Monodromy for Some Rank Two Galois Representations over CM Fields

摘要

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 (vmid l) in (F) at which (pi) is special.