\(\ell\)-integral nonabelian Lubin-Tate theory (Q411721)

For two primes \(\ell\neq p\), the author investigates the \(\mathbb {Z}_{\ell}\)-cohomology of the Lubin-Tate towers of a \(p\)-adic field. It realizes some version of Langlands and Jacquet-Langlands correspondences for flat families of irreducible supercuspidal representations parameterized by a \(\mathbb {Z}_{\ell}\)-algebra \(R\) in a way compatible with the extension of scalars. Applied to \(R=\overline{ \mathbb {F}}_{\ell}\), this gives a cohomological realization of the Langlands-Vigneras correspondence for supercuspidals and a new proof of its existence. Applied to complete local algebras, this provides bijections between deformations of matching \(\overline{ \mathbb {F}}_{\ell}\)-representations. Besides, a virtual realization of both the semi-simple Langlands-Vigneras correspondence and the \(\ell\)-modular Langlands-Jacquet transfer for all representations is obtained, by using the cohomology complex and working in a suitable Grothendieck group.