On a theta correspondence with respect to a quadratic extension

Let F be a totally real algebraic number field, and let E be a totally real quadratic extension of F. In this article we establish a theta corresponde nce between certain automorphic forms defined with respect to a quaternion algebra over E and Hilbert modular Forms defined with respect to F. Given s uch a quaternionic form, say Ir, the main theorem expresses the Fourier coe fficients of its theta lift in terms of periods of h. The results in this p aper generalize some theorems of Shimura. (C) 1999 Academic Press.