A New Proof to the Period Problems of GL(2)

Abstract: We use the relations between the base change representations, theta lifts and Whittaker model, to give a new proof to the period problems of $GL(2)$ over a quadratic local field extension $E/F.$ And we classify both local and global $D^\times(F)-$distinguished representations $\pi^D$ of $D^\times(E),$ where $D^\times$ is an inner form of $GL_2$ defined over a nonarchimedean field or a number field $F.$