Local root numbers, Bessel models, and a conjecture of Guo and Jacquet

Let E/FE/F be a quadratic extension of number fields and D a quaternion algebra over F containing E  . Let πDπD be a cuspidal automorphic representation of GL(n,D)GL(n,D) and π   its Jacquet–Langlands transfer to GL(2n)GL(2n). Guo and Jacquet conjectured that if πDπD is distinguished by GL(n,E)GL(n,E), then π   is symplectic and L(1/2,πE)≠0L(1/2,πE)≠0, where πEπE is the base change of π to E. When n is odd, Guo and Jacquet also conjectured a converse. The converse does not always hold when n   is even, but we conjecture it holds if and only if certain local root number conditions are satisfied, which is if and only if the corresponding generic representation of the split special orthogonal group SO(2n+1)SO(2n+1) has a special E-Bessel model. We use the theta correspondence to relate E  -Bessel periods on SO(5)SO(5) with GL(2,E)GL(2,E)-periods on GL(2,D)GL(2,D), and deduce part of our conjecture when n=2n=2.