The Fourier transform for triples of quadratic spaces

TL;DR

This paper establishes a well-defined Fourier transform for a specific quadratic space scheme using a novel global-to-local approach, extending Poisson summation to boundary terms for the first time in this context.

Contribution

It introduces a new global-to-local method to prove the Fourier transform's well-definedness on Schwartz space for a quadratic space scheme, including boundary terms.

Findings

Proved the Fourier transform is well-defined on Schwartz space.

Extended Poisson summation formula to include boundary terms.

First such summation formula for a non-Braverman-Kazhdan spherical variety.

Abstract

Let $V_1,V_2,V_3$ be a triple of even dimensional vector spaces over a number field $F$ equipped with nondegenerate quadratic forms $\mathcal{Q}_1,\mathcal{Q}_2,\mathcal{Q}_3$, respectively. Let $Y \subset \prod_{i=1}^3 V_i$ be the closed subscheme consisting of $(v_1,v_2,v_3)$ such that $\mathcal{Q}_1(v_1)=\mathcal{Q}_2(v_2)=\mathcal{Q}_3(v_3)$. One has a Poisson summation formula for this scheme under suitable assumptions on the functions involved, but the relevant Fourier transform was previously only defined as a correspondence. In the current paper we employ a novel global-to-local argument to prove that this Fourier transform is well-defined on the Schwartz space of $Y(\mathbb{A}_F).$ To execute the global-to-local argument, we introduce boundary terms and thereby extend the Poisson summation formula to a broader class of test functions. This is the first time a summation formula with boundary terms has been proven for a spherical variety that is not a Braverman-Kazhdan space.