Summation formulae for quadrics
Fuente:
arXiv
Enregistré dans:
| Auteur principal: | |
|---|---|
| Format: | Preprint |
| Publié: |
2022
|
| Sujets: | |
| Accès en ligne: | |
| Tags: |
Ajouter un tag
Pas de tags, Soyez le premier à ajouter un tag!
|
| _version_ | 1866915093715877888 |
|---|---|
| author | Getz, Jayce R. |
| author_facet | Getz, Jayce R. |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2201_02583 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | Summation formulae for quadrics Getz, Jayce R. Number Theory Classical Analysis and ODEs Representation Theory Primary 11F70, Secondary 11E12, 11F27 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. |
| title | Summation formulae for quadrics |
| topic | Number Theory Classical Analysis and ODEs Representation Theory Primary 11F70, Secondary 11E12, 11F27 |
| url | https://arxiv.org/abs/2201.02583 |