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

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 $\ell \ne p$. 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 $0$.