An Ordinary Rank-Two Case of Local-Global Compatibility for Automorphic Representations of Arbitrary Weight Over CM Fields

TL;DR

This paper establishes a specific case of local-global compatibility for automorphic representations over CM fields, using potential automorphy and automorphy lifting theorems for rank-two mod l Galois representations.

Contribution

It proves a rank-two, p ≠ l case of local-global compatibility for automorphic representations of arbitrary weight over CM fields, extending previous results.

Findings

Proves a rank-two potential automorphy theorem for mod l representations.

Establishes a rank-two, p ≠ l local-global compatibility result.

Demonstrates 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 $\iota$-ordinary for some $\iota: \overline{\mathbb{Q}}_l\xrightarrow{\sim}\mathbb{C}$.