Automorphic induction and Leopoldt type conjectures for $GL(N)$

Introduction. Let F be a number field and G0 be a non-split inner form of GL(n) /F split at a rational prime p and ∞. We define G = ResF/QG0. We start with cohomological study of automorphic forms on the reductive algebraic group G /Q which is isomorphic to a product of copies of GL(n) over a number field. In particular, under not so restrictive conditions on G, we shall show that the nearly p–ordinary