Distinguished residual spectrum for GL₂(D)

Let G D GL₂(D) where D is a quaternion division algebra over a number field F and H D Sp₂(D) is the unique inner form of Sp₄(F). We study the period of an automorphic form on G(A[double-struck]) relative to H(A[double-struck]) and we provide a formula, similar to the split case, for an automorphic form in the residual spectrum. We confirm the conjecture due to Dipendra Prasad for noncuspidal automorphic representations, which says that symplectic period is preserved under the global Jacquet-Langlands correspondence.