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 ℓ-adic families of co-Whittaker representations of GL n (F ), where F is a finite extension of Q p and ℓ = 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 ℓadic family. These two theorems are closely related to the local Langlands correspondence in ℓ-adic families.

Characterizing the Mod- Local Langlands Correspondence by Nilpotent Gamma Factors

“…Proof of Theorem 7.1. The only part of the proof of the main theorem in [LM19] that requires reducedness and -torsion freeness is the completeness of Whittaker models statement [LM19, Theorem 4.1]. After replacing Theorem 4.1 in [LM19] with Theorem 5.1 of the present paper and applying the argument of [LM19, Sections 4-6] mutatis mutandis, we obtain Theorem 7.1.…”

“…Recently, the proof of Jacquet's conjecture was extended to the setting of reduced -torsion-free W (k)-algebras in the preprint [LM19] of Liu and the author. As the counterexample in Section 2 of the present article shows, the converse theorem in the -torsion setting requires a different framework from [LM19].…”