  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.
