This paper verifies $n\times 1$ Local Converse Theorem for twisted gamma factors of irreducible cuspidal representations of ${ \rm GL}_n({ \mathbb F}_p)$, for $n\leq 5,$ and of irreducible generic representations, for $n<\frac{ q-1}{ 2\sqrt{ q}}+1$ in the appendix by Zhiwei Yun, where $p$ is a prime and q is a power of $p$. The counterpart of $n\times 1$ converse theorem for level zero cuspidal representations also follows the established relation between gamma factors of ${ \rm GL}_n({ \mathcal F})$ and that of ${ \rm GL}_n({ \mathbb F}_q)$, where ${ \mathcal F}$ denotes a $p$-adic field whose residue field is isomorphic to ${ \mathbb F}_q.$ For $n=6,$ examples failed $n\times 1$ Local Converse Theorem over finite fields are provided and the authors propose a set of primitive representations, for which $n\times 1$ gamma factors should be able to detect a unique element in it. For $m,\ n\in { \mathbb N},$ in the spirit of Langlands functorial lifting, we formulate a conjecture to relate $n\times m$ gamma factors of finite fields with Gauss sums over extended fields.
Categories: math.NT math.RT