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

Yuji Yang

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

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 representations satisfying an ordinary condition. Combined with an ordinary automorphy lifting theorem, we prove a rank-two, case of local-global compatibility for regular algebraic cuspidal automorphic representations of arbitrary weight over CM fields that is -ordinary for some .

Paper Structure

This paper contains 4 sections, 6 theorems, 14 equations.

Table of Contents

  1. Introduction
  2. An Important Lemma
  3. Potential Automorphy
  4. Local-Global Compatibility

Key Result

Theorem 1.3

Let $F$ be a CM field and let $\pi$ be a regular algebraic cuspidal automorphic representation of $\mathop{\mathrm{GL}}\nolimits_2(\mathbb{A}_F)$. Suppose that Then for any finite place $v\nmid l$ in $F$, we have

Theorems & Definitions (15)

  • Conjecture 1.1: Local-global compatibility
  • Remark 1.2
  • Theorem 1.3
  • Definition 2.1
  • Lemma 2.2
  • proof
  • Proposition 3.1
  • proof
  • Theorem 3.2
  • proof
  • ...and 5 more