Summation formulae for quadrics

TL;DR

This paper establishes a Poisson summation formula for quadratic forms in even dimensions, linking it to classical solution counting problems and involving advanced theta lift computations.

Contribution

It introduces a novel Poisson summation formula for quadrics with boundary terms expressed via constants or smaller quadrics, expanding classical analysis methods.

Findings

Derived a new summation formula for zero loci of quadratic forms

Connected the formula to classical solution counting problems

Computed the theta lift of the trivial representation for orthogonal groups

Abstract

We prove a Poisson summation formula for the zero locus of a quadratic form in an even number of variables with no assumption on the support of the functions involved. The key novelty in the formula is that all ``boundary terms'' are given either by constants or sums over smaller quadrics related to the original quadric. We also discuss the link with the classical problem of estimating the number of solutions of a quadratic form in an even number of variables. To prove the summation formula we compute (the Arthur truncated) theta lift of the trivial representation of $\mathrm{SL}_2(\mathbb{A}_F)$. As previously observed by Ginzburg, Rallis, and Soudry, this is an analogue for orthogonal groups on vector spaces of even dimension of the global Schr\"odinger representation of the metaplectic group.