Determination of cusp forms on GL(2) by coefficients restricted to quadratic subfields (with an appendix by Dipendra Prasad and Dinakar Ramakrishnan)

Given E/F a quadratic extension of number fields and a cuspidal representation π of GL2(AE), we give a full description of the fibers of the Asai transfer of π. We then determine the extent to which Fourier coefficients defined by integral ideals of F determine the representation π.