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

The local converse theorem for Rankin-Selberg gamma factors of ${\mathrm{G}\mathrm{L}}_2({\mathbb{F}}_q)$ proved by Piatetski-Shapiro over $\mathbb{C}$ no longer holds after reduction modulo $\mathrm{\ell }\mathrm{\ne }p$. To remedy this, we construct new ${\mathrm{G}\mathrm{L}}_n\times {\mathrm{G}\mathrm{L}}_m$ gamma factors valued in arbitrary $\mathbb{Z}[1\mathrm{/}p,{\zeta }_p]$-algebras for Whittaker-type representations, show that they satisfy a functional equation, and then prove a ${\mathrm{G}\mathrm{L}}_n\times {\mathrm{G}\mathrm{L}}_{n-1}$ converse theorem for irreducible cuspidal representations. In the ${\mathrm{G}\mathrm{L}}_2\times {\mathrm{G}\mathrm{L}}_1$ 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 $0$.