# Monodromy for some rank two Galois representations over CM fields

## TL;DR

This paper proves that for certain automorphic representations over CM fields, the associated Galois representations exhibit nontrivial monodromy at specific places for almost all primes, advancing understanding of local-global compatibility.

## Contribution

It establishes nontrivial monodromy for Galois representations attached to regular algebraic cuspidal automorphic representations over CM fields at special places, for a density-one set of primes.

## Key findings

- Nontrivial monodromy at special places for almost all primes
- Validation of local-global compatibility in new cases
- Extension of known results to CM fields

## Abstract

We investigate local-global compatibility for cuspidal automorphic representations $\pi$ for GL(2) over CM fields that are regular algebraic of weight $0$. We prove that for a Dirichlet density one set of primes $l$ and any $\iota : \overline{\mathbf{Q}}_l \cong \mathbf{C}$, the $l$-adic Galois representation attached to $\pi$ and $\iota$ has nontrivial monodromy at any $v \nmid l$ in $F$ at which $\pi$ is special.