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.