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

TL;DR

This paper establishes a converse theorem for quasi-split non-split even special orthogonal groups over finite fields, addressing challenges from outer automorphisms and non-split tori.

Contribution

It introduces new methods to handle the non-split torus, extending the converse theorem to a broader class of orthogonal groups over finite fields.

Findings

Proves a converse theorem for quasi-split non-split even special orthogonal groups.

Develops techniques to manage outer automorphisms in the proof.

Overcomes difficulties posed by the non-split torus structure.

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 which 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.