Dihedral Iwasawa theory of nearly ordinary quaternionic automorphic forms

Let π ( f ) be a nearly ordinary automorphic representation of the multiplicative group of an indefinite quaternion algebra B over a totally real field F with associated Galois representation ρ f . Let K be a totally complex quadratic extension of F embedding in B . Using families of CM points on towers of Shimura curves attached to B and K , we construct an Euler system for ρ f . We prove that it extends to p -adic families of Galois representations coming from Hida theory and dihedral ℤ d p -extensions. When this Euler system is non-trivial, we prove divisibilities of characteristic ideals for the main conjecture in dihedral and modular Iwasawa theory.