Mod $\ell$ gamma factors and a converse theorem for finite general linear groups
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arXiv
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| Auteurs principaux: | , , , , , |
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| Format: | Preprint |
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2023
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| _version_ | 1866913992319959040 |
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| author | Bakeberg, Jacksyn Gerbelli-Gauthier, Mathilde Goodson, Heidi Iyengar, Ashwin Moss, Gilbert Zhang, Robin |
| author_facet | Bakeberg, Jacksyn Gerbelli-Gauthier, Mathilde Goodson, Heidi Iyengar, Ashwin Moss, Gilbert Zhang, Robin |
| contents | The local converse theorem for Rankin-Selberg gamma factors of $\mathrm{GL}_2(\mathbb{F}_q)$ proved by Piatetski-Shapiro over $\mathbb{C}$ no longer holds after reduction modulo $\ell \neq p$. To remedy this, we construct new $\mathrm{GL}_n \times \mathrm{GL}_m$ gamma factors valued in arbitrary $\mathbb{Z}[1/p, ζ_p]$-algebras for Whittaker-type representations, show that they satisfy a functional equation, and then prove a $\mathrm{GL}_n \times \mathrm{GL}_{n-1}$ converse theorem for irreducible cuspidal representations. In the $\mathrm{GL}_2 \times \mathrm{GL}_1$ case, we define an alternative "new" gamma factor, which takes values in $k$ and satisfies a converse theorem that matches the converse theorem in characteristic $0$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2307_07593 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Mod $\ell$ gamma factors and a converse theorem for finite general linear groups Bakeberg, Jacksyn Gerbelli-Gauthier, Mathilde Goodson, Heidi Iyengar, Ashwin Moss, Gilbert Zhang, Robin Number Theory Representation Theory 20C33 (primary) 20C20, 11L05, 20G40, 11S40 (Secondary) The local converse theorem for Rankin-Selberg gamma factors of $\mathrm{GL}_2(\mathbb{F}_q)$ proved by Piatetski-Shapiro over $\mathbb{C}$ no longer holds after reduction modulo $\ell \neq p$. To remedy this, we construct new $\mathrm{GL}_n \times \mathrm{GL}_m$ gamma factors valued in arbitrary $\mathbb{Z}[1/p, ζ_p]$-algebras for Whittaker-type representations, show that they satisfy a functional equation, and then prove a $\mathrm{GL}_n \times \mathrm{GL}_{n-1}$ converse theorem for irreducible cuspidal representations. In the $\mathrm{GL}_2 \times \mathrm{GL}_1$ case, we define an alternative "new" gamma factor, which takes values in $k$ and satisfies a converse theorem that matches the converse theorem in characteristic $0$. |
| title | Mod $\ell$ gamma factors and a converse theorem for finite general linear groups |
| topic | Number Theory Representation Theory 20C33 (primary) 20C20, 11L05, 20G40, 11S40 (Secondary) |
| url | https://arxiv.org/abs/2307.07593 |