  Let E be a nonarchimedean local field with residue characteristic l, and
suppose we have an n-dimensional representation of the absolute Galois group
G_E of E over a reduced complete Noetherian local ring A with finite residue
field k of characteristic p different from l. We consider the problem of
associating to any such representation an admissible A[GL_n(E)]-module in a
manner compatible with the local Langlands correspondence at characteristic
zero points of Spec A. In particular we give a set of conditions that uniquely
characterise such an A[GL_n(E)]-module if it exists, and show that such an
A[GL_n(E)]-module always exists when A is the ring of integers of a finite
extension of Q_p. We also use these results to define a "modified mod p local
Langlands correspondence" that is more compatible with specialization of Galois
representations than the mod p local Langlands correspondence of Vigneras.
