Monodromy for Some Rank Two Galois Representations over CM Fields

Abstract

We investigate local-global compatibility for cuspidal automorphic representations $\pi$ for $\operatorname{GL}_2$ over CM fields that are regular algebraic of weight $0$. We prove that for a Dirichlet density one set of primes $l$ and any $\iota : \overline{\mathbf{Q}}_l \xrightarrow {\sim} \mathbf{C}$, 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.