Heights of vector bundles and the fundamental group scheme of a curve

Abstract

Let $X$ be a scheme; the fundamental group scheme of $X$, when it exists, is a profinite group scheme that classifies principal homogeneous spaces under finite flat group schemes over $X$. We generalize the construction of the fundamental group scheme given by M. Nori [No] to the case when $X$ is a reduced flat scheme over a Dedekind scheme. We prove that if $X$ is a curve over a $p$-adic field having good reduction, then the prime-to-$p$ part of the fundamental group scheme of $X$ has only finitely many rational representations in ${\rm GL}\sb N$. In the second part of the paper, using tools from Arakelov theory, we construct an intrinsic height on the moduli space of semistable vector bundles (of fixed rank and degree) over a curve defined over a number field. We finally 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 $X$.