Mod $\ell$ gamma factors and a converse theorem for finite general linear groups

TL;DR

This paper develops new mod $\ell$ gamma factors for finite general linear groups, establishing a converse theorem and functional equations, extending classical results to modular representations and matching characteristic zero cases.

Contribution

It constructs gamma factors valued in arbitrary rings for finite general linear groups and proves a converse theorem, extending classical results to modular representations.

Findings

Constructed new gamma factors for $ ext{GL}_n imes ext{GL}_m$ over arbitrary rings.

Proved a $ ext{GL}_n imes ext{GL}_{n-1}$ converse theorem for cuspidal representations.

Defined an alternative gamma factor in the $ ext{GL}_2 imes ext{GL}_1$ case that matches characteristic zero results.

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 $0$.