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

TL;DR

This paper employs a novel trace formula approach combined with analytic number theory to establish the analytic continuation of the Asai L-function and characterize base change automorphic forms over real quadratic fields.

Contribution

It introduces a new trace formula method to prove the analytic continuation of the Asai L-function and characterizes base change automorphic forms, advancing Langlands's Beyond Endoscopy program.

Findings

Proves the analytic continuation of the Asai L-function.

Characterizes when automorphic forms are base change from ield.

Demonstrates the effectiveness of trace formula in this context.

Abstract

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.