Non-vanishing mod p of theta lifts for (O2n+1, Mp4n )

We establish the non-vanishing mod p of global theta lifts from an odd definite orthogonal group <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msub> <m:mi mathvariant="normal">O</m:mi> <m:mrow> <m:mrow> <m:mn>2</m:mn> <m:mo>⁢</m:mo> <m:mi>n</m:mi> </m:mrow> <m:mo>+</m:mo> <m:mn>1</m:mn> </m:mrow> </m:msub> </m:math> {O₂ₙ₊₁} over <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mi>ℚ</m:mi> </m:math> {Q} to a metaplectic group <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msub> <m:mi>Mp</m:mi> <m:mrow> <m:mn>4</m:mn> <m:mo>⁢</m:mo> <m:mi>n</m:mi> </m:mrow> </m:msub> </m:math> {Mp₄ₙ} over <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mi>ℚ</m:mi> </m:math> {Q} under mild conditions. The problem is closely related to non-vanishing modulo p of toric integrals on <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msub> <m:mi mathvariant="normal">O</m:mi> <m:mrow> <m:mrow> <m:mn>2</m:mn> <m:mo>⁢</m:mo> <m:mi>n</m:mi> </m:mrow> <m:mo>+</m:mo> <m:mn>1</m:mn> </m:mrow> </m:msub> </m:math> {O₂ₙ₊₁} . For this, we exploit the distribution properties of toric orbits of unipotent elements on <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:msub> <m:mi mathvariant="normal">O</m:mi> <m:mrow> <m:mrow> <m:mn>2</m:mn> <m:mo>⁢</m:mo> <m:mi>n</m:mi> </m:mrow> <m:mo>+</m:mo> <m:mn>1</m:mn> </m:mrow> </m:msub> <m:mo>⁢</m:mo> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:msub> <m:mi>ℚ</m:mi> <m:mi mathvariant="normal">ℓ</m:mi> </m:msub> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:mrow> </m:math> {O₂ₙ₊₁(Q_{})} using Ratner’s theorems on unipotent flows and we deduce that the toric integral of a p -primitive automorphic form on <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msub> <m:mi mathvariant="normal">O</m:mi> <m:mrow> <m:mrow> <m:mn>2</m:mn> <m:mo>⁢</m:mo> <m:mi>n</m:mi> </m:mrow> <m:mo>+</m:mo> <m:mn>1</m:mn> </m:mrow> </m:msub> </m:math> {O₂ₙ₊₁} is non-zero modulo p for infinitely many characters.