On the local converse theorem and the descent theorem in families

TL;DR

This paper extends the local converse and descent theorems to -adic families of representations of GL_n(F), linking them to the local Langlands correspondence in -adic families.

Contribution

It proves analogues of Jacquet's conjecture for -adic families, advancing understanding of the local Langlands correspondence in this context.

Findings

Established an -adic analogue of Jacquet's local converse theorem.

Proved an -adic analogue of the descent theorem related to gamma factors.

Connected these results to the local Langlands correspondence in -adic families.

Abstract

We prove an analogue of Jacquet's conjecture on the local converse theorem for \ell-adic families of co-Whittaker representations of GL_n(F), where F is a finite extension of Q_p and \ell does not equal 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.