Some local-global non-vanishing results of theta lifts for symplectic-orthogonal dual pairs

Abstract

Following the approach of B. Roberts, we characterize the non-vanishing of the global theta lift for symplectic-orthogonal dual pairs in terms of its local counterpart. In particular, we replace the temperedness assumption present in Robert's work by a certain weaker assumption, and apply our results to small rank similitude groups. Among our applications is a certain instance of Langlands functorial transfer of a (non-generic) cuspidal automorphic representation of GSpð4Þ to GLð4Þ.

On the Bloch-Kato conjecture for the Asai L-function

“…The proof of Theorem 8.5 of [Rob01] provides a suitable four dimensional quadratic space X and representation σ ∈ Irr temp cusp (GO(X, A)) (which is constructed out of π ⊞ χ via Jacquet-Langlands) such that θ 2 (σ v ) ∨ = Π v . That the global theta lift of σ = ⊗ v σ v is non-vanishing follows from [Pau05] Corollary 4.17 (archimedean non-vanishing) and [Tak09] Theorem 1.2 (local-global nonvanishing). To show that Π is cuspidal automorphic we apply [Tak09] Theorem 1.3(2) and [HST93] Lemma 5.…”

“…That the global theta lift of σ = ⊗ v σ v is non-vanishing follows from [Pau05] Corollary 4.17 (archimedean non-vanishing) and [Tak09] Theorem 1.2 (local-global nonvanishing). To show that Π is cuspidal automorphic we apply [Tak09] Theorem 1.3(2) and [HST93] Lemma 5. The statement about the existence of holomorphic representations in the L-packet at infinity follows from [HST93] Lemma 12 and Corollary 3 (see also [Tak11] Proposition 6.5(2)), which shows that it contains two elements, a limit of large and of holomorphic discrete series representation) exactly for χ = χ k K/Q .…”

Descending congruences of theta lifts on GSp4

“…Let p = ℓ be in S,a n dR ′ p be the associated Galois representation to Π ′ p . Firstly, Π ′ p is a local theta lift since Π ′ is a non-zero global theta lift (see also Theorem 1.3 of [25]). The fact that R ′ p has an admissible inertial type ensures that the splitting behaviour of p in L is the same as the one in K.A sΠ ′ is a theta lift and Π ′ ℓ is unramified (as ℓ/ ∈ S), we see that ℓ is unramified in L (Proposition 4.2 of [16]).…”