  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.
