Heights of vector bundles and the fundamental group scheme of a curve
type: Item
instance of: scholarly article
title: Heights of vector bundles and the fundamental group scheme of a curve
zbMATH Open document ID: 1026.11057
DOI: 10.1215/S0012-7094-03-11723-8
author: Carlo Gasbarri
published in: Duke Mathematical Journal
publication date: 2003-12-12T00:00:00Z
review text: \textit{A. Weil} proved in [J. Math. Pures Appl. (9) 17, 47-87 (1938; Zbl 0018.06302)] that every irreducible vector bundle of degree 0 over a Riemann surface \(X\) comes from a representation of the fundamental group of \(X\). Furthermore, he also proved that a vector bundle \(E\) over \(X\) becomes isomorphic to a direct sum of the trivial bundle over an étale Galois cover of \(X\) if and only if there exist \(f,g\in\mathbb{N}[x]\) such that \(f(E)\simeq g(E)\). Vector bundles satisfying this property are called finite vector bundles. More recently, \textit{M. V. Nori} proved [Compos. Math. 33, 29-41 (1976; Zbl 0337.14016)] that if \(X\) is a projective variety defined over a perfect field \(k\), then the finite bundles form an Abelian category that when endowed with the tensor and the ``restriction to the fiber over a rational point'' functor becomes a Tannaka category. Therefore one associates with it a profinite group scheme \(\pi(X/k.x_0)\) which generalizes the ``usual'' fundamental group. In the paper under review the author starts by giving an alternative approach to the fundamental group scheme which works for reduced schemes over Dedekind schemes. Then he analyzes two situations. In the first one he considers the fundamental group scheme of a smooth projective curve defined over a \(p\)-adic field. In the second one the geometric object is an arithmetic surface. In the first setup, if \(K\) denotes the \(p\)-adic field and \(\pi(X_K/K,x_0)^{[p]}\) the prime to \(p\) part of \(\pi(X_K/K,x_0)\), under the hypothesis that \(X_K\) has good reduction, he proves that the set of isomorphism classes of rational representations \(\rho:\pi(X_K/K,x_0)^{[p]}\to\text{GL}_{N,K}\) is finite. In the second half of the paper he considers representations of the fundamental group scheme of an arithmetic surface \(X\). They are related to a special kind of vector bundles which play a role analogue to that of the finite vector bundles in the geometric (function field) case. T…
reviewed by: Amílcar Pacheco
Mathematics Subject Classification ID: 11G50
Mathematics Subject Classification ID: 14G40
Mathematics Subject Classification ID: 14L15
Mathematics Subject Classification ID: 11G35
Mathematics Subject Classification ID: 14H60
Mathematics Subject Classification ID: 11G30
Mathematics Subject Classification ID: 14D20
zbMATH DE Number: 1933166
zbMATH Keywords: height of vector bundle
zbMATH Keywords: fundamental group scheme
zbMATH Keywords: smooth projective curve over a \(p\)-adic field
zbMATH Keywords: arithmetic surface
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 1933166