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, we prove a rank-two, $p\ne l$ case of local-global compatibility for regular algebraic cuspidal automorphic representations of arbitrary weight over CM fields that is ${\hat{I}}^1$-ordinary for some ${\hat{I}}^1:{\overline{\mathbb{Q}}}_l\stackrel{\sim }{\to }\mathbb{C}$.