Monodromy for some rank two Galois representations over CM fields

We investigate local-global compatibility for cuspidal automorphic representations π for GL2 over CM fields that are regular algebraic of weight 0. We prove that for a Dirichlet density one set of primes l and any ι:Ql−→∼C, the l-adic Galois representation attached to π and ι has nontrivial monodromy at any v∤l in F at which π is special.