Lubin-Tate and Drinfeld bundles
type: Item
instance of: scholarly article
title: Lubin-Tate and Drinfeld bundles
zbMATH Open document ID: 1287.11076
author: Jan Kohlhaase
published in: Tôhoku Mathematical Journal. Second Series
publication date: 2011-10-10T00:00:00Z
review text: Period rings play a very important role in the study of \(p\)-adic Galois-representations of \(p\)-adic fields. In view of the \(p\)-adic Langlands programme, it is natural to ask whether it is possible to also study representations of \(p\)-adic reductive groups using suitable rings of periods. The paper under review is a first and promising attempt to this question. Let \(K\) be a nonarchimedean local field and denote by \(\mathfrak{o}\) its ring of integers, \(\pi\in\mathfrak{o}\) a fixed uniformizer and let \(h\geq 1\) be an integer. Let \(\check{K}\) be the completion of the maximal unramified extension of \(K\) and \(\check{\mathfrak{o}}\) its valuation ring. Further denote by \(\check{K}_m\) the field obtained by adjoining the \(\pi^m\)-torsion points of a one-dimensional Lubin-Tate formal \(\mathfrak{o}\)-module to \(\check{K}\). For any integer \(m\geq 0\) let \(\mathcal{Y}_m^{(h)}\) be the generic fibre of the formal \(\check{\mathfrak{o}}\)-scheme parametrizing one dimensional formal \(\mathfrak{o}\)-modules of height \(h\) and level structure \(m\) constructed by \textit{V. G. Drinfeld} [Math. USSR, Sb. 23(1974), 561--592 (1976); translation from Mat. Sb., N. Ser. 94(136), 594--627 (1974; Zbl 0321.14014)]. This space admits a commuting left action of the groups \(G_0^{(h)}=\mathrm{GL}_h(\mathfrak{o})\) and \(H_0^{(h)}=\mathfrak{o}_D^\times\) where \(D\) denotes the central division algebra of invariant \(1/h\) over \(K\) and \(\mathfrak{o}_D\) stands for its ring of integers. On the other hand, let \(\mathcal{X}_0^{(h)}:=\Omega_K^{(h)}\times_K \check{K}\) where \(\Omega_K^{(h)}\) is Drinfeld's upper half space of dimension \(h-1\) over \(K\). As shown by \textit{V. G. Drinfeld} [Funct. Anal. Appl. 10, 107--115 (1976); translation from Funkts. Anal. Prilozh. 10, No. 2, 29--40 (1976; Zbl 0346.14010)], one may interpret \(\mathcal{X}_0^{(h)}\) as the generic fibre of a formal \(\check{\mathfrak{o}}\)-scheme parametrizing special formal \(\mathfrak{o}_D\)-mo…
reviewed by: Gergely Zábrádi
Mathematics Subject Classification ID: 11G18
Mathematics Subject Classification ID: 11F80
Mathematics Subject Classification ID: 14G35
zbMATH DE Number: 5955297
zbMATH Keywords: Lubin-Tate tower
zbMATH Keywords: Drinfeld tower
zbMATH Keywords: period ring
zbMATH Keywords: \(p\)-adic Langlands
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.2748/tmj/1309952087
OpenAlex ID: W2000727533
cites work: Q5611784
cites work: Algèbre commutative
cites work: Q3325851
cites work: Q3483360
cites work: Irreducible components of rigid spaces
cites work: Relative ampleness in rigid geometry
cites work: Espaces symétriques de Drinfeld et correspondance de Langlands locale
cites work: Characters of locally analytic representations of p-adic reductive groups
cites work: Coverings of p-adic symmetric regions
cites work: Berkovich spaces are excellent
cites work: Jacquet modules of locally analytic representations of p-adic reductive groups I. Construction and first properties
cites work: The trace formula and Drinfeld's upper halfplane
cites work: The isomorphism between Lubin-Tate and Drinfeld towers
cites work: Q3980843
cites work: Q4288641
cites work: Fibrés vectoriels sur un polydisque ultramétrique
cites work: The geometry and cohomology of some simple Shimura varieties. With an appendix by Vladimir G. Berkovich
cites work: Crystalline Dieudonné module theory via formal and rigid geometry
cites work: Theorem A und B in der nichtarchimedischen Funktionentheorie
cites work: Formal complex multiplication in local fields
cites work: Formal moduli for one-parameter formal Lie groups
cites work: Q3871006
cites work: Equivariant vector bundles on Drinfeld's upper half space
cites work: Period Spaces for "p"-divisible Groups (AM-141)
cites work: Q2761412
cites work: The cohomology of \(p\)-adic symmetric spaces
cites work: Q4831107
cites work: Locally analytic distributions and \(p\)-adic representation theory, with applications to \(\text{GL}_{2}\)
cites work: Algebras of \(p\)-adic distributions and admissible representations
cites work: Geometrically connected components of Lubin-Tate deformation spaces with level structures
cites work: Deformation spaces of one-dimensional formal modules and their cohomology
cites work: Q3597333
cites work: Local class field theory via Lubin-Tate theory
DOI: 10.2748/TMJ/1309952087
description: scientific article; zbMATH DE number 5955297