Duke Mathematical Journal

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

Carlo Gasbarri

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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 $X$.

Article information

Duke Math. J., Volume 117, Number 2 (2003), 287-311.

First available in Project Euclid: 26 May 2004

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 14G40: Arithmetic varieties and schemes; Arakelov theory; heights [See also 11G50, 37P30]
Secondary: 11G30: Curves of arbitrary genus or genus = 1 over global fields [See also 14H25] 11G50: Heights [See also 14G40, 37P30]


Gasbarri, Carlo. Heights of vector bundles and the fundamental group scheme of a curve. Duke Math. J. 117 (2003), no. 2, 287--311. doi:10.1215/S0012-7094-03-11723-8. https://projecteuclid.org/euclid.dmj/1085598371

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