On the local converse theorem and the descent theorem in 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.

32 pages

References in corpus (3)

• Towards the Jacquet Conjecture on the Local Converse Problem for $p$-adic ${\mathrm{GL}}_n$
• A local converse theorem for GL(n) (archimedean case)
• Converse theorems and the local Langlands correspondence in families