A Rank-Two Case of Local-Global Compatibility for $l = p$

TL;DR

This paper proves a specific case of the local-global compatibility conjecture for automorphic representations over CM fields, linking automorphic and Galois representations with new insights into monodromy and Fontaine-Mazur invariants.

Contribution

It establishes the classical $l=p$ local-global compatibility for certain automorphic representations and introduces a new definition of Fontaine-Mazur $ ext{L}$-invariants in this context.

Findings

Proved local-global compatibility for regular algebraic cuspidal automorphic representations of weight 0.

Showed nontrivial monodromy in Galois representations when automorphic representations have Steinberg components.

Provided a new framework for defining Fontaine-Mazur $ ext{L}$-invariants for automorphic representations.

Abstract

We prove the classical $l = p$ local-global compatibility conjecture for certain regular algebraic cuspidal automorphic representations of weight 0 for GL$_2$ over CM fields. Using an automorphy lifting theorem, we show that if the automorphic side comes from a twist of Steinberg at $v | l$, then the Galois side has nontrivial monodromy at $v$. Based on this observation, we will give a definition of the Fontaine-Mazur $\mathcal{L}$-invariants attached to certain automorphic representations.