Abstract We establish the non-vanishing mod p of global theta lifts from an odd definite orthogonal group O 2 n + 1 O₂₍+₁ over ℚ Q to a metaplectic group Mp 4 n Mp₄₍ over ℚ Q under mild conditions. The problem is closely related to non-vanishing modulo p of toric integrals on O 2 n + 1 O₂₍+₁. For this, we exploit the distribution properties of toric orbits of unipotent elements on O 2 n + 1 (ℚ ℓ) O₂₍+₁ (Q_{) } using Ratner’s theorems on unipotent flows and we deduce that the toric integral of a p -primitive automorphic form on O 2 n + 1 O₂₍+₁ is non-zero modulo p for infinitely many characters.
Xiaoyu Zhang (Thu,) studied this question.
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