On the local converse theorem and the descent theorem in families

Abstract

In this paper, we prove an analogue of Jacquet’s conjecture on the local converse theorem for \(\ell \)-adic families of co-Whittaker representations of \({\mathrm {GL}}_n(F)\), where F is a finite extension of \({\mathbb {Q}}_p\) and \(\ell \ne p\). We also prove an analogue of Jacquet’s conjecture for a descent theorem, which asks for the smallest collection of gamma factors determining the subring of definition of an \(\ell \)-adic family. These two theorems are closely related to the local Langlands correspondence in \(\ell \)-adic families.