A local converse theorem for U(1,1)

In this paper, we define a $\gamma$-factor for generic representations of $\RU(1,1)\times \Res_{E/F}(\GL_1)$ and prove a local converse theorem for $\RU(1,1)$ using the $\gamma$-factor we defined. We also give a new proof of the local converse theorem for $\GL_2$ using a $\gamma$-factor of $\GL_2\times \GL_2$ type which was originally defined by Jacquet in \cite{J}.