On a Theta Correspondence with Respect to a Quadratic Extension, Original Research Article

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 correspondence between certain automorphic forms defined with respect to a quaternion algebra over E and Hilbert modular forms defined with respect to F. Given such a quaternionic form, say h, the main theorem expresses the Fourier coefficients of its theta lift in terms of periods of h. The results in this paper generalize some theorems of Shimura.