逆定理遇到高斯和（附云志伟）
Converse Theorem Meets Gauss Sums (with an appendix by Zhiwei Yun)
摘要：
本文证明了云志伟在附录中给出的关于${\rm GL}_n({\mathbb F}_p)$的不可约尖点表示、$n\leq 5,$的不可约尖点表示和$n<\frac{q-1}{2\sqrt{q}}+1$的不可约一般表示的扭曲Gamma因子的$n\times 1$局部逆定理，其中$p$是素数，q是$p$的幂.零级尖点表示的$n\times 1$逆定理的对应定理也遵循${\rm GL}_n({\mathcal F})$和${\rm GL}_n({\mathbb F}_q)$的伽玛因子之间建立的关系，其中${\mathcal F}$表示其剩余域同构于${\mathbb F}_q.$的$p$ -进域。对于$n=6,$例子，给出了有限域上的$n\times 1$局部逆定理，并提出了一组本原表示，对于$m,\ n\in {\mathbb N},$，基于Langlands函子提升的思想，我们提出了一个猜想，将有限域上的$n\times m$伽玛因子与扩展域上的高斯和联系起来.
英文摘要：
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.
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