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

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 $\overline{\mathbb{Q}}_l \tilde{\to} \mathbb{C}$.

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Received 25 November 2021

Received revised 18 April 2022

Accepted 4 August 2022

Published 17 July 2024