  For two distinct primes p and l, we investigate the Z_l-cohomology of the
Lubin-Tate towers of a p-adic field. We prove that it realizes some version of
Langlands and Jacquet-Langlands correspondences for flat families of
irreducible supercuspidal representations parametrized by a Z_l-algebra R, in a
way compatible with extension of scalars. When R is a field of characteristic
l, this gives a cohomological realization of the Langlands-Vigneras
correspondence for supercuspidals, and a new proof of its existence. When R
runs over complete local algebras, this provides bijections between
deformations of matching mod-l representations. Roughly speaking, we can
decompose "the supercuspidal part" of the l-integral cohomology as a direct
sum, indexed by irreducible supercuspidals \pi mod l, of tensor products of
universal deformations of \pi and of its two mates.
  Besides, we also get a virtual realization of both the semi-simple
Langlands-Vigneras correspondence and the l-modular Langlands-Jacquet transfer
for all representations, by using the cohomology complex and working in a
suitable Grothendieck group.
