Intrinsic heights of vector bundles and representations of the fundamental group scheme of an arithmetic surface

Abstract Let $X$ be a scheme, the fundamental group scheme of $X$, when it exists, is a profinite group scheme which classifies principal homogeneus spaces under finite flat group schemes over $X$. We generalize the construction of the fundamental group scheme given by Nori [No], to the case when $X$ is a reduced flat scheme over a Dedekind scheme. Using tools from Arakelov theory, we construct an intrinsic height on the moduli space of semi--stable vector bundles (of fixed rank and degree) over a curve defined over a number field. We prove that the height of vector bundles over an arithmetic surface $X$, coming from representations of the fundamental group scheme is upper bounded; so we deduce that there are only finitely many isomorphism classes of rational representations of the fundamental group scheme of $X$.