Mod $\ell$ gamma factors and a converse theorem for finite general linear groups
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Bibliographic record
Abstract
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,\zeta_{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 .
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Codex and Gemma teacher scores by category
| Category | Codex | Gemma |
|---|---|---|
| Metaresearch | 0.001 | 0.001 |
| Meta-epidemiology (narrow) | 0.001 | 0.001 |
| Meta-epidemiology (broad) | 0.001 | 0.001 |
| Bibliometrics | 0.000 | 0.001 |
| Science and technology studies | 0.001 | 0.000 |
| Scholarly communication | 0.000 | 0.001 |
| Open science | 0.001 | 0.001 |
| Research integrity | 0.000 | 0.000 |
| Insufficient payload (model declined to judge) | 0.001 | 0.000 |
Machine scores (provisional)
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Baseline scores from an immature model (maturity gate not passed, 7 training rounds). Scores rank; they never assert a category.
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