_{p^f}) cases and sketches a potential automorphy strategy via potential automorphy of residual representations, aiming for broader reach beyond GL_2(Q_p). Overall, the results push automorphy lifting and Breuil–Mézard in new p-adic settings and provide a robust pathway for extending classicality methods to higher-degree p-adic fields.
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.