Algebraic and analytic parallel transport on \(p\)-adic curves
type: Item
instance of: scholarly article
title: Algebraic and analytic parallel transport on \(p\)-adic curves
zbMATH Open document ID: 1277.14025
author: Thomas Ludsteck
published in: Abhandlungen aus dem Mathematischen Seminar der Universität Hamburg
publication date: 2013-06-13T00:00:00Z
review text: This paper deals with the Riemann-Hilbert correspondence and its generalizations. In the classical case, given a Riemann surface \(S\), the Riemann-Hilbert correspondence links complex representations of the fundamental group of \(S\) and \(C^{\infty}\)-vector bundles with flat connections on \(S\). This is a reflection of the idea of parallel transport. Both algebraic and topological \(p\)-adic analogues of the Riemann-Hilbert correspondence have been studied. In the algebraic \(p\)-adic case, Deninger and Werner associated \(\mathbb{C}_p\)-representations of the algebraic fundamental group of a smooth projective curve \(X\) over \(\overline{\mathbb{Q}_p}\) to a certain class \(\mathfrak{B}^s_{X_{\mathbb{C}_p}}\) of vector bundles on \(X_{\mathbb{C}_p}\). In the topological \(p\)-adic case, Berkovich associated \(\mathbb{C}_p\)-representations of the fundamental group of a smooth \(\mathbb{C}_p\)-analytic space to locally unipotent vector bundles on the space. In the paper under review, the starting point is an integral \(\mathbb{C}_p\)-representation of the algebraic fundamental group of a smooth curve \(X\) over \(\mathbb{Q}_p\) that factors over the fundamental group of a normal, flat, projective model. The author shows that one can attach a vector bundle with connection to such a representation, and that this vector bundle lies in the class for which Deninger and Werner's work applies. Furthermore, the construction is inverse to theirs. Secondly, if one starts with a \(\mathbb{C}_p\)-representation of the topological fundamental group of the Berkovich analytification of \(X\), this gives a vector bundle with connection on \(X_{\mathbb{C}_p}^{an}\). Also, taking the profinite completion gives a representation of the algebraic fundamental group of \(X\) for which the previous paragraph applies, thus yielding an algebraic vector bundle with connection. The author shows that the analytic vector bundle with connection is the analytification of the algebraic vector b…
reviewed by: Andrew Obus
Mathematics Subject Classification ID: 14H30
Mathematics Subject Classification ID: 14H60
Mathematics Subject Classification ID: 14H70
Mathematics Subject Classification ID: 14G22
Mathematics Subject Classification ID: 11G20
zbMATH DE Number: 6175552
zbMATH Keywords: \(p\)-adic geometry, Riemann-Hilbert correspondence
zbMATH Keywords: berkovich analytic space
zbMATH Keywords: parallel transport
MaRDI profile type: Publication
full work available at URL: s12188-013-0076-0
OpenAlex ID: W2067161967
cites work: Étale cohomology for non-Archimedean analytic spaces
cites work: Integration of One-forms on P-adic Analytic Spaces. (AM-162)
cites work: Formal and rigid geometry. I: Rigid spaces
cites work: Q5518494
cites work: The monodromy pairing.
cites work: Vector bundles on p-adic curves and parallel transport
cites work: A \(p\)-adic Simpson correspondence
cites work: Rigid analytic geometry and its applications
cites work: Theory of tubular neighborhood in étale topology
cites work: Q2551490
cites work: \(p\)-adic representations and vector bundles on Mumford curves
cites work: Unipotent Schottky bundles on Riemann surfaces and complex tori
cites work: Tempered Fundamental Group and Graph of the Stable Reduction
cites work: Q5484940
cites work: Q5748213
cites work: Homogeneous \(p\)-adic vector bundles on abelian varieties that are analytic tori. II
cites work: Galois Groups and Fundamental Groups
cites work: The direct image theorem in formal and rigid geometry
DOI: 10.1007/S12188-013-0076-0
description: scientific article; zbMATH DE number 6175552