Fibrés vectoriels semi-stables sur une courbe de Mumford. (Semistable vector fibers on a Mumford curve) (Q1059121)

Semi-stable vector bundles of degree zero on a compact Riemann surface of genus \(\geq 2\) correspond to unitary representations of the fundamental group. This theorem of M. Narasimhan and C. S. Seshadri has a rigid analytic version recently proved by G. Faltings. In this version the curve is a Mumford curve and the unitary representation is replaced by a \(\Phi\)-bounded representation of a Schottky group. In the present paper this theorem of Faltings is proved for arbitrary non-archimedean valued fields. New ideas and simplifications are introduced. In particular sheaves of normed vector spaces and their cohomology are introduced. The cohomology groups of a sheaf of normed vector spaces corresponding to a vector-bundle on the Mumford curve turn out to be free modules of finite rank over a certain non commutative ring. Using these cohomology groups one finds in an explicit way the r-dimensional \(\Phi\)-bounded representations which correspond to a given semi-stable vector bundle on the Mumford curve.