2025/12/04 by Matsumoto, Kojiro
#Algebraic Geometry (math.AG) #FOS: Mathematics #Number Theory (math.NT) #Representation Theory (math.RT)
paper · doi:10.48550/arxiv.2512.04641
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 = ℚp2 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\acuteezard conjecture in some GL2(ℚp2) 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 ℚp for any p-adic place v of F+ so that we can use the p-adic Langlands correspondence of GL2(ℚp). Moreover, we propose a conjectural strategy to prove such results in some GL2(ℚpf) cases.