The local converse theorem for quasi-split $\mathrm {O}_{2n}$ and $\mathrm {SO}_{2n}$

Abstract Let F be a non-archimedean local field of characteristic not equal to 2. In this article, we prove the local converse theorem for quasi-split $\mathrm {O}_{2n}(F)$ and $\mathrm {SO}_{2n}(F)$ , via the description of the local theta correspondence between $\mathrm {O}_{2n}(F)$ and $\mathrm {Sp}_{2n}(F)$ . 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 $\mathrm {O}_{2n}(\mathbb {A})$ and $\mathrm {SO}_{2n}(\mathbb {A})$ , respectively, where $\mathbb {A}$ is a ring of adele of a global number field L .