Converse theorem on gamma factors

Chu-Feng Nien

College of Mathematics and Statistics

Hunan Normal University

nienpig@hotmail.com

The talk is about converse theorem of gamma factors. After Jacquet's conjecture (`n\times[\frac{n}{2}]` converse theorem) is confirmed, we wonder what information about representations is encoded in `n\times 1` gamma factors in finite field case and its counterpart of level zero cuspidal representations in p-adic case. In a joint work with Lei Zhang, we use number-theoretic result of Gauss sums to verify `n\times 1` Local Converse Theorem of cuspidal representations of `\text{GL}_n(\mathbb{F}_p)`, for prime `p` and `n\leq 5`. After the communication with Zhiwei Yun, he applied geometric method and established `n\times 1` Local Converse Theorem for generic representations of `\text{GL}_n(\mathbb{F}_q)`, when `n<\frac{q-1}{2\sqrt{q}}+1` and `q` is a prime power.