A summation formula for triples of quadratic spaces

TL;DR

This paper establishes a Poisson summation formula analogue for a space of triples of quadratic forms over number fields, inspired by conjectures in harmonic analysis and automorphic forms.

Contribution

It introduces a new summation formula for triples of quadratic spaces, extending harmonic analysis techniques to this geometric setting.

Findings

Proved a Poisson summation formula analogue for quadratic space triples

Connected the result to conjectures of Braverman and Kazhdan

Related work to Langlands program and automorphic forms

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 \begin{align*} Y \subset \prod_{i=1}V_i \end{align*} be the closed subscheme consisting of $(v_1,v_2,v_3)$ on which $\mathcal{Q}_1(v_1)=\mathcal{Q}_2(v_2)=\mathcal{Q}_3(v_3)$. Motivated by conjectures of Braverman and Kazhdan and related work of Lafforgue, Ng\^o, and Sakellaridis we prove an analogue of the Poisson summation formula for certain functions on this space.