Converse Theorem Meets Gauss Sums (with an appendix by Zhiwei Yun)

Abstract

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.