The local converse theorem for quasi-split $O_{2n}$ and $SO_{2n}$

Abstract: Let $F$ be a non-archimedean local field of characteristic not equal to 2. In this paper, we prove the local converse theorem for quasi-split $\O_{2n}(F)$ and $\SO_{2n}(F)$, via the description of the local theta correspondence between $\O_{2n}(F)$ and $\Sp_{2n}(F)$. More precisely, as a main step, we explicitly describe the precise behavior of the $\gamma$-factors under the correspondence. Furthermore, we apply our results to prove the weak rigidity theorems for irreducible generic cuspidal automorphic representations of $\O_{2n}(\A)$ and $\SO_{2n}(\mathbb{A})$, respectively, where $\A$ is a ring of adele of a global number field $L$.