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.

We give sharp point-wise bounds in the weight-aspect on fourth moments of modular forms on arithmetic hyperbolic surfaces associated to Eichler orders. Thereby, we strengthen a result of Xia and extend it to co-compact lattices. We realize this fourth moment by constructing a holomorphic theta kernel on $\mathbf {G} \times \mathbf {G} \times \mathbf {SL}_{2}$, for $\mathbf {G}$ an indefinite inner form of $\mathbf {SL}_2$ over $\mathbb {Q}$, based on the Bergman kernel, and considering its $L^2$-norm in the Weil variable. The constructed theta kernel further gives rise to new elementary theta series for integral quadratic forms of signature $(2,2)$.

A crucial ingredient in the theory of theta liftings of Kudla and Millson is the construction of a $q$-form $\varphi_{KM}$ on an orthogonal symmetric space, using Howe's differential operators. This form can be seen as a Thom form of a real oriented vector bundle. We show that the Kudla-Millson form can be recovered from a canonical construction of Mathai and Quillen. A similar result was obtaind by Garcia for signature $(2,q)$ in case the symmetric space is hermitian and we extend it to arbitrary signature.

In this note, assuming the nonvanishing result of explicit theta correspondence for the symplectic–orthogonal dual pair over quaternion algebra $\mathbb {H}$, we show that, for metapletic–orthogonal dual pair over $\mathbb {R}$ and the symplectic–orthogonal dual pair over quaternion algebra $\mathbb {H}$, the theta correspondence is compatible with tempered condition by directly estimating the matrix coefficients, without using the classification theorem.

In an earlier paper of Wee Teck Gan and Gordan Savin, the local Langlands correspondence for metaplectic groups over a nonarchimedean local field of characteristic zero was established. In this paper, we formulate and prove a local intertwining relation for metaplectic groups assuming the local intertwining relation for non-quasi-split odd special orthogonal groups.

We show a Siegel–Weil formula in the setting of exceptional theta correspondence. Using this, together with a new Rankin–Selberg integral for the Spin L-function of $\text{PGSp}_{6}$ discovered by Pollack, we prove that a cuspidal representation of $\text{PGSp}_{6}$ is a (weak) functorial lift from the exceptional group $G_{2}$ if its (partial) Spin L-function has a pole at $s=1$.

Using theta correspondence, we classify the irreducible representations of Mp2n in terms of the irreducible representations of SO2n+1 and determine many properties of this classification. This is a local Shimura correspondence which extends the well-known results of Waldspurger for n=1.

We determine essentially completely the theta correspondence arising from the dual pair ${\it PGL}_3 \times G_2 \subset E_6$ over a p-adic field. Our first result determines the theta lift of any non-supercuspidal representation of PGL3 and shows that the lifting respects Langlands functoriality. Our second result shows that the theta lift $\theta(\pi)$ of a (non-self-dual) supercuspidal representation $\pi$ of PGL3 is an irreducible generic supercuspidal representation of G2; we also determine $\theta(\pi)$ explicitly when $\pi$ has depth zero.