Projective crystalline representations of étale fundamental groups and twisted periodic Higgs-de Rham flow
type: Item
instance of: scholarly article
title: Projective crystalline representations of étale fundamental groups and twisted periodic Higgs-de Rham flow
zbMATH Open document ID: 1493.14031
DOI: 10.4171/JEMS/1116
author: Ruiran Sun
author: Jinbang Yang
author: Kang Zuo
published in: Journal of the European Mathematical Society (JEMS)
publication date: 2022-04-19T00:00:00Z
full work available at URL: https://arxiv.org/abs/1709.01485
review text: Let \(k\) be the algebraic closure of a finite field and let \(X\) be a smooth proper scheme over the Witt ring \(W(k)\). \textit{G. Lan} et al. [J. Eur. Math. Soc. (JEMS) 21, No. 10, 3053--3112 (2019; Zbl 1444.14048)] introduced periodic Higgs-de Rham flows on \(X/W(k)\) and they proved that it is equivalent to the category of crystalline representations of the étale fundamental group of the generic fiber of \(X\). Here the authors introduce the twisted version of periodic Higgs-de Rham flows and they prove that they correspond to projective crystalline representations of the étale fundamental group of the generic fiber of \(X\). They also study the base change of these objects over very ramified valuation rings and show that a stable periodic Higgs bundle gives rise to a geometrically absolutely irreducible crystalline representation. These results are also generalized to the logarithmic setting, when \(X\) is replaced by a pair consisting of \(X\) and a simple normal crossing divisor \(D\) relative to \(W(k)\). Finally, the authors study the dynamics of the map provided by Higgs-de Rham flow on the moduli spaces of rank \(2\) stable Higgs bundles of degree \(1\) on \({\mathbb P}^1\) with logarithmic structure on \(n\) marked points \(D\) for \(n\ge 4\). Using Hrushovski's theorem they produce infinitely many geometrically absolutely irreducible \(\mathrm{PGL}_2({\mathbb Z}^{\mathrm{ur}}_p)\)-crystalline representations of \(\pi _1^{\mathrm{ét}} ({\mathbb P}^1-D)\). They also provide an explicit formula for the self-map of the moduli space in case \(m=4\). Then they pose a few conjectures relating periodic Higgs bundles on \({\mathbb P}^1\) with \(4\) marked points to torsion points on the associated elliptic curve. These results and conjectures are remarkably explicit and can be expressed in an elementary way. They also consider some questions relating periodic Higgs bundles in the \(p\)-adic case to the Hecke eigenforms in the \(\ell\)-adic case posing another c…
reviewed by: Adrian Langer
Mathematics Subject Classification ID: 14F30
Mathematics Subject Classification ID: 14H60
zbMATH DE Number: 7509412
zbMATH Keywords: \(p\)-adic Hodge theory
zbMATH Keywords: crystalline representation
MaRDI profile type: Publication
arXiv ID: 1709.01485
cites work: Langlands correspondence for isocrystals and the existence of crystalline companions for curves
cites work: Q4537720
cites work: Vector bundles on p-adic curves and parallel transport
cites work: On the pro-semisimple completion of the fundamental group of a smooth variety over a finite field
cites work: Stratified bundles and étale fundamental group
cites work: Q4023663
cites work: Q4027533
cites work: Integral crystalline cohomology over very ramified valuation rings
cites work: A \(p\)-adic Simpson correspondence
cites work: Construction de représentations $p$-adiques
cites work: Q4143433
cites work: Q3579591
cites work: Logarithmic structures of Fontaine-Illusie. II
cites work: Nilpotent connections and the monodromy theorem: Applications of a result of Turrittin
cites work: Notes on Motives in Finite Characteristic
cites work: Uniformization of \(p\)-adic curves via Higgs-de Rham flows
cites work: Nonabelian Hodge theory in positive characteristic via exponential twisting
cites work: Semistable Higgs bundles, periodic Higgs bundles and representations of algebraic fundamental groups
cites work: Vektorbündel auf Kurven und Darstellungen der algebraischen Fundamentalgruppe
cites work: Semistable sheaves in positive characteristic
cites work: Semistable modules over Lie algebroids in positive characteristic
cites work: The action of the Frobenius map on rank 2 vector bundles in characteristic 2
cites work: Lectures on an introduction to Grothendieck's theory of the fundamental group
cites work: Nonabelian Hodge theory in characteristic \(p\)
cites work: The generalized Verschiebung map for curves of genus 2
cites work: Constructing Variations of Hodge Structure Using Yang-Mills Theory and Applications to Uniformization
cites work: Galois Groups and Fundamental Groups
OpenAlex ID: W3198513875
description: scientific article