Summation formulae for quadrics
Jayce R. Getz
TL;DR
The paper proves a Poisson-type summation formula for the vanishing locus $X_\ell$ of a quadratic form in an even number of variables by exploiting the Arthur-truncated theta lift of the trivial representation of $\mathrm{SL}_2(\mathbb{A}_F)$ and the Weil representation. It expresses the global sum as a balance between boundary terms and bulk contributions, with all terms given in geometric language and no test-function restrictions, and shows that boundary terms decompose into constants or sums over smaller quadrics via explicit operators $I$, $c_i$, $d_i$, and the Fourier transform $\mathcal{F}_{X_\ell}$. The work provides a canonical, invariant framework by introducing framed flags and a universal flag, and proves key invariance and canonicity properties of the Schwartz spaces $\mathcal{S}(X_i(\mathbb{A}_F))$. By connecting to the circle method, minimal representations, and theta correspondences, the results yield a robust tool for counting or estimating points on quadrics and for exploring poles of related $L$-functions in analytic number theory.
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 Schrödinger representation of the metaplectic group.