Local tame lifting for GL(n) III:: explicit base change and Jacquet-Langlands correspondence

Let F be a finite extension of Q(p) with p not equal 2, and D a central F-division algebra of dimension p(2m). Let pi be an irreducible supercuspidal representation of GL(pm) (F). The Jacquet-Langlands correspondence associates to 7r an irreducible smooth representation pi D of D-x, determined up to isomorphism by a character relation. Using a variant of the description of irreducible supercuspidal representations of GL(n)(F) as induced representations, due to Bushnell and Kutzko, along with a parallel description for D-x due to Broussous, we give an explicit realization of the correspondence pi -> pi D in the case where pi is totally ramified. This is a step towards our main result. Let KIF be a finite unramified extension, and pi a totally ramified supercuspidal representation of GL(pm) (F). Base change, in the sense of Arthur and Clozel, gives a totally ramified supercuspidal representation b(K/F)pi of GL(pm) (K). In earlier work, the authors gave an explicit definition of a representation l(K/F)pi and showed that l(K/F)pi = b(K/F)pi when p does not divide the degree of KIF. We complete this by showing that l(K/F)pi = b(K/F)pi for all K/F. The proof relies on evaluating the twisted character of l(K/F)pi in terms of the character of pi(D) and then using the explicit Jacquet-Langlands correspondence. Many of the central arguments remain valid when F is a non-Archimedean local field of odd positive characteristic.