A converse theorem for quasi-split even special orthogonal groups over finite fields

Abstract

We prove a converse theorem for the case of quasi-split non-split even special orthogonal groups over finite fields. There are two main difficulties that arise from the outer automorphism and non-split part of the torus. The outer automorphism is handled similarly to the split case, while new ideas are developed to overcome the non-split part of the torus.