An ordinary rank-two case of local-global compatibility for automorphic representations of arbitrary weight over CM fields

Abstract

We prove a rank-two potential automorphy theorem for $\mod l$ representations satisfying an ordinary condition. Combined with an ordinary automorphy lifting theorem from $\href{https://doi.org/10.4007/annals.2023.197.3.2}{\textrm{[1]}}$, we prove a rank-two, $p \neq l$ case of local-global compatibility for regular algebraic cuspidal automorphic representations of arbitrary weight over CM fields that is $\iota$-ordinary for some $\overline{\mathbb{Q}}_l \tilde{\to} \mathbb{C}$.