Quadratic Base Change and the Analytic Continuation of the Asai L-function: A new Trace formula approach

Using Langlands's {\it Beyond Endoscopy} idea and analytic number theory techniques, we study the Asai L-function associated to a real quadratic field $\mathbf{K}/\Q.$ If the Asai L-function associated to an automorphic form over $\mathbf{K}$ has a pole, then the form is a base change from $\Q$. We prove this and further prove the analytic continuation of the L-function. This is one of the first examples of using a trace formula to get such information. A hope of Langlands is that general L-functions can be studied via this method.